ARTICLE | doi:10.20944/preprints201807.0182.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: degenerate Bernstein polynomials, degenerate Bernstein operators, degenerate Euler polynomials.
Online: 10 July 2018 (13:56:13 CEST)
In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.
ARTICLE | doi:10.20944/preprints201807.0566.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: q-Bernoulli numbers; q-Bernoulli polynomials; Bernstein polynomials; q-Bernstein polynomials; p-adic integral on Zp
Online: 30 July 2018 (08:17:45 CEST)
In this paper, we study the p-adic integral representation on Zp of q-Bernoulli numbers arising from two variable q-Bernstein polynomials and investigate some properties for the q-Bernoulli numbers. In addition, we give some new identities of q-Bernoulli numbers.
ARTICLE | doi:10.20944/preprints202008.0529.v2
Subject: Life Sciences, Biophysics Keywords: membrane potential; Nernst; Bernstein; action potential; propagation; theory
Online: 9 September 2020 (09:24:15 CEST)
Man has always been interested in animal electricity, which seems to be measured in every living cell. He has been fascinated by trying to elucidate the mechanisms by which this potential is created and maintained. Biology is the science that seeks to explain this mystery. Biology is based on basic sciences such as physics or chemistry. The latter, in turn, make systematic use of mathematics to measure, evaluate and predict certain phenomena and to develop "laws" and models that are as general as possible while respecting, as closely as possible, observations and facts. The Nernst equation was one of the pillars of electrochemistry. Biology also uses this same equation as one of the indispensable bases for the computation of membrane potential. Man has established a cellular model that highlights this equation in several forms. However, we are going to show by various means that this model is inadequate or even inapplicable.
ARTICLE | doi:10.20944/preprints202208.0481.v1
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: Random normalization; thinning operators; Bernstein Theorem; problem of moments; Sibuya distribution
Online: 29 August 2022 (09:43:57 CEST)
Different variants of thinning for discrete random variables are studied. The thinning procedure allows to introduce an analog of scale parameter for positive integer-valued random variables. Sufficient and necessary conditions for the existence of such a scale are given.
ARTICLE | doi:10.20944/preprints202106.0356.v1
Subject: Life Sciences, Biochemistry Keywords: cell model; Bernstein; Nernst equation; membrane potential; GHK equation; HH model
Online: 14 June 2021 (11:50:34 CEST)
The cellular model we teach and have theorized assumes that the cell is the basic unit of multicellular living beings. This fundamental element has been the subject of many theories concerning its properties and the exchanges that exist with its environment. In this article, we demonstrate that certain functional aspects, in particular the electrical aspects related to diffusion, have not been correctly assumed or that certain initial conditions have been purely ignored and are in contradiction with physics, chemistry and thermodynamics.
ARTICLE | doi:10.20944/preprints201901.0202.v2
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: Concentration Inequality, Empirical Bernstein Bound, Stratified Random Sampling, Shapley Value Approximation
Online: 31 May 2019 (10:37:48 CEST)
We derive a concentration inequality for the uncertainty in the mean computed by stratified random sampling, and provide an online sampling method based on this inequality. Our concentration inequality is versatile and considers a range of factors including: the data ranges, weights, sizes of the strata, the number of samples taken, the estimated sample variances, and whether strata are sampled with or without replacement. Sequentially choosing samples to minimize this inequality leads to a online method for choosing samples from a stratified population. We evaluate and compare the effectiveness of our method against others for synthetic data sets, and also in approximating the Shapley value of cooperative games. Results show that our method is competitive with the performance of Neyman sampling with perfect variance information, even without having prior information on strata variances. We also provide a multidimensional extension of our inequality and discuss future applications.
ARTICLE | doi:10.20944/preprints201712.0116.v1
Subject: Keywords: (p, q)-calculus; hermite polynomials; bernstein polynomials; generating function; hyperbolic functions
Online: 18 December 2017 (08:32:22 CET)
In this paper, we introduce a new generalization of the Hermite polynomials via (p, q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also de ne a (p, q)-analogue of the Bernstein polynomials and acquire their some formulas. We then provide some (p, q)-hyperbolic representations of the (p, q)-Bernstein polynomials. In addition, we obtain a correlation between (p, q)-Hermite polynomials and (p, q)-Bernstein polynomials.
ARTICLE | doi:10.20944/preprints202206.0227.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: Chebfun; differential equation; non-linearity; singularity; convergence; Bernstein growth; improper integrals; boundary layer
Online: 16 June 2022 (02:53:41 CEST)
The Chebyshev collocation method (ChC) implemented as Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for toughest problems we have to loosen the default Chebfun tolerance in Newton's solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).
ARTICLE | doi:10.20944/preprints202010.0117.v1
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: Biased Continuous-time Random Walks, Bernstein matrix functions, Space-time fractional Poisson process, General fractional Calculus, Prabhakar fractional calculus
Online: 6 October 2020 (10:34:24 CEST)
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.
ARTICLE | doi:10.20944/preprints201709.0112.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: multivariate logarithmic polynomial; generating function; completely monotonic function; Bernstein function; integral representation; Lévy-Khintchine representation; real part; imaginary part; uniform convergence; recurrence relation; mathematical induction
Online: 23 September 2017 (10:55:57 CEST)
In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.
ARTICLE | doi:10.20944/preprints201611.0146.v1
Subject: Mathematics & Computer Science, Analysis Keywords: explicit form; inhomogeneous linear ordinary differential equation; derivative; Lerch transcendent; absolute monotonicity; complete monotonicity; Bernstein function; inequality; diagonal recurrence relation; Stirling numbers of the first kind; logarithmic function
Online: 29 November 2016 (08:00:53 CET)
In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.
ARTICLE | doi:10.20944/preprints201703.0119.v1
Subject: Mathematics & Computer Science, Analysis Keywords: Lévy–Khintchine representation; integral representation; Bernstein function; Stieltjes function; Toader–Qi mean; weighted geometric mean; Bessel function of the first kind; probabilistic interpretation; probabilistic interpretation; application in engineering; inequality
Online: 16 March 2017 (11:31:31 CET)
In the paper, by virtue of a Lévy–Khintchine representation and an alternative integral representation for the weighted geometric mean, the authors establish a Lévy–Khintchine representation and an alternative integral representation for the Toader–Qi mean. Moreover, the authors also collect an probabilistic interpretation and applications in engineering of the Toader–Qi mean.