A Proof of Convergence for the Tridiagonal QL Algorithm in Floating-Point Arithmetic

Numerous routines are available to find the eigenvalues of a real symmetric tridiagonal matrix. Since it is known to converge in exact arithmetic, the tridiagonal QL algorithm with origin shift is widely used. Here we analyze the algorithm in floating-point arithmetic. This analysis suggests two modifications to the EISPACK implementation TQLl that enable one to prove correctness and hence convergence of the routine. Also, it is known that the implicit and explicit versions of the QL algorithm produce the same results in exact arithmetic. A counter-example to the floating-point analog of this theorem is presented

Factors governing the reactivity of the unsymmetrical benzils

Several mixed benzils containing one substituent in the benzene ring have been prepared, and their reaction iith various ':etonic reagents considered. In addition the absorption spectra of these benzils have been recorded.p -Nitro acetylbenzoin and p- nitrobenzil have been prepared by a greatly improved method.Tvo methods for the determination of the structure of ketonic derivatives of benzils have been described and the structure of p- nethoxybenzil hydrazone determined.The action of potash and of alcoholic potassium cyanide on the benzils has been described.The mixed 'oenzoin. condensation has been discussed v ith the aid of a diagram and the importance of quantitative uor:- in this connection is emphasised.The reactivity of the mixed benzils eras con- sidered, and i t vins seen that though the effect of the reagent employed vas very important, in many cases it teas found that only one ketonic group reacted.It z;as sho1n that the main band in the spect um of benzil is probably intimately connected z',ith the effect of substituents on the reactivity of the carbo yl. The spectra of the benzils :,,ere found to shove conside able shifts due to the substituents.The feasibility of chemical and of physical methods for the determination of the reactivity of a carbonyl groin: has been co_ZE idered, and it has been shown that different methods give different values for the reactivity.The lack of accurate auantitati ve data obtained under comparable conditions was felt in almos every phase of this work.It was conclusively shown that the only reage which gave parallel results for all the benzils, was dinitrophenylhydrazine, and that in conseoue,nce it is the most suitable reagent for the characterisation of carbonyl compounds

Molecular structure of exudate gums with special reference to gums of the sterculia genus

The term 'gum', in its broadest sense, refers to both hydrophobic and hydrophilic substances of high molecular weight which usually exhibit colloidal properties when dispersed in an appropriate solvent. Hydrophobic substances often called gums include high molecular weight hydrocarbons and other petroleum products, rubbers, certain synthetic polymers and resinous saps which often exude from evergreens.More specifically, the term gum applies to plant polysaccharides or their derivatives which are dispersible in either cold or hot water to produce viscous solutions or suspensions. As much as three- quarters of the dry weight of plants may be polysaccharide and, consequently, such substances are of wide occurrence. The most important gums, however, are those which are readily obtainable in large amounts from the plant. Some of these gums are used industrially and, indeed, many have been known since ancient times. One of the chief sources of such polysaccharides is seaweed which furnishes agar, algin and carrageenin while seed gums, such as gum guar and locust bean gum, are also important, particularly from the point of view that the plant which produces the seeds is often grown extensively as a food crop. In contrast to these naturally occurring gums, other gums are obtained from cellulose, one of the main components of the plant cell wall, and starch, a food reserve polysaccharide, by esterification and etherification. Commercially, however, the most important gums are plant exudates and most plant families have been found to include species which exude gums to a greater or lesser degree.In this context, the term 'exudate gum' strictly refers to those commercially important gums which exude in copious amounts from shrubs or low- growing trees, forming, on exposure to the atmosphere, glossy nodules or flakes which are usually brown or yellow in colour. These gum producing trees grow predominantly in Africa or Asia indicating the climatic requirements for their growth.The most likely function of gum,formation is to prevent infection of the plant tissue and to prevent loss of moisture (1). This is evident from the fact that tapping of the trees immediately results in the sealing off of the exposed area by the gum and the tree continues to grow without apparent damage.Several suggestions have been made as to the origin of these gums. They may be products of normal plant metabolism but healthy Acacia trees, grown under favourable conditions of moisture, soil and temperature do not produce any gum, while trees which are grown under adverse conditions do secrete gum (2).A most challenging problem to the chemist is the mode of formation of these gums. This requires a knowledge of the complex processes whereby they are first of all synthesised in the plant from simple compounds and then transported as required from the site of formation to the injured site. Work on the early stages of biosynthesis concerning the reactions involved in the transformations of monosaccharides, the building units of polysaccharides, into one another and into derivatives capable of enzymic polymerisation has been reviewed by J.K.N. Jones (3) and by Neufeld and Hassid (4). At present, little is known of the reactions involved in the enzymic polymerisation stage of complex heteropolysaccharide biosynthesis although, in this respect, recent advances have been made in the biosynthesis of complex lipopolysaccharides from the cell wall of Gram -negative bacteria (5). Clearly, a better understanding of the problem would be obtained from a knowledge of the structural relationships between different polysaccharides such as cellulose, hemicelluloses, pectins and gum exudates and between individual polysaccharides at different stages in their formation

Les sables de Fontainebleau: a natural quartz reference sample and its characterisation

Fundamental studies on luminescence production in natural quartz require samples which can be studied by groups of laboratories using complementary methods. In the framework of a European collaboration studying quartz luminescence, a sample originating from the Fontainebleau Sandstone Formation in France was selected for characterisation and distribution to establish a starting point for interlaboratory work. Here we report on the preparation and characterisation work undertaken before distribution with the aim of ensuring that each laboratory received comparable material. Material was purified to enrich the quartz concentration, followed by mineralogical screening by SEM and ICP-MS analyses. Luminescence screening measurements were undertaken at a single laboratory (SUERC) to verify the suitability of the sample for use within the study, and to establish the level of homogeneity of subsamples prepared for distribution. The sample underwent minimal non-chemical pre-treatment by multiple cycles of magnetic separation and annealing. SEM analysis showed that the sample consists mainly of SiO2. The luminescence characterisation confirmed a dose sensitivity of ca. 22,000–160,000 cts K−1 Gy−1 per 260– 290 grains for the 110◦C UV TL peak, well developed low (here: 100–300◦C) temperature (pre-dose) TL signals and high OSL sensitivities. The grain to grain OSL response varies by more than one order of magnitude. No significant IRSL signal was observed. In summary, the results from luminescence characterisation confirm the suitability of the sample for the luminescence experiments envisaged and have established a basis for comparability in studies conducted by a network of laboratories

A guide for understanding and designing Mendelian randomization studies in the musculoskeletal field

Mendelian randomization (MR) is an increasingly popular component of an epidemiologist's toolkit, used to provide evidence of a causal effect of one trait (an exposure, eg, body mass index [BMI]) on an outcome trait or disease (eg, osteoarthritis). Identifying these effects is important for understanding disease etiology and potentially identifying targets for therapeutic intervention. MR uses genetic variants as instrumental variables for the exposure, which should not be influenced by the outcome or confounding variables, overcoming key limitations of traditional epidemiological analyses. For MR to generate a valid estimate of effect, key assumptions must be met. In recent years, there has been a rapid rise in MR methods that aim to test, or are robust to violations of, these assumptions. In this review, we provide an overview of MR for a non‐expert audience, including an explanation of these key assumptions and how they are often tested, to aid a better reading and understanding of the MR literature. We highlight some of these new methods and how they can be useful for specific methodological challenges in the musculoskeletal field, including for conditions or traits that share underlying biological pathways, such as bone and joint disease. © 2022 The Authors. JBMR Plus published by Wiley Periodicals LLC on behalf of American Society for Bone and Mineral Research

Nitrogen and harvest date affect developmental morphology and biomass yield of warm-season grasses

Information on the growth and development of warm-season grasses in response to management is required to use them successfully as a biomass crop. Our objectives were to determine optimum harvest periods and effect of N fertilization rates on the biomass production of four warm-season grasses, and to investigate if traits of canopy structure can explain observed yields with varying harvest dates and N rates. A field study was conducted at Sorenson Research Farm near Ames, IA, during 2006 and 2007. The experimental design was split-split plot arranged in a randomized complete block with four replications. Big bluestem (Andropogon gerardii Vitman), eastern gamagrass (Tripsacum dactyloides L.), indiangrass (Sorghastrum nutrans L. Nash), and switchgrass (Panicum virgatum L.) were main plots. Three N application rates (0, 65, and 140 kg ha−1) were subplots, and 10 harvest dates were sub-sub plots. Biomass of warm-season grasses increased with advanced maturity, but differently among species. The maximum yield of eastern gamagrass occurred at the highest MSC (1.6 and 2.2) when the largest seed ripening tillers were present. Big bluestem, switchgrass, and indiangrass obtained the maximum yields at MSC 3.5, 3.9, and 2.9, respectively when the largest reproductive tillers were present. In terms of a biomass supply strategy, eastern gamagrass may be used during early summer, while big bluestem and switchgrass may be best used between mid- and late- summer, and indiangrass in early fall. Nitrogen fertilization increased yield by increasing tiller development. Optimum biomass yields were obtained later in the season when they were fertilized with 140 kg ha−1