q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various ...probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the noncommutative q-binomial theorem), where the q is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent.
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Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in
obtained from the partitions of the fixed positive integer n. These ...distributions arise naturally when considering equally-likely random permutations on the set of n letters. For one of the distributions, the expectation vector and covariance matrix is derived. For the other distribution, conjectures for several elements of the expectation vector are provided.
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MacMahon showed that the generating function for partitions into at most
k
parts can be decomposed into a partial fraction-type sum indexed by the partitions of
k
. In the present work, a ...generalization of MacMahon’s result is given, which in turn provides a full combinatorial explanation.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Grubbs and Weaver (
1947
) suggest a minimum-variance unbiased estimator for the population standard deviation of a normal random variable, where a random sample is drawn and a weighted sum of the ...ranges of subsamples is calculated. The optimal choice involves using as many subsamples of size eight as possible. They verified their results numerically for samples of size up to 100, and conjectured that their "rule of eights" is valid for all sample sizes. Here we examine the analogous problem where the underlying distribution is exponential and find that a "rule of fours" yields optimality and prove the result rigorously.
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Abstract
Objective
Assess the effectiveness of providing Logical Observation Identifiers Names and Codes (LOINC®)-to-In Vitro Diagnostic (LIVD) coding specification, required by the United States ...Department of Health and Human Services for SARS-CoV-2 reporting, in medical center laboratories and utilize findings to inform future United States Food and Drug Administration policy on the use of real-world evidence in regulatory decisions.
Materials and Methods
We compared gaps and similarities between diagnostic test manufacturers’ recommended LOINC® codes and the LOINC® codes used in medical center laboratories for the same tests.
Results
Five medical centers and three test manufacturers extracted data from laboratory information systems (LIS) for prioritized tests of interest. The data submission ranged from 74 to 532 LOINC® codes per site. Three test manufacturers submitted 15 LIVD catalogs representing 26 distinct devices, 6956 tests, and 686 LOINC® codes. We identified mismatches in how medical centers use LOINC® to encode laboratory tests compared to how test manufacturers encode the same laboratory tests. Of 331 tests available in the LIVD files, 136 (41%) were represented by a mismatched LOINC® code by the medical centers (chi-square 45.0, 4 df, P < .0001).
Discussion
The five medical centers and three test manufacturers vary in how they organize, categorize, and store LIS catalog information. This variation impacts data quality and interoperability.
Conclusion
The results of the study indicate that providing the LIVD mappings was not sufficient to support laboratory data interoperability. National implementation of LIVD and further efforts to promote laboratory interoperability will require a more comprehensive effort and continuing evaluation and quality control.
The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of the main descriptors that correlate a chemical compound’s molecular structure with ...experimentally gathered data regarding the compound’s characteristics. In 2008, Wang and Zhang independently characterized trees with specified degree sequence that minimize the Wiener index. In the paper of Wang, a corollary on maximizing the Wiener index was pointed out to be incorrect by Zhang et. al. in 2010. Zhang et. al. also provided partial results and noted that the question turns out to be complicated. Later, Çela et. al. considered this question as a quadratic assignment problem and provided a polynomial time algorithm. We make some progress in this contribution, providing information on the candidate trees for the maximum Wiener index. Some interesting combinatorial relations to other objects arose from this study. We also consider the bound of this maximum value as well as study this question for trees with small diameter and for chemical trees with specified degree sequence.
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In this note, we consider the trees (caterpillars) that minimize the number of subtrees among trees with a given degree sequence. This is a question naturally related to the extremal structures of ...some distance based graph invariants. We first confirm the expected fact that the number of subtrees is minimized by some caterpillar. As with other graph invariants, the specific optimal caterpillar is nearly impossible to characterize and depends on the degree sequence. We provide some simple properties of such caterpillars as well as observations that will help finding the optimal caterpillar.
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The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as ...sums of quasi-polynomials) for pm(n) for any desired m. We do this to demonstrate the power of “rigorous guessing” as facilitated by the quasi-polynomial ansatz.
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