Evaluation of

Graham Jameson and Nick Lord (Math. Gazette 97 (2013), 504–505 Let ∑∞ n=1 1 n2 = S. Numerous proofs are known of the well-known identity S = π/6. Robin Chapman’s website [1] contains a very useful collation of 14 such proofs. Here we give a short one based on a certain double integral. The first two proofs in [1] give alternative double integral proofs from [2, 3] and another can be seen in [4]. Note first that

[1]  K. Manly,et al.  Map Manager QTX, cross-platform software for genetic mapping , 2001, Mammalian Genome.

[2]  T. Mackay.  The genetic architecture of quantitative traits. , 2001, Annual review of genetics.

[3]  P. Langridge,et al.  Trends in genetic and genome analyses in wheat: a review , 2001 .

[4]  R. Eastwood,et al.  Establishment and characterisation of wheat genetic mapping populations , 2001 .

[5]  S. Ullrich,et al.  QTL analysis of agronomic traits in barley based on the doubled haploid progeny of two elite North American varieties representing different germplasm groups , 2001, Theoretical and Applied Genetics.

[6]  O. Merah,et al.  Carbon isotope discrimination and productivity in field-grown barley genotypes , 2001 .

[7]  D. This,et al.  New QTLs identified for plant water status, water-soluble carbohydrate and osmotic adjustment in a barley population grown in a growth-chamber under two water regimes , 2001, Theoretical and Applied Genetics.

[8]  M. Havaux,et al.  Drought and Heat Responses in the Wild Wheat Relative Aegilops geniculata Roth: Potential Interest for Wheat Improvement , 2001 .

[9]  J. Ehleringer,et al.  Water use trade‐offs and optimal adaptations to pulse‐driven arid ecosystems , 2001 .

[10]  J. Angus.  Nitrogen supply and demand in Australian agriculture , 2001 .

[11]  O. Merah,et al.  Stability of Carbon Isotope Discrimination and Grain Yield in Durum Wheat , 2001 .

[12]  John Angus,et al.  Increasing Water Use and Water Use Efficiency in Dryland Wheat , 2001 .

[13]  A. Kilian,et al.  Diversity arrays: a solid state technology for sequence information independent genotyping. , 2001, Nucleic acids research.

[14]  O. Merah,et al.  Effect of Glaucousness on Carbon Isotope Discrimination and Grain Yield in Durum Wheat , 2000 .

[15]  H. Miura,et al.  Mapping QTLs controlling grain yield and its components on chromosome 5A of wheat , 2000, Theoretical and Applied Genetics.

[16]  A B Korol,et al.  Molecular genetic maps in wild emmer wheat, Triticum dicoccoides: genome-wide coverage, massive negative interference, and putative quasi-linkage. , 2000, Genome research.

[17]  J. Dubcovsky,et al.  Epistatic interaction between vernalization genes Vrn-Am1 and Vrn-Am2 in diploid wheat. , 2000, The Journal of heredity.

[18]  D G Brown,et al.  Selective mapping: a strategy for optimizing the construction of high-density linkage maps. , 2000, Genetics.

[19]  L. Alegre,et al.  Changes in carotenoids, tocopherols and diterpenes during drought and recovery, and the biological significance of chlorophyll loss in Rosmarinus officinalis plants , 2000, Planta.

[20]  K Maxwell,et al.  Chlorophyll fluorescence--a practical guide. , 2000, Journal of experimental botany.

[21]  S. Tanksley,et al.  Fine mapping of a quantitative trait locus (QTL) from Lycopersicon hirsutum chromosome 1 affecting fruit characteristics and agronomic traits: breaking linkage among QTLs affecting different traits and dissection of heterosis for yield , 2000, Theoretical and Applied Genetics.

[22]  M. Reynolds,et al.  Avenues for genetic modification of radiation use efficiency in wheat. , 2000, Journal of experimental botany.

[23]  O. Merah,et al.  Grain yield, carbon isotope discrimination, mineral and silicon content in durum wheat under different precipitation regimes , 1999 .

[24]  Z B Zeng,et al.  Estimating the genetic architecture of quantitative traits. , 1999, Genetical research.

[25]  H. Miura,et al.  Detection of an earliness per se quantitative trait locus in the proximal region of wheat chromosome 5AL , 1999 .

[26]  L. Kumar,et al.  DNA markers in plant improvement: an overview. , 1999, Biotechnology advances.

[27]  P. Christou,et al.  ‘Green revolution’ genes encode mutant gibberellin response modulators , 1999, Nature.

[28]  Jianhua Zhang,et al.  Effects of water stress on photosystem II photochemistry and its thermostability in wheat plants , 1999 .

[29]  Z. Zeng,et al.  Multiple interval mapping for quantitative trait loci. , 1999, Genetics.

[30]  Mathias Neumann Andersen,et al.  Root signalling and osmotic adjustment during intermittent soil drying sustain grain yield of field grown wheat , 1999 .

[31]  B. Keller,et al.  Genetic linkage map of a wheat×spelt cross , 1999, Theoretical and Applied Genetics.

[32]  Gustavo A. Slafer,et al.  Wheat: Ecology and Physiology of Yield Determination , 1999 .

[33]  K. Tsunewaki,et al.  Production of near-isogenic lines of common wheat for glaucousness and genetic basis of this trait clarified by their use , 1999 .

[34]  H. Nguyen,et al.  Genetic analysis of osmotic adjustment in crop plants , 1999 .

[35]  David A. Stephens,et al.  BAYESIAN ANALYSIS OF QUANTITATIVE TRAIT LOCUS DATA USING REVERSIBLE JUMP MARKOV CHAIN MONTE CARLO , 1998 .

[36]  S. Salvi,et al.  RFLP mapping of quantitative trait loci controlling abscisic acid concentration in leaves of drought-stressed maize (Zea mays L.) , 1998, Theoretical and Applied Genetics.

[37]  Jianhua Zhang,et al.  Changes in photosystem II function during senescence of wheat leaves , 1998 .

[38]  P. Visscher,et al.  A nonparametric bootstrap method for testing close linkage vs. pleiotropy of coincident quantitative trait loci. , 1998, Genetics.

[39]  M. Ganal,et al.  A microsatellite map of wheat. , 1998, Genetics.

[40]  M. Brick,et al.  Comparison of selection indices to identify productive dry bean lines under diverse environmental conditions , 1998 .

[41]  M. Ganal,et al.  Genetic analysis of the dwarfing gene (Rht8) in wheat. Part I. Molecular mapping of Rht8 on the short arm of chromosome 2D of bread wheat (Triticum aestivum L.) , 1998, Theoretical and Applied Genetics.

[42]  M. Ganal,et al.  Genetic analysis of the dwarfing gene Rht8 in wheat. Part II. The distribution and adaptive significance of allelic variants at the Rht8 locus of wheat as revealed by microsatellite screening , 1998, Theoretical and Applied Genetics.

[43]  J. Giraudat,et al.  ABSCISIC ACID SIGNAL TRANSDUCTION. , 1998, Annual review of plant physiology and plant molecular biology.

[44]  P. Goldsbrough,et al.  Evaluation of near-isogenic sorghum lines contrasting for QTL markers associated with drought tolerance , 1998 .

[45]  M. Khairallah,et al.  Several QTLs involved in osmotic-adjustment trait variation in barley (Hordeum vulgare L.) , 1998, Theoretical and Applied Genetics.

[46]  R. Percy,et al.  Stomatal conductance predicts yields in irrigated Pima cotton and bread wheat grown at high temperatures , 1998 .

[47]  S. Lin,et al.  Fine mapping of quantitative trait loci Hd-1, Hd-2 and Hd-3, controlling heading date of rice, as single Mendelian factors , 1998, Theoretical and Applied Genetics.

[48]  C. N. Law,et al.  Genetic analysis of some flowering time and adaptive traits in wheat , 1997 .

[49]  C. Kao,et al.  General formulas for obtaining the MLEs and the asymptotic variance-covariance matrix in mapping quantitative trait loci when using the EM algorithm. , 1997, Biometrics.

[50]  Graeme L. Hammer,et al.  The role of physiological understanding in plant breeding; From a breeding perspective , 1996 .

[51]  J. M. Lilley,et al.  Expression of osmotic adjustment and dehydration tolerance in diverse rice lines , 1996 .

[52]  J. Lilley,et al.  Locating QTL for osmotic adjustment and dehydration tolerance in rice , 1996 .

[53]  R. A. Fischer,et al.  Canopy Temperature Depression Association with Yield of Irrigated Spring Wheat Cultivars in a Hot Climate , 1996 .

[54]  Z B Zeng,et al.  Multiple trait analysis of genetic mapping for quantitative trait loci. , 1995, Genetics.

[55]  M. Sorrells,et al.  Molecular mapping of wheat. Homoeologous group 2. , 1995, Genome.

[56]  E. Zamski,et al.  Short- and Long-eared High-yielding Hexaploid Wheat Cultivars: which has Unexpressed Potential for Higher Yield? , 1995 .

[57]  R. B. Austin,et al.  Carbon isotope discrimination and grain yield for three bread wheat germplasm groups grown at different levels of water stress , 1995 .

[58]  Z. Zeng.  Precision mapping of quantitative trait loci. , 1994, Genetics.

[59]  R. Jansen,et al.  University of Groningen High Resolution of Quantitative Traits Into Multiple Loci via Interval Mapping , 2022 .

[60]  Z B Zeng,et al.  Theoretical basis for separation of multiple linked gene effects in mapping quantitative trait loci. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[61]  Kenneth G. Cassman,et al.  Adjustment for Specific Leaf Weight Improves Chlorophyll Meter's Estimate of Rice Leaf Nitrogen Concentration , 1993 .

[62]  R. Jansen,et al.  Interval mapping of multiple quantitative trait loci. , 1993, Genetics.

[63]  P. Stam,et al.  Construction of integrated genetic linkage maps by means of a new computer package: JOINMAP. , 1993 .

[64]  L. Andersson,et al.  The abundance of various polymorphic microsatellite motifs differs between plants and vertebrates. , 1993, Nucleic acids research.

[65]  E. Barlow,et al.  Drought Stress Induces Changes in the Non-Structural Carbohydrate Composition of Wheat Stems , 1991 .

[66]  J. Clarke,et al.  Phenotypic Diversity and Associations of Some Potentially Drought-responsive Characters in Durum Wheat , 1991 .

[67]  R. Richards.  Crop improvement for temperate Australia: Future opportunities , 1991 .

[68]  R. Belford,et al.  Growth, development and light interception of old and modern wheat cultivars in a Mediterranean-type environment , 1989 .

[69]  S. Tanksley,et al.  RFLP Mapping in Plant Breeding: New Tools for an Old Science , 1989, Bio/Technology.

[70]  M. Daly,et al.  MAPMAKER: an interactive computer package for constructing primary genetic linkage maps of experimental and natural populations. , 1987, Genomics.

[71]  Jianhua Zhang,et al.  Control of Stomatal Behaviour by Abscisic Acid which Apparently Originates in the Roots , 1987 .

[72]  N. Turner.  Adaptation to water deficits: a changing perspective , 1986 .

[73]  R. Blackwell,et al.  Some effects of leaf posture on the yield and water economy of winter wheat , 1983, The Journal of Agricultural Science.

[74]  Tom M. Apostol,et al.  A proof that euler missed: evaluating ζ(2) the easy way , 1983 .

[75]  R. Richards,et al.  Yield, Water Relations, Gas Exchange, and Surface Reflectances of Near‐Isogenic Wheat Lines Differing in Glaucousness1 , 1983 .

[76]  F. Bidinger,et al.  Introduction: the human consequences of drought and crop research priorities for their alleviation , 1981 .

[77]  A. Rosielle,et al.  Theoretical Aspects of Selection for Yield in Stress and Non-Stress Environment 1 , 1981 .

[78]  J. Morgan.  Possible role of abscisic acid in reducing seed set in water-stressed wheat plants , 1980, Nature.

[79]  J. Morgan.  Osmotic Adjustment in the Spikelets and Leaves of Wheat , 1980 .

[80]  C. N. Law,et al.  A Genetic study of day-length response in wheat , 1978, Heredity.

[81]  J. Zadoks.  A decimal code for the growth stages of cereals , 1974 .

[82]  J. B. Powell,et al.  Components of Height Inheritance of the Semidwarf Straw Character in Wheat Triticum aestivum L. 1 , 1967 .

[83]  A. Willis,et al.  The estimation of carbohydrates in plant extracts by anthrone. , 1954, The Biochemical journal.

[84]  K. V. SRINATH,et al.  Mechanism of Crossing-over , 1946, Nature.

[85]  K. Sax,et al.  The Association of Size Differences with Seed-Coat Pattern and Pigmentation in PHASEOLUS VULGARIS. , 1923, Genetics.

[86]  J. Mitra.  Genetics and genetic improvement of drought resistance in crop plants , 2001 .

[87]  S. Belenko.  Research on drug courts: a critical review 2001 update , 2001 .

[88]  S. Belenko.  Research on Drug Courts: a Critical Review I Research on Drug Courts: a Critical Review 2001 Update Research on Drug Courts: a Critical Review 2001 Update Article Summaries , 2001 .

[89]  J. Araus,et al.  Cuticular conductance, water use efficiency and drought tolerance of durum wheat isolines of differing glaucousness , 2000 .

[90]  G. McDonald,et al.  Shoot growth, root growth and grain yield of bread and durum wheat in South Australia , 1999 .

[91]  Mark E. Westgate,et al.  Reproductive Development in Grain Crops during Drought , 1999 .

[92]  A. Scartazza,et al.  Carbon isotope discrimination in leaf and stem sugars, water-use efficiency and mesophyll conductance during different developmental stages in rice subjected to drought , 1998 .

[93]  T. E. Miller,et al.  The influence of the group 1 chromosomes of wheat on ear-emergence times and their involvement with vernalization and day length , 1998, Heredity.

[94]  R. Richards,et al.  'Haying-off', the negative grain yield response of dryland wheat to nitrogen fertiliser II.Carbohydrate and protein dynamics , 1998 .

[95]  F. Salamini,et al.  Catalogue of gene symbols for wheat , 1998 .

[96]  R. Munns,et al.  Water status and ABA content of floral organs in drought-stressed wheat , 1996 .

[97]  M. Tan,et al.  Chromosomal Location of a Wheat Osmoregulation Gene Using RFLP Analysis , 1996 .

[98]  R. Jansen.  Mapping of quantitative trait loci by using genetic markers: an overview of biometrical models used , 1994 .

[99]  K. Siddique,et al.  Morphological and physiological traits associated with wheat yield increases in Mediterranean environments , 1994 .

[100]  J. Morgan.  Osmotic components and properties associated with genotypic differences in osmoregulation in wheat , 1992 .

[101]  R. Richards.  The Effect of Dwarfing Genes in Spring Wheat in Dry Environments. II.* Growth, Water Use and Water-use Efficiency , 1992 .

[102]  R. J. Baker,et al.  Estimation of heritability and prediction of selection response in plant populations , 1991 .

[103]  R. C. Muchow,et al.  A critical evaluation of traits for improving crop yields in water-limited environments. , 1990 .

[104]  R. A. Fischer,et al.  Genotype, sowing date and plant spacing influence on high-yielding irrigated wheat in southern New South Wales. I. Phasic development, canopy growth and spike production , 1990 .

[105]  E. Lander,et al.  Mapping mendelian factors underlying quantitative traits using RFLP linkage maps. , 1989, Genetics.

[106]  H. Rawson,et al.  Nocturnal transpiration in wheat , 1988 .

[107]  E. R. Sears,et al.  Evolution in the Genus Triticum and the Origin of Cultivated Wheat , 1987 .

[108]  J. Morgan,et al.  Water Use, Grain Yield, and Osmoregulation in Wheat , 1986 .

[109]  R. Richards,et al.  Glaucousness in Wheat: Its Development and Effect on Water-use Efficiency, Gas Exchange and Photosynthetic Tissue Temperatures* , 1986 .

[110]  J. Morgan,et al.  OSMOREGULATION AND WATER STRESS IN HIGHER PLANTS , 1984 .

[111]  Taylor Francis Online.  Critical reviews in plant sciences , 1983 .

[112]  G. Wright,et al.  Differences between Two Grain Sorghum Genotypes in Adaptation to Drought Stress. III* Physiological Responses , 1983 .

[113]  H. Saini,et al.  Sterility in Wheat (Triticum aestivum L.) Induced by Water Deficit or High Temperature: Possible Mediation by Abscisic Acid , 1982 .

[114]  常脇 恒一郎.  Comparative gene analysis of common wheat and its ancestral species , 1965 .