Pearls of discrete mathematics

I read with interest Thomas Byrne's article: "An Extension of Pascal's Triangle and Work concerning Binomial Expansion Coefficients" in the February 1990 issue. On page 28 Mr. Byrne states: "I was seeking a mechanism to generate multinomial expansion coefficients but this doesn't do it . . . Maybe a three dimensional extension will prove useful."

Picking binomial coefficients which cannot be divided by a given prime from Pascal's triangle, we find that they form a set with self-similarity. Essouabri studied on a class of meromorphic functions associated to the above set. These functions are related to fractal geometry and it is a problem whether such a function has a non-real pole on its axis of absolute convergence. Essouabri gave a proof of existence of such a non-real pole in the simplest case. The keys of his proof are Stein's and Wilson's estimates on how fast the points multiply in Pascal's triangle modulo a prime. This article will give an extension of Essouabri's result to some cases with certain ways to count the points in Pascal's triangle modulo a prime which are different from the traditional one.

(1999). Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part I: Binomial Coefficients, Extended Binomial Coefficients and Preparation for Further Work. The College Mathematics Journal: Vol. 30, No. 3, pp. 170-186.

Permutation tests are based on all possible arrangements of observed data sets. Consequently, such tests yield exact probability values obtained from discrete probability distributions. An exact nondirectional method to combine independent probability values that obey discrete probability distributions is introduced. The exact method is the discrete analog to Fisher's classical method for combining probability values from independent continuous probability distributions. If the combination of probability values includes even one probability value that obeys a sparse discrete probability distribution, then Fisher's classical method may be grossly inadequate.

Mass spectrometry (MS) is widely used for isotopic studies of metabolism in which detailed information about biochemical processes is obtained from the analysis of isotope incorporation into metabolites. The biological value of such experiments is dependent on the accuracy of the isotopic measurements. Using MS, isotopologue distributions are measured from the quantitative analysis of isotopic clusters. These measurements are prone to various biases, which can occur during the experimental workflow and/or MS analysis. The lack of relevant standards limits investigations of the quality of the measured isotopologue distributions. To meet that need, we developed a complete theoretical and experimental framework for the biological production of metabolites with fully controlled and predictable labeling patterns. This strategy is valid for different isotopes and different types of metabolisms and organisms, and was applied to two model microorganisms, Pichia augusta and Escherichia coli, cultivated on (13)C-labeled methanol and acetate as sole carbon source, respectively. The isotopic composition of the substrates was designed to obtain samples in which the isotopologue distribution of all the metabolites should give the binomial coefficients found in Pascal's triangle. The strategy was validated on a liquid chromatography-tandem mass spectrometry (LC-MS/MS) platform by quantifying the complete isotopologue distributions of different intracellular metabolites, which were in close agreement with predictions. This strategy can be used to evaluate entire experimental workflows (from sampling to data processing) or different analytical platforms in the context of isotope labeling experiments.

The Pascal triangle is a geometric representation of binomial coefficients in triangular form. We utilize this formalism to deterministically arrange silver nanocylinders of different sizes (30, 60, and 90 nm) on a triangle and numerically study their near-field optical properties. We show that near-field intensities at specific points on this triangle depend on the wavelength and angle of incidence. From the wavelength-dependent studies at various junctions of nanocylinders, we obtain maximum near-field intensity at 350 and 380 nm. By varying the angle of incidence of the TM-polarized plane wave, we find systematic variation in the near-field intensity at different junctions of the geometry. Our study will lead to insights in designing controllable electromagnetic hot spots for chip-based plasmonic devices.

Generalized Pascal triangle for binomial coefficients of words

A certain discrete probability distribution was considered in [“A discrete probability distribution and some applications”, Mediterr. J. Math., 2023]. Its basic properties were investigated and some applications were presented. We now embed this distribution into a family of discrete distributions depending on two parameters and investigate the properties of the new distributions.

At first in Chap. 3 are described such important notions as characteristic functions of a probability distribution defined as Fourier transform of a probability density function and constants characterizing random variables: an average value, a median value describing the middle of a distribution, a variation ratio closely connected with a standard deviation. Other characteristic constants characterizing random variables are: an asymmetry coefficient and an excess coefficient. Other parameters describing shape of a probability density functions called quantiles are also defined. A number of useful probability distributions are presented, among them are discrete probability distributions: a binomial distribution, a multinomial distribution, a Poisson distribution, and continuous probability distributions: a rectangular distribution, an exponential distribution, a Weibull distribution and a Student distribution are described. Finally are given some introductory remarks concerning multidimensional probability distributions, developed further in Chaps. 5 and 7 for analyzing various practical problems.

This paper addresses a fundamental problem in random variate generation: given access to a random source that emits a stream of independent fair bits, what is the most accurate and entropy-efficient algorithm for sampling from a discrete probability distribution ( p 1 , …, p n ), where the probabilities of the output distribution ( p̂ 1 , …, p̂ n ) of the sampling algorithm must be specified using at most k bits of precision? We present a theoretical framework for formulating this problem and provide new techniques for finding sampling algorithms that are optimal both statistically (in the sense of sampling accuracy) and information-theoretically (in the sense of entropy consumption). We leverage these results to build a system that, for a broad family of measures of statistical accuracy, delivers a sampling algorithm whose expected entropy usage is minimal among those that induce the same distribution (i.e., is “entropy-optimal”) and whose output distribution ( p̂ 1 , …, p̂ n ) is a closest approximation to the target distribution ( p 1 , …, p n ) among all entropy-optimal sampling algorithms that operate within the specified k -bit precision. This optimal approximate sampler is also a closer approximation than any (possibly entropy-suboptimal) sampler that consumes a bounded amount of entropy with the specified precision, a class which includes floating-point implementations of inversion sampling and related methods found in many software libraries. We evaluate the accuracy, entropy consumption, precision requirements, and wall-clock runtime of our optimal approximate sampling algorithms on a broad set of distributions, demonstrating the ways that they are superior to existing approximate samplers and establishing that they often consume significantly fewer resources than are needed by exact samplers.

In socio-economics sciences, researchers need to model behaviour of ordinal variables in simultaneous equation system involving multiple endogenous ordinal variables. In existing literature the tedious solution of such likelihood function is misunderstood by applied researchers see O’Brien and Homer (Qual Quant 21(4):349–360, 1987). This paper is an attempt to convert ordinal variables into continuous format for the common use in standard simultaneous regression models. The proposed method involves generating random numbers from continuous probability distributions (say, uniform and/or truncated normal distributions) within a given discrete probability distribution. The new method can be used in ordered-logit and probit models by avoiding existing restrictive assumptions. However, the proposed method can be used under certain restrictions.

Computationally modelling the affective content of images has been extensively studied recently because of its wide applications in entertainment, advertisement, and education. Significant progress has been made on designing discriminative features to bridge the affective gap. Assuming that viewers can reach a consensus on the emotion of images, most existing works focused on assigning the dominant emotion category or the average dimension values to an image. However, the image emotions perceived by viewers are subjective by nature with the influence of personal and situational factors. In this paper, we propose a novel machine learning approach that characterizes the categorical image emotions as a discrete probability distribution (DPD). To associate emotion with the visual features extracted from images, we present shared sparse learning to learn the combination coefficients, with which the DPD of an unseen image is predicted by linearly combining the DPDs of the training images. Furthermore, we extend our method to the setup where multi-features are available and learn the optimal weights for each feature to reflect the importance of different features. Extensive experiments are carried out on Abstract, Emotion6 and IESN datasets and the results demonstrate the superiority of the proposed method, as compared to the state-of-the-art approaches.

In many real-world applications, the phenomena of interest are continuous in nature and modeled through continuous probability distributions, but their observed values are actually discrete and hence it would be more reasonable and convenient to choose an appropriate (multivariate) discrete distribution generated from the underlying continuous model preserving one or more important features. In this paper, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely (1) the joint density function, or (2) the joint survival function. Examples of applications are presented, which involve two types of bivariate exponential distributions, in order to illustrate how the discretization procedures work and show whether and to which extent they alter the dependence structure of the original model. We also prove that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. Two real datasets, considering correlated discrete recurrence times (the former) and counts (the latter) are eventually fitted using two discrete analogues of a bivariate exponential distribution.

A theoretical probability distribution conveys more information than the mean in summarizing data. We investigated the ability of two discrete probability distributions, binomial and β–binomial, to describe the incidence (proportion) of seed-borne fungi among seed lots. The fit of the distributions to 185 data sets was assessed by either χ2 analysis or a dithered Kolmogorov-Smirnov goodness-of-fit test. The data sets represented a range of fungi, crops, and geographic regions. The binomial distribution was an adequate fit to only 36% of the data sets, whereas the β–binomial distribution adequately fit 85% of the data sets (P > 0.05). The β–binomial was a better fit than the binomial in 72% of data sets (P < 0.01) based on the likelihood-ratio test, indicating that there was greater variability in seed infection than expected for a binomial (i.e., random) distribution. For a subset of 25 data sets on wheat-seed infection by Fusarium graminearum Schwabe, a binary power law analysis indicated that heterogeneity of seed infection (summarized by the θ parameter of the β–binomial) was a function of mean incidence. Therefore, in most instances, the β–binomial captures the observed heterogeneity in the incidence of seed-borne fungi.

Combining ensemble forecasts from different models into a multi-model ensemble (MME) have been shown to improve forecast skill at different time-scales, including the sub-seasonal to seasonal (S2S) one. Here, we investigate a new method to build such MME based on barycenter. Recognizing ensemble forecasts as discrete probability distributions, we work directly in the probability distribution space. This allows us to use existing tools in this space, and in particular the concept of barycenter. The barycenter of a collection of distributions (or the ensemble forecasts here) is the distribution that best represents them, based on a given metric. The barycenter can thus be seen as the combination of these distributions, and so used to build a MME. We compare two barycenters based on different metrics: the L2 and the Wasserstein distances. The Wasserstein distance corresponds to the cost of the optimal transport between two distributions and has interesting properties in the distribution space. We compare it to the L2-barycenter which is in fact shown to be equivalent to the well-known &amp;#8220;pooling&amp;#8221; MME method (i.e. the concatenation of the different ensembles members). Another interesting point of the barycenters is that they allow you to give different weights to the models and so to easily build a weighted-MME. The weights have an important impact on the skill of the MMEs. We are thus optimizing the weights by learning them from the data using cross-validation on the forecasts. The two barycenter-based MMEs are applied to the combination of the models from the S2S project&amp;#8217;s database. Their performances are evaluated for the prediction of weekly 2m-temperature during European winter with respect to different metrics. As a proof of concept, we first start with the combination of two models, namely the European Centre Medium-Range Weather Forecasts (ECMWF) and the National Center for Environmental Prediction (NCEP) models. We show that the two MMEs are generally able to perform as well or better than both the single-models, but that the best combination method depends on the chosen metric. We then extend the barycenter approach to the combination of more models, of which we will discuss preliminary results.