Nonhomogeneous Case Research Articles

Minkowski spacetime does not apply to a homogeneously accelerating medium

Home and comoving inertial frame parameters of an individual point of an idealized medium of launch length L uniformly co-accelerating between identical fixed-thrust rockets, are well known. This is not the case with the varying inter-rocket radar periods and related implications regarding a changing ‘noninertial own-length’ Λ which differs from a front rocket’s retrospective separationL from the simultaneously relatively moving rear rocket. On the other hand, the nonhomogeneous acceleration case involving every comoving frame’s unchanging perception of a contrived ‘rigor mortis’ medium (so-called ‘rigid motion’ traditionally associated with ‘Rindler coordinates’) whereby Λ=L=L, constitutes the sole extended accelerating medium scenario where the entrenched Minkowski metric is actually applicable. Paraphrasing Wolfgang Pauli, not only is Minkowski spacetime not correct [in the general sense], it is not even wrong [in the restricted sense].

Modelling clinical trial recruitment using poisson processes

Forecasting recruitment to clinical trials is often too simplistic and rarely statistical. This is unfortunate because modelling trial recruitment is simple using Poisson processes. The simplest case is to assume patients arrive randomly at a constant rate, using homogeneous Poisson processes to simulate sample recruitment curves. In contrast, time-varying complexity can be incorporated using non-homogeneous Poisson processes. This necessitates a recruitment intensity function, 0 ≤ p(t) ≤1, (t ≥ 0), to describe the instantaneous rate of recruitment potential at time t, as a proportion of the maximum rate possible. Using this method, one can incorporate important factors like seasonal trends, staggered opening of recruitment centres, accelerating and decelerating recruitment and planned suspensions (for instance, due to interim assessments), to reflect more accurately the possible recruitment patterns over time. There are many benefits to investing the time to model trial recruitment. Guidance on plausible recruitment scenarios can inform inventory needs and forewarn the impact of recruitment shocks, both welcome and unwelcome. In designs with time-to-event outcomes, the rate of patient recruitment can affect statistical operating characteristics. In complicated trial designs, it may be especially important to model recruitment to demonstrate trial feasibility. We present the mathematics of the homogeneous and non-homogeneous cases, along with simple, versatile algorithms for simulating each using open-source R code. We discuss the insights we garnered by modelling recruitment to the National Lung Matrix trial, a multi-arm, multi-drug, phase II adaptive design stratified by random genetic changes in non-small cell lung cancer patients.

The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions

In a previous work, we have derived the general solution of the state space linear fractional system of commensurate order for real simple and multiple eigenvalues of the state space matrix. The obtained solutions of the homogeneous and non-homogeneous cases have been expressed as a linear combination of introduced fundamental functions. In this paper, the above work has been extended to solve the state space linear fractional system of commensurate order for complex eigenvalues of the state space matrix. First, suitable fundamental functions corresponding to the different types of complex eigenvalues of the state space matrix are introduced. Then, the derived formulations of the resolution approach are presented for the homogeneous and the non-homogeneous cases. The solutions are expressed in terms of a linear combination of the proposed fundamental functions which are in the form of exponentials, sine, cosine, damped sine and damped cosine functions depending on the commensurate fractional order. The results are validated by solving an illustrative example to demonstrate the effectiveness of the proposed analytical tool for the solution of the state space linear fractional system of commensurate order.

Urban inundation simulation considering road network and building configurations

AbstractThe main difference between rural and urban area inundation is the existence of an impermeable layer and the complexity of the road and building configuration. Therefore, although surface overland flow can be appropriately simulated by the two‐dimensional diffusive model in a rural area, there also is a need for a more accurate and elaborate simulation model. Dual drainage concept has been popular to simulate urban drainage system. Although the concept has immense advantages, through which overland flow and pipe flow can be simulated simultaneously, many researchers set manholes as discharge exchange spots between ground surface and sewerage system, which causes lack of reality. Therefore, in this study, we developed a new urban inundation model that can reflect real field situation. Also, two kinds of simulations are carried out with homogeneous meshes and non‐homogeneous ones using manholes and storm drains as discharge exchange spots, respectively, and those results are compared to estimate the effect of road networks and building groups according to different discharge exchange spots. Two‐dimensional run‐off flow model and one‐dimensional slot‐model are used to simulate ground surface run‐off flow and sewer pipe flow. To connect both models, we used a newly suggested bi‐directional model and its coefficients. The simulated results showed that more reasonable results are obtained in the nonhomogeneous case; the results further reveal the importance of using storm drains as exchange spots in the model.

Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the “elastic” operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution. What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term. We also provide counterexamples in order to show the optimality of our results.

Fast and asymptotic computation of the fixation probability for Moran processes on graphs

Evolutionary dynamics has been classically studied for homogeneous populations, but now there is a growing interest in the non-homogeneous case. One of the most important models has been proposed in Lieberman et al. (2005), adapting to a weighted directed graph the process described in Moran (1958). The Markov chain associated with the graph can be modified by erasing all non-trivial loops in its state space, obtaining the so-called Embedded Markov chain (EMC). The fixation probability remains unchanged, but the expected time to absorption (fixation or extinction) is reduced. In this paper, we shall use this idea to compute asymptotically the average fixation probability for complete bipartite graphs Kn,m. To this end, we firstly review some recent results on evolutionary dynamics on graphs trying to clarify some points. We also revisit the ‘Star Theorem’ proved in Lieberman et al. (2005) for the star graphs K1,m. Theoretically, EMC techniques allow fast computation of the fixation probability, but in practice this is not always true. Thus, in the last part of the paper, we compare this algorithm with the standard Monte Carlo method for some kind of complex networks.

On the existence of stationary solutions for higher-order p-Kirchhoff problems

In this paper, we establish the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains. The p-polyharmonic operators [Formula: see text] were recently introduced in [F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal.74 (2011) 5962–5974] for all orders L and independently, in the same volume of the journal, in [V. F. Lubyshev, Multiple solutions of an even-order nonlinear problem with convex-concave nonlinearity, Nonlinear Anal.74 (2011) 1345–1354] only for L even. In Sec. 3, the results are then extended to non-degeneratep(x)-polyharmonic Kirchhoff operators. The main tool of the paper is a three critical points theorem given in [F. Colasuonno, P. Pucci and Cs. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal.75 (2012) 4496–4512]. Several useful properties of the underlying functional solution space [Formula: see text], endowed with the natural norm arising from the variational structure of the problem, are also proved both in the homogeneous case p ≡ Const. and in the non-homogeneous case p = p(x). In the latter some sufficient conditions on the variable exponent p are given to prove the positivity of the infimum λ1of the Rayleigh quotient for the p(x)-polyharmonic operator [Formula: see text].

On some properties of polynomials measurable with respect to a Gaussian measure

As it is known (see [1]), for every measurable poly� nomial f with respect to a Gaussian measure γ there exists a version with the following property: for almost each x, the function (x + h) is a continuous polynomial in h on the Cameron–Martin space of the measure γ. It is shown in this paper that there exists a version with this property which is a homogeneous algebraic polynomial, moreover, for a homogeneous polynomial, any version that is an algebraic and homogeneous will be suitable. At the same time, in the nonhomogeneous case, not every such version will be suitable, which will be seen in the examples below. We recall some concepts used below. Let X be a locally convex space with the topological dual space X* and the Borel σ�field � (X). A Borel probability measure γ on X is called Radon (see [1, 2]) if for every B and every e > 0 there exists a compact set K ⊂ B such that γ(B\K) < e. We also assume throughout that the mea� sure γ is a centered Gaussian measure (see [1, 3, 4]), i.e., for every functional f ∈ X*, its image γ ° f –1 is a centered Gaussian measure on the real line, in other words, f is a centered Gaussian random variable on (X, � (X), γ). We use the symbol H(γ) to denote the Cameron–Martin space of the measure γ, which con� sists of all vectors with the finite Cameron–Martin norm

On Volterra-type integral equations in noncompact metric space

In the present work, we consider one of the possible generalizations of linear and nonlinear Volterra integral equations of the second kind in the case when the independent variable belongs to an arbitrary noncompact metric space. Sufficient conditions are obtained for the existence of solutions of Volterra-type integral equations in the nonhomogeneous case. Some applications of the obtained results to the integral inequalities are given. MSC:35Q99, 35R35, 65M12, 65M70.

Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns

The generalized Pólya urn has been extensively studied and is widely applied in many disciplines. An important application of urn models is in the development of randomized treatment allocation schemes in clinical studies. The randomly reinforced urn was recently proposed, but, although the model has some intuitively desirable properties, it lacks theoretical justification. In this paper we obtain important asymptotic properties for multicolor reinforced urn models. We derive results for the rate of convergence of the number of patients assigned to each treatment under a set of minimum required conditions and provide the distributions of the limits. Furthermore, we study the asymptotic behavior for the nonhomogeneous case.

Asymptotic Properties of Multicolor Randomly Reinforced Pólya Urns

The generalized Pólya urn has been extensively studied and is widely applied in many disciplines. An important application of urn models is in the development of randomized treatment allocation schemes in clinical studies. The randomly reinforced urn was recently proposed, but, although the model has some intuitively desirable properties, it lacks theoretical justification. In this paper we obtain important asymptotic properties for multicolor reinforced urn models. We derive results for the rate of convergence of the number of patients assigned to each treatment under a set of minimum required conditions and provide the distributions of the limits. Furthermore, we study the asymptotic behavior for the nonhomogeneous case.

Existence results for second-order monotone differential inclusions on the positive half-line

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t)+q(t)u′(t)∈Au(t)+f(t) a.e. in (0,∞), with the condition (B): u(0)=x∈D(A)¯, where A:D(A)⊂H→H is a (possibly set-valued) maximal monotone operator whose range contains 0; p,q∈L∞(0,∞), with essinfp>0 and q+∈L1(0,∞). More than four decades ago, V. Barbu established the existence of a unique bounded solution to (E), (B), in the particular case p≡1, q≡0 and f≡0. Subsequently the existence and uniqueness of bounded solutions in the homogeneous case (f≡0) have been further investigated by H. Brezis (1972), N. Pavel (1976), L. Véron (1974–1976), and by E.I. Poffald and S. Reich (1986) when A is an m-accretive operator in a Banach space. The non-homogeneous case has received less attention from this point of view. In this paper, we prove existence and uniqueness of bounded solutions to (E), (B) in the general case of non-constant functions p, q satisfying the mild conditions above, thus compensating for the lack of existence theory for such kind of second order problems. Note that our results open up the possibility to apply Lions' method of artificial viscosity towards approximating the solutions of some nonlinear parabolic and hyperbolic problems, as shown in the last section of the paper.

Nonhomogeneous atherosclerotic plaque analysis via enhanced 1D structural models

The static analysis of structures with arbitrary cross-section geometry and material lamination via a refined one-dimensional (1D) approach is presented in this paper. Higher-order 1D models with a variable order of expansion for the displacement field are developed on the basis of Carrera Unified Formulation (CUF). Classical Euler-Bernoulli and Timoshenko beam theories are obtained as particular cases of the first-order model. Numerical results of displacement, strain and stress are provided by using the finite element method (FEM) along the longitudinal direction for different configurations in excellent agreement with three-dimensional (3D) finite element solutions. In particular, a layered thin-walled cylinder is considered as first assessment with a laminated conventional cross-section. An atherosclerotic plaque is introduced as a typical structure with arbitrary cross-section geometry and studied for both the homogeneous and nonhomogeneous material cases through the 1D variable kinematic models. The analyses highlight limitations of classical beam theories and the importance of higher-order terms in accurately detecting in-plane cross-section deformation without introducing additional numerical problems. Comparisons with 3D finite element solutions prove that 1D CUF provides remarkable three-dimensional accuracy in the analysis of even short and nonhomogeneous structures with arbitrary geometry through a significant reduction in computational cost.

Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight

In this paper we continue our research from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) on continuous dependence on a parameter k of solutions to linear integral equations of the form x(t)= x ˜ k + ∫ a t d[ A k ]x+ f k (t)− f k (a), t∈[a,b], k∈N, where −∞<a<b<∞, X is a Banach space, L(X) is the Banach space of linear bounded operators on X, x ˜ k ∈X, A k :[a,b]→L(X) have bounded variations on [a,b], f k :[a,b]→X are regulated on [a,b]. The integrals are understood as the abstract Kurzweil-Stieltjes integral and the studied equations are usually called generalized linear differential equations (in the sense of Kurzweil, cf. (Kurzweil in Czechoslov. Math. J. 7(82):418-449, 1957) or (Kurzweil in Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions, 2012)). In particular, we are interested in the situation when the variations var a b A k need not be uniformly bounded. Our main goal here is the extension of Theorem 4.2 from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) to the nonhomogeneous case. Applications to second-order systems and to dynamic equations on time scales are included as well.MSC:45A05, 34A30, 34N05.

Dynamics of a nonlinear mathematical model for three interacting populations

In this paper, we derive and analyze both analytically and numerically a mathematical model for three interacting populations. These take the form of a herbivore, a plant, and a pollinator. The full model is a nonlinear reaction–diffusion–advection system, which is derived on the basis of a series of plausible and widely supported ecological hypothesis. The study we present here deals with the conditions for the coexistence of the three interacting species. The analysis is carried out in two stages. For the homogeneous case, the mathematical model reduces to a nonlinear three-dimensional autonomous ODE system, which, as the ecological parameters change, exhibits different dynamical behaviors, namely a limit cycle and a positive attractor. The non-homogeneous case is studied mainly by means of numerical simulations of the full model defined on a rectangular region and considering appropriate initial and boundary conditions. Our results strongly suggest the stabilizing role played by the herbivore population which, in turn means that the introduction of this population into the mutualistic pollinator–plant interaction, favors the coexistence of the three interacting species.

A Semi-Analytical Method for Solutions of a Certain Class of Second Order Ordinary Differential Equations

This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous and nonhomogeneous cases. For the robustness and efficiency of the method, four examples are considered. The results indicate that the DTM is reliable and accurate when compared to the exact solutions of the solved problems.

Optimization of Force Between Cylindrical Permanent Magnets

We calculated analytically and with the finite-element method the force between two identical coaxial cylindrical magnets for nine different magnetic materials at separations of 1, 3, 5, 7, and 9 mm and for height-to-diameter ratios between 0.1 and 2. For the analytical calculations, we assumed homogeneous magnetization. For the nonhomogeneous case, we used the finite-element method taking into consideration the magnetic flux density vs. field (B-H) curves of the materials. The results show that, similarly for cuboid magnets, the maximum force for a given volume of permanent-magnet material was achieved for a height-to-diameter ratio around 0.4 for homogeneously magnetized (ideal) permanent magnets. We also show that this is not always true for the nonhomogeneous cases. The resulting forces are dependent on the B-H curve of the magnets. As the B-H curves of the materials deviate from their ideal counterparts, the aspect ratio that yields maximum force increases.

Periodicity in the [formula omitted]-norm for partial functional differential equations in fading memory spaces

The aim of this work is to study the existence of periodic solutions in the α-norm for some partial functional differential equations of neutral type in fading memory spaces. We assume that a linear part is densely defined and is the generator of an analytic semigroup. The delayed part is assumed to be periodic with respect to the first argument. In the nonhomogeneous linear case, we show that the existence of a bounded solution in R+ implies the existence of the periodic solution. In the nonlinear case, we use two approaches, the first one is based on the ultimate boundedness of solutions and the second one is based on the multivalued fixed point theorem.

Theory of Linear Hahn Difference Equations

Hahn introduced the dierence operator Dq;!f(t) = 􀀀 f(qt + !) 􀀀 f(t)=􀀀 t(q 􀀀 1) + ! in 1949, where 0 0 are fixed real numbers. This operator extends the classical dierence operator M! f(t) = (f(t + !) 􀀀 f(t))=! as well as Jackson q􀀀 dierence operator Dqf(t) = (f(qt)􀀀f(t))=(t(q 􀀀1)). In this paper, our target is to give a rigorous study of the theory of linear Hahn dierence equations of the form a0(t)Dn q;!x(t) + a1(t)Dn􀀀1 q;! x(t) + ::: + an(t)x(t) = 0: We introduce its fundamental set of solutions when the coecients are constant and the Wronskian associated with Dq;!. Hence, we obtain the corresponding Liouville's formula. Also, we derive solutions of the first and second order linear Hahn dierence equations with non-constant coffiecients. Finally, we present the analogues of the variation of parameter technique and the annihilator method for the non-homogeneous case.

Boundary element approaches for filling simulations of anisotropic reinforced preforms used in the resin transfer molding process

The boundary element method (BEM) is implemented in the simulation of the filling of anisotropic fiber reinforced preforms used in the resin transfer molding process (RTM). They are analyzed both as a homogeneous and a non-homogeneous domain. In the former case, two injection regimes are considered: constant pressure and constant flow. The BEM results of flow front positions, pressure history and pressure profiles in the two main directions of permeability are compared to analytical and experimental results. In the second case of non-homogeneous domain, the domain integral appearing in the boundary integral formulation is treated by the dual reciprocity (DR) technique; the DR-BEM results of the curve pressure versus radial distance Pressure vs. Radial distance in the main directions of permeability are compared to numerical results obtained from the solution of the equation of pressure using MacLaurin series. In this case, the influence of the linear radial change of the fiber volume fraction on the flow front positions and on the curves of pressure versus radial distance is analyzed Pressure vs. Radial distance. For the homogeneous case, an acceptable coincidence between BEM and analytical results is appreciated as the circular inlet effect, which is considered in the BEM formulation and is not considered in the analytical method, lessens with the time. In non-homogeneous case, the coincidence among the DR-BEM and the numerical MacLaurin series results indicates that the DR-BEM is a promising technique to deal with the infiltration phenomenon of anisotropic reinforced preforms in non-homogeneous domains.