Comparison of turbulence models in simulating swirling pipe flows

Swirling flow is a common phenomenon in engineering applications. A numerical study of the swirling flow inside a straight pipe was carried out in the present work with the aid of the commercial CFD code fluent. Two-dimensional simulations were performed, and two turbulence models were used, namely, the RNG k–ε model and the Reynolds stress model. Results at various swirl numbers were obtained and compared with available experimental data to determine if the numerical method is valid when modeling swirling flows. It has been shown that the RNG k–ε model is in better agreement with experimental velocity profiles for low swirl, while the Reynolds stress model becomes more appropriate as the swirl is increased. However, both turbulence models predict an unrealistic decay of the turbulence quantities for the flows considered here, indicating the inadequacy of such models in simulating developing pipe flows with swirl.     

Effects of a Degeneracy in the Competition Model: Part I. Classical and Generalized Steady-State Solutions

We study the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. We show that there exists a critical number λ* such that if λ <λ*, then the model behaves similarly to the well-studied classical competition model where all the coefficient functions are positive constants, but when λ>λ*, new phenomena occur. Our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative, but can be qualitative as well. In part I here, we mainly study two kinds of steady-state solutions which determine the dynamics of the model: one consists of finite functions while the other consists of generalized functions which satisfy (u, v)=(∞, 0) on the part of the domain that b(x) vanishes, but are positive and finite on the rest of the domain, and are determined by certain boundary blow-up systems. The research is continued in part II, where these two kinds of steady-state solutions will be used to determine the dynamics of the model.     

Effects of a Degeneracy in the Competition Model: Part II. Perturbation and Dynamical Behaviour

This is the second part of our study on the competition model where the coefficient functions are strictly positive over the underlying spatial region Ω except b(x), which vanishes in a nontrivial subdomain of Ω, and is positive in the rest of Ω. In part I, we mainly discussed the existence of two kinds of steady-state solutions of this system, namely, the classical steady-states and the generalized steady-states. Here we use these solutions to determine the dynamics of the model. We do this with the help of the perturbed model where b(x) is replaced by b(x)+ε, which itself is a classical competition model. This approach also reveals the interesting relationship between the steady-state solutions (both classical and generalized) of the above system and that of the perturbed system.     

Enumeration of (p,q)-parking functions

Parking functions are central in many aspects of combinatorics. We define in this communication a generalization of parking functions which we call (p₁,…,p_k)-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent configurations in the sandpile model for some graphs. We also establish a correspondence with a Lukasiewicz language, which enables to enumerate (p₁,…,p_k)-parking functions as well as increasing ones.     

Joint modelling of the total amount and the number of claims by conditionals

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple (S,N), where N is a count variable and S=X₁+?+X_N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S|N and N|S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson-Gamma and the Poisson-Binomial conditionals models. Other conditionals models are proposed, including the Poisson-Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.     

Multiplicative models for configuration spaces of algebraic varieties

Fulton and MacPherson (Ann. Math. 139 (1994) 183) found a Sullivan dg-algebra model for the space of n-configurations of a smooth complex projective variety X. K?í? (Ann. Math. 139 (1994) 227) gave a simpler model, E_n(H), depending only on the cohomology ring, H?H^*X.We construct an even simpler and smaller model, J_n(H). We then define another new dg-algebra, E_n(H^°), and use J_n(H) to prove that E_n(H^°) is a model of the space of n-configurations of the non-compact punctured manifold X^°, when X is 1-connected. Following an idea of Drinfel’d (Leningrad Math. J. 2 (1991) 829), we put a simplicial bigraded differential algebra structure on {E_n(H^°)}_n?0.     

Characteristic properties of equivalent structures in compositional models

Compositional model theory serves as an alternative approach to multidimensional probability distribution representation and processing. Every compositional model over a finite non-empty set of variables N is uniquely defined by its generating sequence - an ordered set of low-dimensional probability distributions. A generating sequence structure induces a system of conditional independence statements over N valid for every multidimensional distribution represented by a compositional model with this structure.The equivalence problem is how to characterise whether all independence statements induced by structure P are induced by a second structure P^′ and vice versa. This problem can be solved in several ways. A partial solution of the so-called direct characterisation of an equivalence problem is represented here. We deduce and describe three properties of equivalent structures necessary for equivalence of the respective structures. We call them characteristic properties of classes of equivalent structures.     

Global stability of a stage-structured epidemic model with a nonlinear incidence

In this paper, a stage-structured epidemic model with a nonlinear incidence with a factor S^p is investigated. By using limit theory of differential equations and Theorem of Busenberg and van den Driessche, global dynamics of the model is rigorously established. We prove that if the basic reproduction number R₀ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model.     

Estimation of time-varying mortality rates using continuous models for Daphnia magna

Structured population models that make the assumption of constant demographic rates do not accurately describe the complex life histories seen in many species. We investigated the accuracy of using constant versus time-varying mortality rates within discrete and continuously structured models for Daphnia magna. We tested the accuracy of the models we considered using density-independent survival data for 90 daphnids. We found that a continuous differential equation model with a time-varying mortality rate was the most accurate model for describing our experimental D. magna survival data. Our results suggest that differential equation models with variable parameters are an accurate tool for estimating mortality rates in biological scenarios in which mortality might vary significantly with age.     

Existence of weak solutions to a degenerate time-dependent semiconductor equations with temperature effect

In this paper, we consider a degenerate time-dependent drift-diffusion model for semiconductors. The electric conductivity in the system is assumed to be temperate-dependent. And the pressure function we use in this paper is φ(s)=sα(α>1). We present existence results for general nonlinear diffusivities for the degenerate Dirichlet-Neumann mixed boundary value problem.     

Assortment optimization with repeated exposures and product-dependent patience cost

In this paper, we study the assortment optimization problem faced by many online retailers such as Amazon. We develop a cascade multinomial logit model, based on the classic multinomial logit model, to capture the consumers' purchasing behavior across multiple stages. Unlike most of existing studies, our model allows for repeated exposures of a product. In addition, each consumer has a patience budget that is sampled from a known distribution and each product is associated with a patience cost, which is the required amount of the cognitive efforts on browsing that product. We propose an approximation solution to the assortment optimization problem under cascade multinomial logit model.     

A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients

This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R⁺. We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L¹ norm was used, we apply the flat metric, similar to the Wasserstein W₁ distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.     

On phase separation points for one-dimensional models

In the paper, the one-dimensional model with nearest-neighbor interactions I _n , n ∈ Z, and the spin values ±1 is considered. It is known that, under some conditions on parameters of I _n , a phase transition occurs for this model. We define the notion of a phase separation point between two phases. We prove that the expectation value of this point is zero and its mean-square fluctuation is bounded by a constant C(β) which tends to ¼ as β → ∞, where β = 1/T and T is the temperature.     

The rotor-router model on regular trees

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We show that the set of occupied sites for this model on an infinite regular tree is a perfect ball whenever it can be, provided the initial rotor configuration is acyclic (that is, no two neighboring vertices have rotors pointing to one another). This is proved by defining the rotor-router group of a graph, which we show is isomorphic to the sandpile group. We also address the question of recurrence and transience: We give two rotor configurations on the infinite ternary tree, one for which chips exactly alternate escaping to infinity with returning to the origin, and one for which every chip returns to the origin. Further, we characterize the possible “escape sequences” for the ternary tree, that is, binary words a₁…an for which there exists a rotor configuration so that the kth chip escapes to infinity if and only if ak=1.     

On the Chern-Simons limit for a Maxwell-Chern-Simons model on bounded domains

This paper concerns the Chern-Simons limit for the Abelian Maxwell-Chern-Simons model on bounded domains with vanishing gauge fields. We prove that every sequence of solutions of the Maxwell-Chern-Simons equations has a subsequence converging to a solution of the Chern-Simons equation in any Ck norms. We also show that the Maxwell-Chern-Simons equations with the nontopological type boundary condition do not admit any nontrivial solutions on star-shaped domains.     

Analysis of a two-phase cell division model

In this paper we mainly study an equal mitosis two-phase cell division model. By using the C₀-semigroup theory, we prove that this model is the well-posed in L¹[0, 1] × L¹[0, 1], and exhibits asynchronous exponential growth phenomenon as time goes to infinity. We also give a comparison between this two-phase model with the classical one-phase model. Finally, we briefly study the asymmetric two-phase cell division model, and show that similar results hold for it.     

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+auq−bur, subject to the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t→∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.     

Elementary submodels of parametrizable models

We introduce the notion of F-parametrizable model and prove some general results on elementary submodels of F-parametrizable models. Using this notion, we can uniformly characterize all elementary submodels for the field of real numbers and for the group of all permutations on natural numbers in the first order language as well as in the language of hereditarily finite superstructures. Assuming the constructibility axiom, we obtain a simpler characterization of elementary submodels of F-parametrizable models and prove some additional properties of the structure of their elementary submodels.     

Mean-field limit for the stochastic Vicsek model

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number N of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space R^d. As part of the study we give existence and uniqueness results for both the particle system and the PDE.