Fibonacci series from power series

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.

Hankel determinants of sequences related to Bernoulli and Euler polynomials

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

Some Symmetry Identities for Carlitz’s Type Degenerate Twisted (p,q)-Euler Polynomials Related to Alternating Twisted (p,q)-Sums

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted (p,q)-Euler polynomials by using computer.

EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

Quality control strategies for brain MRI segmentation and parcellation: practical approaches and recommendations - insights from The Maastricht Study

ABSTRACTBackgroundQuality control of brain segmentation is a fundamental step to ensure data quality. Manual quality control is the current gold standard, despite unfeasible in large neuroimaging samples. Several options for automated quality control have been proposed, providing potential time efficient and reproducible alternatives. However, those have never been compared side to side, which prevents to reach consensus in the appropriate QC strategy to use. This study aims to elucidate the changes manual editing of brain segmentations produce in morphological estimates, and to analyze and compare the effects of different quality control strategies in the reduction of the measurement error.MethodsWe used structural MR images from 259 participants of The Maastricht Study. Morphological estimates were automatically extracted using FreeSurfer 6.0. A subsample of the brain segmentations with inaccuracies was manually edited, and morphological estimates were compared before and after editing. In parallel, 11 quality control strategies were applied to the full sample. Those included: a manual strategy, manual-QC, in which images were visually inspected and manually edited; five automated strategies where outliers were excluded based on the tools MRIQC and Qoala-T, and the metrics morphological global measures, Euler numbers and Contrast-to-Noise ratio; and five semi-automated strategies, were the outliers detected through the mentioned tools and metrics were not excluded, but visually inspected and manually edited. We used a regression of morphological brain measures against age as a test case to compare the changes in relative unexplained variance that each quality control strategy produces, using the reduction of relative unexplained variance as a measure of increase in quality.ResultsManually editing brain surfaces produced changes particularly high in subcortical brain volumes and moderate in cortical surface area, thickness and hippocampal volumes. The exclusion of outliers based on Euler numbers yielded a larger reduction of relative unexplained variance for measurements of cortical area, subcortical volumes and hippocampal subfields, while manual editing of brain segmentations performed best for cortical thickness. MRIQC produced a lower, but consistent for all types of measures, reduction in relative unexplained variance. Unexpectedly, the exclusion of outliers based on global morphological measures produced an increase of relative unexplained variance, potentially removing more morphological information than noise from the sample.ConclusionOverall, the automatic exclusion of outliers based on Euler numbers or MRIQC are reliable and time efficient quality control strategies that can be applied in large neuroimaging cohorts.

Symmetric Properties for Dirichlet-Type Multiple (p,q)-L-Function

The main aim of this article is to investigate some interesting symmetric identities for the Dirichlet-type multiple (p,q)-L function. We use this function to examine the symmetry of the generalized higher-order (p,q)-Euler polynomials related to χ. First, the generalized higher-order (p,q)-Euler numbers and polynomials related to χ are defined. We also give a few new symmetric properties for the Dirichlet-type multiple (p,q)-L-function and generalized higher-order (p,q)-Euler polynomials related to χ.

With Respect to Quantum Perspective Model, Can Euler Numbers be Related to Biochemistry?

This article researches whether there is a link between Euler’s numbers and genetic codes. At first, the sum of the numbers of the first fifteen "15" digits of Euler’s numbers after the comma are converted to bases in genetic codes. Secondly, after the comma, Euler’s numbers with eighteen fifteen groups are converted to nucleotide bases. So, the results obtained by this way are expressed as nucleotide bases ( A, T, C, G, U). (A)Adenine, (T)Thymine, (C)Cytosine, (G),Guanine, (U)Uracil. Thirdly, the search result is similar to ZEBRAFISH-DANIO RERIO, and even bat coronavirus after the NCBI (National Biotechnology Information Center) searched this sequence ”AUGUUGAUAUTAAUCATC”. Fourtly, the genetic codes of Zebrafish have been proven to be very similar to human genetic codes. Fifthly, multiple spawning of these fish species also means that Euler's numbers are increasing. In sum, the relationship between the Euler’s numbers in mathematical science and the atomic weights of atomic elements in genetic codes also shed lights on Biochemistry.