Showing papers on "Bernoulli's principle published in 2020"

A Note on a New Type of Degenerate Bernoulli Numbers

Abstract: Studying degenerate versions of various special polynomials has became an active area of research and has yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of the polylogarithm function, the so-called degenerate polylogarithm function. Then we construct a new type of degenerate Bernoulli polynomial and number, the so-called degenerate poly-Bernoulli polynomial and number, by using the degenerate polylogarithm function, and derive several properties concerning the degenerate poly-Bernoulli numbers.

Solving a new design of nonlinear second-order Lane-Emden pantograph delay differential model via Bernoulli collocation method

Abstract: The present study is related to the design of a new mathematical model based on the Lane–Emden pantograph delay differential equation. The new model is obtained by using the sense of delay differential equation and standard Lane–Emden second-order equation. For the numerical solutions of the designed model, a well-known Bernoulli collocation method is implemented. In order to check the perfection and exactness of the designed model, three different nonlinear examples have been solved by using the Bernoulli collocation scheme. Furthermore, the comparison of the numerical results obtained by the Bernoulli collocation scheme with the exact solutions is also presented. Moreover, some numerical tables and the graphs of absolute error are plotted using different values of N for all problems.

Computationally Efficient Distributed Multi-Sensor Fusion With Multi-Bernoulli Filter

Abstract: This paper proposes a computationally efficient algorithm for distributed fusion in a sensor network in which multi-Bernoulli (MB) filters are locally running in every sensor node for multi-object tracking. The generalized Covariance Intersection (GCI) fusion rule is employed to fuse multiple MB random finite set densities. The fused density comprises a set of fusion hypotheses that grow exponentially with the number of Bernoulli components. Thus, GCI fusion with MB filters can become computationally intractable in practical applications that involve tracking of even a moderate number of objects. In order to accelerate the multi-sensor fusion procedure, we derive a theoretically sound approximation to the fused density. The number of fusion hypotheses in the resulting density is significantly smaller than the original fused density. It also has a parallelizable structure that allows multiple clusters of Bernoulli components to be fused independently. By carefully clustering Bernoulli components into isolated clusters using the GCI divergence as the distance metric, we propose an alternative to build exactly the approximated density without exhaustively computing all the fusion hypotheses. The combination of the proposed approximation technique and the fast clustering algorithm can enable a novel and fast GCI-MB fusion implementation. Our analysis shows that the proposed fusion method can dramatically reduce the computational and memory requirements with small bounded $L_1$ -error. The Gaussian mixture implementation of the proposed method is also presented. In various numerical experiments, including a challenging scenario with up to forty objects, the efficacy of the proposed fusion method is demonstrated.

On the complex solutions to the (3+1)-dimensional conformable fractional modified KdV–Zakharov–Kuznetsov equation

Abstract: This paper presents some exponential function solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (mKdV–ZK). The improved Bernoulli sub-equation function method...

A Fast Labeled Multi-Bernoulli Filter Using Belief Propagation

Abstract: We propose a fast labeled multi-Bernoulli (LMB) filter that uses belief propagation for probabilistic data association. The complexity of our filter scales only linearly in the numbers of Bernoulli components and measurements, while the performance is comparable to or better than that of the Gibbs sampler-based LMB filter.

Entropy of Bernoulli convolutions and uniform exponential growth for linear groups

Abstract: The exponential growth rate of non-polynomially growing subgroups of GLrf is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture from number theory. In this note, we show that it is equivalent to the Lehmer conjecture. This is done by establishing a lower bound for the entropy of the random walk on the semi-group generated by the maps x → λ · x ± 1, where λ is an algebraic number. We give a bound in terms of the Mahler measure of λ. We also derive a bound on the dimension of Bernoulli convolutions.

A new Bernoulli–Euler beam model based on a reformulated strain gradient elasticity theory:

Abstract: A new non-classical Bernoulli–Euler beam model is developed using a reformulated strain gradient elasticity theory that incorporates both couple stress and strain gradient effects. This reformulate...

On some wavelet solutions of singular differential equations arising in the modeling of chemical and biochemical phenomena

Abstract: This paper is concerned with the Lane–Emden boundary value problems arising in many real-life problems. Here, we discuss two numerical schemes based on Jacobi and Bernoulli wavelets for the solution of the governing equation of electrohydrodynamic flow in a circular cylindrical conduit, nonlinear heat conduction model in the human head, and non-isothermal reaction–diffusion model equations in a spherical catalyst and a spherical biocatalyst. These methods convert each problem into a system of nonlinear algebraic equations, and on solving them by Newton’s method, we get the approximate analytical solution. We also provide the error bounds of our schemes. Furthermore, we also compare our results with the results in the literature. Numerical experiments show the accuracy and reliability of the proposed methods.

A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem

Abstract: In this paper we propose and analyze, utilizing mainly tools and abstract results from Banach spaces rather than from Hilbert ones, a new fully-mixed finite element method for the stationary Boussinesq problem with temperature-dependent viscosity. More precisely, following an idea that has already been applied to the Navier–Stokes equations and to the fluid part only of our model of interest, we first incorporate the velocity gradient and the associated Bernoulli stress tensor as auxiliary unknowns. Additionally, and differently from earlier works in which either the primal or the classical dual-mixed method is employed for the heat equation, we consider here an analogue of the approach for the fluid, which consists of introducing as further variables the gradient of temperature and a vector version of the Bernoulli tensor. The resulting mixed variational formulation, which involves the aforementioned four unknowns together with the original variables given by the velocity and temperature of the fluid, is then reformulated as a fixed point equation. Next, we utilize the well-known Banach and Brouwer theorems, combined with the application of the Babuska-Brezzi theory to each independent equation, to prove, under suitable small data assumptions, the existence of a unique solution to the continuous scheme, and the existence of solution to the associated Galerkin system for a feasible choice of the corresponding finite element subspaces. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the performance of the fully-mixed scheme and confirming the theoretical rates of convergence.

Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice

Abstract: We consider a Hamiltonian given by the Laplacian plus a Bernoulli potential on the two dimensional lattice. We prove that, for energies sufficiently close to the edge of the spectrum, the resolvent on a large square is likely to decay exponentially. This implies almost sure Anderson localization for energies sufficiently close to the edge of the spectrum. Our proof follows the program of Bourgain–Kenig, using a new unique continuation result inspired by a Liouville theorem of Buhovsky–Logunov–Malinnikova–Sodin.

New critical exponent inequalities for percolation and the random cluster model

Abstract: We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin-Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following: The critical exponent inequalities $\gamma \leq \delta-1$ and $\Delta \leq \gamma +1$ hold for percolation and the random cluster model on any transitive graph. These inequalities are new even in the context of Bernoulli percolation on $\mathbb{Z}^d$, and are saturated in mean-field for Bernoulli percolation and for the random cluster model with $q \in [1,2)$. The volume of a cluster has an exponential tail in the entire subcritical phase of the random cluster model on any transitive graph. This proof also applies to infinite-range models, where the result is new even in the Euclidean setting.

Stratified equatorial flows in cylindrical coordinates

Abstract: We construct an exact solution modelling the geophysical dynamics of an inviscid and incompressible fluid which possesses a variable density stratification, where the fluid density may vary with both the depth and latitude. Our solution pertains to the large-scale equatorial dynamics of a fluid body with a free surface propagating steadily in a purely azimuthal direction, and is expressed in terms of cylindrical coordinates. Allowing for general fluid stratification greatly complicates the Bernoulli relation—which relates the imposed pressure to the reciprocal fluid distortion at the free-surface—thereby acting as a constraint on the existence of a solution. Employing the implicit function theorem, we establish the existence of solutions and determine that the requisite monotonicity properties hold for the flow solutions we found. Furthermore, since the fluid velocity and pressure are prescribed by explicit formulae in the framework of cylindrical coordinates, our solution is amenable to analysis by the short-wavelength stability approach, which we investigate.

A simple finite element for the geometrically exact analysis of Bernoulli-Euler rods

Abstract: This work develops a simple finite element for the geometrically exact analysis of Bernoulli–Euler rods. Transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. A straight reference configuration is assumed for the rod. The cross-section undergoes a rigid body motion. A rotation tensor with the Rodrigues formula is used to describe the rotation, which makes the updating of the rotational variables very simple. A formula for the Rodrigues parameters in function of the displacements derivative and the torsion angle is for the first time settled down. The consistent connection between elements is thoroughly discussed, and an appropriate approach is developed. Cubic Hermitian interpolation for the displacements together with linear Lagrange interpolation for the torsion incremental angle were employed within the usual Finite Element Method, leading to adequate C1 continuity. A set of numerical benchmark examples illustrates the usefulness of the formulation and numerical implementation.

On the hausdorff dimension of bernoulli convolutions

Abstract: We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution νβ to arbitrary given accuracy whenever β is algebraic. In particular, if the Garsia entropy H(β) is not equal to log(β) then we have a finite time algorithm to determine whether or not dimH(νβ) = 1.

Regularity of the free boundary for the vectorial bernoulli problem

Abstract: We study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure D⊂ℝd, Λ>0, and ϕi∈H1∕2(∂D), we deal with min { ∑ i = 1 k ∫ D | ∇ v i | 2 + Λ | ⋃ i = 1 k { v i ≠ 0 } | : v i = ϕ i on ∂ D } . We prove that, for any optimal vector U=(u1,…,uk), the free boundary ∂(⋃i=1k{ui≠0})∩D is made of a regular part, which is relatively open and locally the graph of a C∞ function, a (one-phase) singular part, of Hausdorff dimension at most d−d∗, for a d∗∈{5,6,7}, and by a set of branching (two-phase) points, which is relatively closed and of finite ℋd−1 measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.

Planar random-cluster model: fractal properties of the critical phase

Abstract: This paper is studying the critical regime of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4)$. More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on said boundary. Additionally, they imply that any sub-sequential scaling limit of the collection of interfaces between primal and dual clusters is made of loops that are non-simple. We also obtain a number of properties of so-called arm-events: three universal critical exponents (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), quasi-multiplicativity and well-separation properties (even when arms are not alternating between primal and dual), and the fact that the four-arm exponent is strictly smaller than 2. These results were previously known only for Bernoulli percolation ($q = 1$) and the FK-Ising model ($q = 2$). Finally, we prove new bounds on the one, two and four arms exponents for $q\in[1,2]$. These improve the previously known bounds, even for Bernoulli percolation.

An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation

Abstract: The nonlinear conformable time-fractional Symmetric Regularized Long Wave (SRLW) equation plays an important role in physics. This equation is an interesting model to describe ion-acoustic and space change waves with weak nonlinearity. In this paper, we solve the SRLW equation via the Improved Bernoulli Sub-Equation Function Method (IBSEFM). New exact solutions are constructed and by the aid of software mathematics programme, 2D and 3D graphs of the solutions according to the parameters are plotted. The results show that IBSEFM is a powerful mathematical tool to solve nonlinear conformable time-fractional equations arising in mathematical physics.

A Reduced-Order Flow Model for Fluid-Structure Interaction Simulation of Vocal Fold Vibration.

Abstract: We present a novel reduced-order glottal airflow model that can be coupled with the three-dimensional (3D) solid mechanics model of the vocal fold tissue to simulate the fluid-structure interaction (FSI) during voice production. This type of hybrid FSI models have potential applications in the estimation of the tissue properties that are unknown due to patient variations and/or neuromuscular activities. In this work, the flow is simplified to a one-dimensional (1D) momentum equation-based model incorporating the entrance effect and energy loss in the glottis. The performance of the flow model is assessed using a simplified yet 3D vocal fold configuration. We use the immersed-boundary method-based 3D FSI simulation as a benchmark to evaluate the momentum-based model as well as the Bernoulli-based 1D flow models. The results show that the new model has significantly better performance than the Bernoulli models in terms of prediction about the vocal fold vibration frequency, amplitude, and phase delay. Furthermore, the comparison results are consistent for different medial thicknesses of the vocal fold, subglottal pressures, and tissue material behaviors, indicating that the new model has better robustness than previous reduced-order models.

Stochastic Comparisons between the Extreme Claim Amounts from Two Heterogeneous Portfolios in the Case of Transmuted-G Model

Abstract: Let Xλ1,…,Xλn be independent and non-negative random variables belong to the transmuted-G model and let Yi=IpiXλi,i=1,…,n, where Ip1,…,Ipn are independent Bernoulli random variables independent of ...

Nonfragile and Nonsynchronous Synthesis of Reachable Set for Bernoulli Switched Systems

Abstract: In this paper, the problems of nonfragile and nonsynchronous synthesis for Bernoulli switched systems are studied. The disturbances in the systems are assumed to have bounded peak in the sense of mean square. With the interval additive gain variations consideration, a nonfragile and nonsynchronous controller is designed subject to a nonsynchronous assumption that the switching signals of the system are not accurately gained for synthesis. A sufficient condition that is dependent on the known sojourn probabilities of switching signals is proposed to ascertain the reachable set of the closed-loop system, which is a hidden Bernoulli model, mean square contained in an ellipsoid-like set. Several nonfragile and nonsynchronous controllers are parameterized to guarantee the corresponding requirements to be satisfied for the closed-loop systems. The effectiveness and advantages of the proposed new design techniques are illustrated by a numerical simulation example.

Planar random-cluster model: scaling relations

Abstract: This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $\beta$, $\gamma$, $\delta$, $\eta$, $ u$, $\zeta$ as well as $\alpha$ (when $\alpha\ge0$). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the ``mixing rate'' critical exponent $\iota$ replacing the four-arm event exponent $\xi_4$.

Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Abstract: In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

The broad-crested weir

Abstract: The purpose of this practical is to investigate: the rapidly varying flow in the vicinity of a broad-crested weir, the flow transition between sub-critical and super-critical flows, and the basic concepts of hydraulic controls, specific energy and Bernoulli principle applied to open channel flows. A key feature of the broad-crested weir practical is the smooth and short transition from subcritical to supercritical flow with small energy losses, i.e. in contrast to a hydraulic jump.

A Deep Neural Network Based Glottal Flow Model for Predicting Fluid-Structure Interactions during Voice Production.

Abstract: This paper proposes a machine-learning based reduced-order model that can provide fast and accurate prediction of the glottal flow during voice production. The model is based on the Bernoulli equation with a viscous loss term predicted by a deep neural network (DNN) model. The training data of the DNN model is a Navier-Stokes (N-S) equation-based three-dimensional simulation of glottal flows in various glottal shapes generated by a synthetic shape function, which can be obtained by superimposing the instantaneous modal displacements during vibration on the prephonatory geometry of the glottal shape. The input parameters of the DNN model are the geometric and flow parameters extracted from discretized cross sections of the glottal shapes and the output target is the corresponding flow resistance coefficient. With this trained DNN-Bernoulli model, the flow resistance coefficient as well as the flow rate and pressure distribution in any given glottal shape generated by the synthetic shape function can be predicted. The model is further coupled with a finite-element method based solid dynamics solver for simulating fluid-structure interactions (FSI). The prediction performance of the model for both static shape and FSI simulations is evaluated by comparing the solutions to those obtained by the Bernoulli and N-S model. The model shows a good prediction performance in accuracy and efficiency, suggesting a promise for future clinical use.