We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the "runs" conjecture (Kolpakov & Kucherov (FOCS '99)), which states that the maximum number of runs ρ(n) in a string of length n is less than n. The proof is remarkably simple, considering the numerous endeavors to tackle this problem in the last 15 years, and significantly improves our understanding of how runs can occur in strings. In addition, we obtain an upper bound of 3n for the maximum sum of exponents σ(n) of runs in a string of length n, improving on the best known bound of 4.1n by Crochemore et al. (JDA 2012), as well as other improved bounds on related problems. The characterization also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string. We also establish a relationship between runs and nodes of the Lyndon tree, which gives a simple optimal solution to the 2-Period Query problem that was recently solved by Kociumaka et al. (SODA 2015). * A preliminary version of this paper has appeared in [1].

No abstract

In this study, we aimed to assess how sample sizes affect confidence of stand-scale transpiration (E) estimates calculated from sap flux (F(d)) and sapwood area (A(S_tree)) measurements of individual trees. In a Japanese cypress plantation, we measured F(d) and A(S_tree) in all trees (n = 58) within a 20 x 20 m study plot, which was divided into four 10 x 10 subplots. We calculated E from stand A(S_tree) (A(S_stand)) and mean stand F(d) (J(S)) values. Using Monte Carlo analyses, we examined the potential errors associated with sample sizes in E, A(S_stand) and J(S) using the original A(S_tree) and F(d) data sets. Consequently, we defined the optimal sample sizes of 10 and 15 for A(S_stand) and J(S) estimates, respectively, in the 20 x 20 m plot. Sample sizes larger than the optimal sample sizes did not decrease potential errors. The optimal sample sizes for J(S) changed according to plot size (e.g., 10 x 10 and 10 x 20 m), whereas the optimal sample sizes for A(S_stand) did not. As well, the optimal sample sizes for J(S) did not change in different vapor pressure deficit conditions. In terms of E estimates, these results suggest that the tree-to-tree variations in F(d) vary among different plots, and that plot size to capture tree-to-tree variations in F(d) is an important factor. The sample sizes determined in this study will be helpful for planning the balanced sampling designs to extrapolate stand-scale estimates to catchment-scale estimates.

Understanding radial and azimuthal variation, and tree-to-tree variation, in sap flux density (Fd) as sources of uncertainty is important for estimating transpiration using sap flow techniques. In a Japanese cedar (Cryptomeria japonica D. Don.) forest, Fd was measured at several depths and aspects for 18 trees, using heat dissipation (Granier-type) sensors. We observed considerable azimuthal variation in Fd. The coefficient of variation (CV) calculated from Fd at a depth of 0-20 mm (Fd1) and Fd at a depth of 20-40 mm (Fd2) ranged from 6.7 to 37.6% (mean = 28.3%) and from 19.6 to 62.5% (mean = 34.6%) for the -azimuthal directions. Fd at the north aspect averaged for nine trees, for which azimuthal measurements were made, was -obviously smaller than Fd at the other three aspects (i.e., west, south and east) averaged for the nine trees. Fd1 averaged for the nine trees was significantly larger than Fd2 averaged for the nine trees. The error for stand-scale transpiration (E) estimates caused by ignoring the azimuthal variation was larger than that caused by ignoring the radial variation. The error caused by ignoring tree-to-tree variation was larger than that caused by ignoring both radial and azimuthal variations. Thus, tree-to-tree variation in Fd would be more important than both radial and azimuthal variations in Fd for E estimation. However, Fd for each tree should not be measured at a consistent aspect but should be measured at various aspects to make accurate E estimates and to avoid a risk of error caused by the relationship of Fd to aspect.

No abstract

No abstract

The symptoms of idiopathic normal pressure hydrocephalus (iNPH) can be improved by shunt surgery, but prediction of treatment outcome is not established. We investigated changes of the corticospinal tract (CST) in iNPH before and after shunt surgery by using diffusion microstructural imaging, which infers more specific tissue properties than conventional diffusion tensor imaging. Two biophysical models were used: neurite orientation dispersion and density imaging (NODDI) and white matter tract integrity (WMTI). In both methods, the orientational coherence within the CSTs was higher in patients than in controls, and some normalization occurred after the surgery in patients, indicating axon stretching and recovery. The estimated axon density was lower in patients than in controls but remained unchanged after the surgery, suggesting its potential as a marker for irreversible neuronal damage. In a Monte-Carlo simulation that represented model axons as undulating cylinders, both NODDI and WMTI separated the effects of axon density and undulation. Thus, diffusion MRI may distinguish between reversible and irreversible microstructural changes in iNPH. Our findings constitute a step towards a quantitative image biomarker that reflects pathological process and treatment outcomes of iNPH.

BackgroundNeurite orientation dispersion and density imaging (NODDI) is a diffusion magnetic resonance imaging (MRI) technique with the potential to visualize the microstructure of the brain. Revolutionary histological methods to render the mouse brain transparent have recently been developed, but verification of NODDI by these methods has not been reported.PurposeTo confirm the concordance of NODDI with histology in terms of density and orientation dispersion of neurites of the brain.Material and MethodsWhole brain diffusion MRI of a thy-1 yellow fluorescent protein mouse was acquired with a 7-T MRI scanner, after which transparent brain sections were created from the same mouse. NODDI parameters calculated from the MR images, including the intracellular volume fraction (Vic) and the orientation dispersion index (ODI), were compared with histological findings. Neurite density, Vic, and ODI were compared between areas of the anterior commissure and the hippocampus containing crossing fibers (crossing areas) and parallel fibers (parallel areas), and the correlation between fiber density and Vic was assessed.ResultsThe ODI was significantly higher in the crossing area compared to the parallel area in both the anterior commissure and the hippocampus (P = 0.0247, P = 0.00022, respectively). Neurite density showed a similar tendency, but was significantly different only in the hippocampus (P = 7.91E−07). There was no significant correlation between neurite density and Vic.ConclusionNODDI was verified by histology for quantification of the orientation dispersion of neurites. These results indicate that the ODI is a suitable index for understanding the microstructure of the brain in vivo.