Classical Bernoulli Research Articles

On normalized tails of series expansion of generating function of Bernoulli numbers

In the paper, the authors present the positivity and decreasing property of the normalized tails of the series expansion of the generating function of the classical Bernoulli numbers and prove the increasing property of the ratio between two normalized tails of the series expansion of the generating function of the classical Bernoulli numbers by showing the increasing property of the ratio between two Bernoulli polynomials.

Dynamics and Stability of Double-Walled Carbon Nanotube Cantilevers Conveying Fluid in an Elastic Medium

The paper concerns the dynamics and stability of double-walled carbon nanotubes conveying fluid. The equations of motion adopted in the current study to describe the dynamics of such nano-pipes stem from the classical Bernoulli–Euler beam theory. Several additional terms are included in the basic equations in order to take into account the influence of the conveyed fluid, the impact of the surrounding medium and the effect of the van der Waals interaction between the inner and outer single-walled carbon nanotubes constituting a double-walled one. In the present work, the flow-induced vibrations of the considered nano-pipes are studied for different values of the length of the pipe, its inner radius, the characteristics of the ambient medium and the velocity of the fluid flow, which is assumed to be constant. The critical fluid flow velocities are obtained at which such a cantilevered double-walled carbon nanotube embedded in an elastic medium loses stability.

Finding the optimal exploration-exploitation trade-off online through Bayesian risk estimation and minimization

We propose endogenous Bayesian risk minimization (EBRM) over policy sets as an approach to online learning across a wide range of settings. Many real-world online learning problems have complexities such as action- and belief-dependent rewards, time-discounting of reward, and heterogeneous costs for actions and feedback; we find that existing online learning heuristics cannot leverage most problem-specific information, to the detriment of their performance. We introduce a belief-space Markov decision process (BMDP) model that can capture these complexities, and further apply the concepts of aleatoric, epistemic, and process risks to online learning. These risk functions describe the risk inherent to the learning problem, the risk due to the agent's lack of knowledge, and the relative quality of its policy, respectively. We demonstrate how computing and minimizing these risk functions guides the online learning agent towards the optimal exploration-exploitation trade-off in any stochastic online learning problem, constituting the basis of the EBRM approach. We also show how Bayes' risk, the minimization objective in stochastic online learning problems, can be decomposed into the aforementioned aleatoric, epistemic, and process risks.In simulation experiments, EBRM algorithms achieve state-of-the-art performance across various classical online learning problems, including Gaussian and Bernoulli multi-armed bandits, best-arm identification, mixed objectives with action- and belief-dependent rewards, and dynamic pricing, a finite partial monitoring problem. To our knowledge, it is also the first computationally efficient online learning approach that can provide online bounds on an algorithm's Bayes' risk. Finally, because the EBRM approach is parameterized by a set of policy algorithms, it can be extended to incorporate new developments in online learning algorithms, and is thus well-suited as the foundation for developing real-world learning agents.

Nonlinear grey Bernoulli model with physics-preserving Cusum operator

The Cusum (cumulative sum) operator is a fundamental prerequisite for the nonlinear grey Bernoulli model. Traditionally, it is believed to visually identify the underlying dynamic pattern of the original time series. This paper presents the misconceptions concerning the Cusum operation and the over-optimization of the initial condition in the classical nonlinear grey Bernoulli model, both of which inspire the proposal of a physics-preserving Cusum operator. Under a state space framework, separable nonlinear least squares and nonlinear least squares are formulated to generate simultaneous estimates of structural parameters and initial condition. By combining the Cusum operators and parameter estimation methods, four modeling paradigms are generated and comprehensively compared. The simulation results show that (i) the physics-preserving Cusum outperforms the traditional Cusum and (ii) nonlinear least squares outperforms separable nonlinear least squares, especially in irregular sampling settings with large time intervals and high noise levels. Finally, the proposed approach is used to identify the underlying dynamics from short-term traffic flow data, and the results validate its effectiveness.

Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

Estimates for negative eigenvalues of Schrödinger operators on unbounded fractal spaces

We study an asymptotic formula for the number of negative eigenvalues of Schrödinger operators on unbounded fractal spaces, which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpiński gasket and unbounded generalized Sierpiński carpets. Finally, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.

Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.

A vectorial problem with thin free boundary

We consider the vectorial analogue of the thin free boundary problem introduced by Caffarelli et al. (J Eur math Soc 12:1151-1179, 2010) as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a “regular” and a “singular” part. Finally we use a viscosity approach to prove C^{1,alpha } regularity of the regular part of the free boundary.

Stochastic Bernoulli equation on the algebra of generalized functions

UDC 519.21 Based on the topological dual space ℱ θ * ( S ' ℂ ) of the space of entire functions with θ -exponential growth of finite type, we introduce the generalized stochastic Bernoulli–Wick differential equation (or the stochastic Bernoulli equation on the algebra of generalized functions) by using the Wick product of elements in ℱ θ * ( S ' ℂ ) . This equation is an infinite-dimensional stochastic distributions analog of the classical Bernoulli differential equation. This stochastic differential equation is solved and exemplified by several examples.

Forecasting education expenditure with a generalized conformable fractional-order nonlinear grey system model

As an important human capital investment, education is an effective means to improve the comprehensive quality of people. Education expenditure is an important material guarantee for the development of educational undertakings. Education expenditure data is highly susceptible to numerous economic and social factors that complicate its nonlinear structure. In order to model the complex nonlinear problems of the system, this paper proposes a generalized conformable fractional-order nonlinear grey prediction model for the first time by analyzing the traditional time series-based modeling method in a nonlinear grey domain. The proposed model expands on the classical grey Bernoulli model by introducing the generalized conformable fractional accumulation as a new accumulation generator and utilizes error minimization principles in the modeling process. By altering the optimal order of the model and the cumulative generation operator, this model can adapt to various time series and reduce errors. Finally, the model is applied to education expenditure forecasting, and it is proved that the proposed model achieved good results and has higher accuracy than other models.

On Some Probability Distributions Related to the Classical Bernoulli Scheme. II

On Some Probability Distributions Related to the Classical Bernoulli Scheme. II

On some probability distributions, related to the classical Bernoulli scheme. II

The classic scheme of independent trials by Bernoulli has been one of the most popular topics in probability theory for more than three centuries, starting with the work of Jacob Bernoulli. It is ideal for setting and solving various practical problems. Many results have been obtained related to modifications of this scheme, but new situations, new problems appear that require further progress in the study of various random variables related to one degree or another with independent Bernoulli tests. In this paper, we continue to study some problems related to the series of successes and failures in sequences of Bernoulli random variables (c. v.). This work is a direct continuation of the authors’ article “On some probability distributions related to the classical Bernoulli scheme”, published in 2022.

On Some Probabilistic Distributions Related to the Classical Bernoulli Scheme

On Some Probabilistic Distributions Related to the Classical Bernoulli Scheme

Predictive analysis of the industrial water-waste-energy system using an optimised grey approach: A case study in China

To estimate the dynamic trend of industrial water-waste-energy (hereinafter referred to as WWE) system, this paper proposes a new method for forecasting specific indicators in such a system. First, the fractional accumulated generation operator, fractional derivative and classic nonlinear grey Bernoulli model are simultaneously coupled to develop an optimised nonlinear grey Bernoulli model that identifies the nonlinear trends in industrial WWE systems. Second, the particle swarm optimization algorithm is employed to determine the optimal model parameters in the newly-designed model. Based on this, simulation studies are conducted to examine the stability of the proposed model. Finally, the model is applied in the industrial WWE system. The results demonstrate that (1) the proposed model outperforms other competitive models in terms of error-value metrics and (2) industrial water use and industrial energy consumption will increase, whereas industrial wastewater discharge will decline. Furthermore, the rationality of the predicted results redis analyzed from a policy perspective.

A new nonlinear 5-parameter beam model accounting for the Poisson effect

In this paper, a new kinematic beam model based on a five-parameter displacement field is proposed in a geometric nonlinear framework. The proposed displacement field enriches the classical Timoshenko beam displacement field with two additional parameters, accounting for the Poisson effect. The equilibrium equations have been derived through a variational approach, and the linearized equations are solved analytically. The adoption of the linear solution as approximation functions for the nonlinear case allows prediction of nonlinear response of problems involving complex geometries with a relatively small computational effort. Several numerical examples of benchmark problems are analyzed, highlighting the characteristic features of the proposed five-parameter model and comparing the results with those obtained using the classical Bernoulli beam model and 3D finite element model. Numerical results show that, although for thin beams the differences in generalized displacement are generally negligible, the proposed model predicts a comparable stress field only when the Poisson ratio is equal to zero. Conversely, the stress field is meaningfully enriched by the new parameters, with significant differences where the Poisson effect is more pronounced.

On some probability distributions, related to the classical Bernoulli scheme

For more than three centuries, the classical scheme of independent Bernoulli trials, starting with the work of Jacob Bernoulli, has been one of the most popular topics in probability theory. It is ideal for setting and solving various practical problems. Many results have been obtained related to modifications of this scheme, but new situations and new problems appear that require further progress in the study of various random variables associated to one degree or another with independent Bernoulli trials. Solutions to some of these problems are presented in this paper.

General quantum Bernoulli factory: framework analysis and experiments

The unremitting pursuit for quantum advantages gives rise to the discovery of a quantum-enhanced randomness processing named quantum Bernoulli factory (QBF). This quantum enhanced process can show its priority over the corresponding classical process through readily available experimental resources, thus in the near term it may be capable of accelerating the applications of classical Bernoulli factories, such as the widely used sampling algorithms. In this work, we provide the framework analysis of the QBF. We thoroughly analyze the quantum state evolution in this process, discovering the field structure of the constructible quantum states. Our framework analysis shows that naturally, the previous works can be described as specific instances of this framework. Then, as a proof of principle, we experimentally demonstrate this framework via an entangled two-photon source along with a reconfigurable photonic logic, and show the advantages of the QBF over the classical model through a classically infeasible instance. These results may stimulate the discovery of advantages of the quantum randomness processing in a wider range of tasks, as well as its potential applications.

Self-similar measures and the Rajchman property

For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erdos [2] and Salem [12]. We prove weak forms of their results for general self-similar measures associated to affine contractions of the real line.

Analytical solution for a 5-parameter beam displacement model

In this paper, a new beam model based on a 5-parameter displacement field, accounting for an enhanced kinematics and able to reproduce the Poisson effect, is proposed. The displacement field enriches the classical three-parameter Timoshenko beam with two new parameters capable to simulate the shortening effect over the thickness. The related differential equations, derived from a variational formulation, are analytically solved and implemented in a deformation method approach, capable of solving structural problems of beams with general geometry, boundary and load conditions. Several numerical applications are developed, highlighting the characteristic of the proposed 5-parameters model and comparing the results with those obtained using the classical Bernoulli, Timoshenko and Reddy beam models. Numerical results show that, although for homogeneous beam the differences in terms of generalized stress and displacement are generally very small, the proposed model returns local stress enriched by the new parameters, with more significant differences where the Poisson effect is more pronounced.

Beam Theory of Thermal-Electro-Mechanical Coupling for Single-Wall Carbon Nanotubes.

The potential application field of single-walled carbon nanotubes (SWCNTs) is immense, due to their remarkable mechanical and electrical properties. However, their mechanical properties under combined physical fields have not attracted researchers’ attention. For the first time, the present paper proposes beam theory to model SWCNTs’ mechanical properties under combined temperature and electrostatic fields. Unlike the classical Bernoulli–Euler beam model, this new model has independent extensional stiffness and bending stiffness. Static bending, buckling, and nonlinear vibrations are investigated through the classical beam model and the new model. The results show that the classical beam model significantly underestimates the influence of temperature and electrostatic fields on the mechanical properties of SWCNTs because the model overestimates the bending stiffness. The results also suggest that it may be necessary to re-examine the accuracy of the classical beam model of SWCNTs.