SUMMARYThe Persian walnut (Juglans regia L.), a diploid species native to the mountainous regions of Central Asia, is the major walnut species cultivated for nut production and is one of the most widespread tree nut species in the world. The high nutritional value of J. regia nuts is associated with a rich array of polyphenolic compounds, whose complete biosynthetic pathways are still unknown. A J. regia genome sequence was obtained from the cultivar 'Chandler' to discover target genes and additional unknown genes. The 667-Mbp genome was assembled using two different methods (SOAPdenovo2 and MaSuRCA), with an N50 scaffold size of 464 955 bp (based on a genome size of 606 Mbp), 221 640 contigs and a GC content of 37%. Annotation with MAKER-P and other genomic resources yielded 32 498 gene models. Previous studies in walnut relying on tissue-specific methods have only identified a single polyphenol oxidase (PPO) gene (JrPPO1). Enabled by the J. regia genome sequence, a second homolog of PPO (JrPPO2) was discovered. In addition, about 130 genes in the large gallate 1-b-glucosyltransferase (GGT) superfamily were detected. Specifically, two genes, JrGGT1 and JrGGT2, were significantly homologous to the GGT from Quercus robur (QrGGT), which is involved in the synthesis of 1-O-galloyl-b-D-glucose, a precursor for the synthesis of hydrolysable tannins. The reference genome for J. regia provides meaningful insight into the complex pathways required for the synthesis of polyphenols. The walnut genome sequence provides important tools and methods to accelerate breeding and to facilitate the genetic dissection of complex traits.

We describe a technique for fast compression of time series, indexing of compressed series, and retrieval of series similar to a given pattern. The compression procedure identifies "important" points of a series and discards the other points. We use the important points not only for compression, but also for indexing a database of time series. Experiments show the effectiveness of this technique for indexing of stock prices, weather data and electroencephalograms.

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean ¿-space E ¿ to the context of circle polyhedra in the ¾-sphere S ¾ . We prove that any two convex and proper non-unitary c-polyhedra with Möbiuscongruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in H ¿ .

We describe a procedure for identifying major minima and maxima of a time series, and present two applications of this procedure. The first application is fast compression of a series, by selecting major extrema and discarding the other points. The compression algorithm runs in linear time and takes constant memory. The second application is indexing of compressed series by their major extrema, and retrieval of series similar to a given pattern. The retrieval procedure searches for the series whose compressed representation is similar to the compressed pattern. It allows the user to control the trade-off between the speed and accuracy of retrieval. We show the effectiveness of the compression and retrieval for stock charts, meteorological data, and electroencephalograms.

Given nonnegative integers n and d, where n ≫ d, what is the minimum number r such that there exist linear forms ℓ 1 , . . . , ℓ r ∈ C[x 1 , . . . , x n ] so that ℓ d 1 + · · · + ℓ d r * kpratt@andrew.cmu.eduRemark 59. If we take f = x 1 x 2 · · · x d and use the upper bound on R(x 1 · · · x d ) given by Equation (5), it follows from Theorem 56 that