Monotone Subsequences in the Sequence of Fractional Parts of Multiples of an Irrational

Hammersley [7] showed that if X1, X2, . . . is a sequence of independent identically distributed random variables whose common distribution is continuous, and if ln+(ln-) denotes the length of the longest increasing (decreasing) subsequence of X1, X2, . . ., Xn, then there is a constant c such that ln-⁄n½→ c and ln+⁄n½→ c in probability, as n → ∞. Kesten [8] showed that in fact there is almost sure convergence. Logan and Shepp [11] proved that c ≧ 2, and recently Versik and Kerov [13] have announced that c = 2

Lower Bounds for Nonparametric Density Estimation Rates

In Wegman\u27s paper [5] on nonparametric density estimation, he states that it would be interesting to show that there is no density estimator which has mean integrated square rate better than O(n-1). The object of this note is to prove such a result, making no arbitrary assumptions about the specific form of the estimator. This proof is given in Section 2. Our method applies to some other measures of error, as we point out in Section 3

Relative kerf and sawing variation values for some hardwood sawing machines

Information on the conversion efficiency of sawing machines is important to those involved in the management, maintenance, and design of sawmills. Little information on the conversion characteristics of hardwood sawing machines has been available. This study, based on 266 studies of 6 machine types, provides an analysis of the machine characteristics of kerf width, within-board, between-board, and total sawing variations and wood loss per sawline. Machine conversion efficiency was found to be explained by feedworks and setworks type, and sawblade thickness and type. This analysis of machine characteristics provides information for a rational choice of sawing machines for hardwood sawmills

Estimating hardwood sawmill conversion efficiency based on sawing machine and log characteristics

Increased problems of hardwood timber availability have caused many sawmiller, industry analysts, and planners to recognize the importance of sawmill conversion efficiency. Conversion efficiency not only affects sawmill profits, but is also important on a much broader level. Timber supply issues have caused resource planners and policy makers to consider the effects of conversion efficiency on the utilization and depletion of the timber resource. Improvements in sawmill conversion efficiency would favorably impact sawmill profits, and would be equivalent in effect to extending existing supplies of standing timber. An equation was developed to estimate lumber recovery factor for hardwood sawmills based on the characteristics of sawing machines and log resources. Variables included in the model were headrig type, headrig kerf, average log diameter and length, and the influence of total sawing variation. The estimated coefficients significantly influenced lumber recovery factor. The model should be helpful in assessing conversion efficiency trends and potential benefits from gains in sawmill efficiency

Online Selection of Alternating Subsequences from a Random Sample

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence

Association between surgical volume and failure of primary total hip replacement in England and Wales: findings from a prospective national joint replacement register

Objective To investigate the association of volume of total hip arthroplasty (THA) between consultants and within the same consultant in the previous year and the hazard of revision using multilevel survival models. Design Prospective cohort study using data from a national joint replacement register. Setting Elective THA across all private and public centres in England and Wales between April 2003 and February 2017. Participants Patients aged 50 years or more undergoing THA for osteoarthritis. Intervention The volume of THA conducted in the preceding 365 days to the index procedure. Main outcome and measure Revision surgery (excision, addition or replacement) of a primary THA. Results Of the 579 858 patients undergoing primary THA (mean baseline age 69.8 years (SD 10.2)), 61.1% were women. Multilevel survival found differing results for between and within-consultant effects. There was a strong volume–revision association between consultants, with a near-linear 43.3% (95% CI 29.1% to 57.4%) reduction of the risk of revision comparing consultants with volumes between 1 and 200 procedures annually. Changes in individual surgeons (within-consultant) case volume showed no evidence of an association with revision. Conclusion Separation of between-consultant and within- consultant effects of surgical volume reveals how volume contributes to the risk of revision after THA. The lack of association within-consultants suggests that individual changes to consultant volume alone will have little effect on outcomes following THA. These novel findings provide strong evidence supporting the practice of specialisation of hip arthroplasty. It does not support the practice of low-volume consultants increasing their personal volume as it is unlikely their results would improve if this is the only change. Limiting the exposure of patients to consultants with low volumes of THA and greater utilisation of centres with higher volume surgeons with better outcomes may be beneficial to patients