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This paper develops a Failure Mode, Effects and Criticality Analysis (FMECA) for a heating, ventilation and air conditioning (HVAC) system in railway. HVAC is a safety critical system which must ensure emergency ventilation in case of fire and in case of loss of primary ventilation functions. A study of the HVAC’s critical areas is mandatory to optimize its reliability and availability and consequently to guarantee a low operation and maintenance cost. The first part of the paper describes the FMECA which is performed and reported to highlight the main criticalities of the HVAC system under analysis. Secondly, the paper deals with the problem of the evaluation of a threshold risk value, which can distinguish negligible and critical failure modes. Literature barely considers the problem of an objective risk threshold estimation. Therefore, a new analytical method based on finite difference is introduced to find a univocal risk threshold value. The method is then tested on two Risk Priority Number datasets related to the same HVAC. The threshold obtained in both cases is a good tradeoff between the risk mitigation and the cost investment for the corrective actions required to mitigate the risk level. Finally, the threshold obtained with the proposed method is compared with the methods available in literature. The comparison shows that the proposed finite difference method is a well-structured technique, with a low computational cost. Furthermore, the proposed approach provides results in line with the literature, but it completely deletes the problem of subjectivity.

This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.

We investigate the existence of positive radial solutions for a nonlinear elliptic equation with p -Laplace operator and sign-changing weight, both in superlinear and sublinear case. We prove the existence of solutions u , which are globally defined and positive outside a ball of radius R , satisfy fixed initial conditions u ( R ) = c > 0 , u ′ ( R ) = 0 and tend to zero at infinity. Our method is based on a fixed point result for boundary value problems on noncompact intervals and on asymptotic properties of suitable auxiliary half-linear differential equations. The results are new also for the classical Laplace operator and may be used for proving the existence of ground state solutions and decaying solutions with exactly k -zeros which are defined in the whole space. Some examples illustrate our results.

A boundary value problem associated to the difference equation with advanced argument \begin{equation} \label{*}\Delta\bigl (a_{n}\Phi(\Delta x_{n})\bigr)+b_{n}\Phi(x_{n+p} )=0,\ \ n\geq1 \tag{$*$} \end{equation} is presented, where $\Phi(u)=|u|^{\alpha}$sgn $u,$ $\alpha>0,p$ is a positive integer and the sequences $a,b,$ are positive. We deal with a particular type of decaying solutions of (\ref{*}), that is the so-called intermediate solutions (see below for the definition) . In particular, we prove the existence of these type of solutions for (\ref{*}) by reducing it to a suitable boundary value problem associated to a difference equation without deviating argument. Our approach is based on a fixed point result for difference equations, which originates from existing ones stated in the continuous case. Some examples and suggestions for future researches complete the paper., Comment: accepted for publication on Philosophical Transactions of the Royal Society A. Issue: Topological degree and fixed point theories in differential and difference equations Editors: Maria Patrizia Pera and Marco Spadini

We study the existence of positive solutions on the half-line of a second order ordinary differential equation subject to functional boundary conditions. Our approach relies on a combination between the fixed point index for operators on compact intervals, a fixed point result for operators on noncompact sets, and some comparison results for principal and nonprincipal solutions of suitable auxiliary linear equations., 13 pages

The paper deals with the nonlinear differential equation \begin{document}$\bigl(a(t)\Phi(x^{\prime})\bigr)^{\prime}+b(t)F(x)=0,\ \ \ t\in\lbrack1,\infty),$ \end{document} in the case when the weight \begin{document}$b$\end{document} has indefinite sign. In particular, the problem of the existence of the so-called globally positive Kneser solutions, that is solutions \begin{document}$x$\end{document} such that \begin{document}$x(t)>0, {{x}'}(t) on the whole closed interval \begin{document}$[1,\infty )$\end{document} , is considered. Moreover, conditions assuring that these solutions tend to zero as \begin{document}$t\rightarrow\infty$\end{document} are investigated by a Schauder's half-linearization device jointly with some properties of the principal solution of an associated half-linear differential equation. The results cover also the case in which the weight \begin{document}$b$\end{document} is a periodic function or it is unbounded from below.

A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is examined by proving new Sturm comparison theorems for linear difference equations and using a fixed point approach based on a linearization device. %The process from the continuous problem to discrete one is examined, too. The process of discretization of the boundary value problem on the unbounded domain is examined, and some discrepancies between the discrete and the continuous case are pointed out, too., Comment: final version accepted for publication on Electronic Journal of Qualitative Theory of Differential Equations

We establish conditions guaranteeing that all eventually positive increasing solutions of a half-linear delay differential equation are regularly varying and derive precise asymptotic formulae for them. The results here presented are new also in the linear case and some of the observations are original also for non-functional equations. A substantial difference between the delayed and non-delayed case for eventually positive decreasing solutions is pointed out., Comment: submitted version

Some nonlocal boundary value problems, associated to a class of functional difference equations on unbounded domains, are considered by means of a new approach. Their solvability is obtained by using properties of the recessive solution to suitable half-linear difference equations, a half-linearization technique and a fixed point theorem in Frechet spaces. The result is applied to derive the existence of nonoscillatory solutions with initial and final data. Examples and open problems complete the paper.

We study the existence of positive solutions on the half-line $[0,\infty)$ for the nonlinear second order differential equation \[ \bigl(a(t)x^{\prime}\bigr)^{\prime}+b(t)F(x)=0,\quad t\geq0, \] satisfying Dirichlet type conditions, say $x(0)=0$, $\lim_{t\rightarrow\infty}x(t)=0$. The function $b$ is allowed to change sign and the nonlinearity $F$ is assumed to be asymptotically linear in a neighborhood of zero and infinity. Our results cover also the cases in which $b$ is a periodic function for large $t$ or it is unbounded from below., Comment: 21 pages, 2015 (submitted)

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n-th order nonlinear differential equations, equations with a generalized Laplacian, and nonlinear partial differential systems.

Under the assumption that the coefficients are regularly varying functions, existence and asymptotic form of strongly decreasing solutions are here studied for a system of two coupled nonlinear second-order equations of Emden–Fowler type, satisfying a subhomogeneity condition. Several examples of application of the main result and a comparison with existing literature complete the paper.

The paper deals with a degenerate reaction-diffusion equation, including aggregative movements and convective terms. The model also incorporates a real parameter causing the change from a purely diffusive to a diffusive-aggregative and to a purely aggregative regime. Existence and qualitative properties of traveling wave solutions are investigated, and estimates of their threshold speeds are furnished. Further, the continuous dependence of the threshold wave speed and of the wave profiles on a real parameter is studied, both when the process maintains its diffusion-aggregation nature and when it switches from it to another regime.

Necessary and sufficient conditions for regular or slow variation of all positive solutions of a second-order linear difference equation with arbitrary sign coefficient are established. Relations with the so-called -classification are also analyzed and a generalization of the results to the half-linear case completes the paper.

In this paper a necessary and sufficient condition is derived for all positive decreasing solutions of a half-linear second order difference equation to be rapidly varying of index -~. Relations with the standard classification of nonoscillatory solutions and with the notion of recessive solutions are also discussed. The results of this paper are complementary to those of a previous paper by the authors, and lead to a complete characterization of positive decreasing solutions with respect to their regularly or rapidly varying behavior.

Ultrawideband radar is commonly used in the frequency range of 50-500 MHz to detect buried pipes at a depth of about 1-2 m depending on the soil characteristics. The typical feature used to locate the pipes is the hyperbolic pattern of the time of flight generated by a linear scan of the antenna above the surface. When the pipes are close together, the hyperbolas overlap, and a straightforward least squares fit is not possible. The Hough transform provides one possible solution. This paper extends the Hough transform by introducing a weighting factor depending on the differentials of the unknown parameters with respect to the experimental errors, namely, the probe position error and the time-of-flight error. This enables optimally placed sets of data pairs to be given greater weight than "ill-conditioned" sets, as for example when all data pairs lie near one end of the arc. The result is a decrease in the background amplitude with respect to the maximum of the peaks in the Hough accumulator space. It is shown that this improvement persists even when many arcs are present. A mathematical analysis with analytical results is given for the case of four unknowns: pipe radius R, pipe center position (Y, Z), and soil propagation velocity V. The results are presented through simulations introducing controlled uncertainties in the probe position, the time of flight, and its bin size. The simulations demonstrate the correlations that occur between the radius, depth, and velocity for given experimental uncertainties.

Some basic properties of regularly varying sequences are presented here and applied to the study of the asymptotic behavior of decreasing solutions of half-linear difference equations. Relations with the classification of nonoscillatory solutions and with the notion of recessive solutions are also discussed.

A complete characterization for nonoscillatory solutions of a coupled nonlinear difference system is presented. Nonoscillatory solutions are classified into several classes according to their limit behavior, and necessary and sufficient conditions for the existence in each of these classes are established. The multiplicity of solutions is also examined and explicit asymptotic formulas are derived. Possible generalizations and comparisons of our results with existing ones are discussed as well.

The generalised Hough transform method is applied to the measurement of the diameters of buried cylindrical pipes by Ground Penetration Radar (GPR). 600 MHz radar scans across long pipes, buried in one metre or so of soil, show complex reflection patterns consisting of a series of inverted hyperbolic arcs. The time of flight t(y) as the probe is scanned along an axis, y, perpendicular to the pipe, shows an arc whose shape depends on four unknown variables: y 0 , the position of the centre of the pipe along the scan, z 0 , the depth of the pipe centre, R, its radius and V the velocity in the medium. Analytic expressions for the solution of these variables have been obtained. They use sets of between 1 and 4 times t, at corresponding positions y i , along the arc, depending on the number of variables to be determined. In the generalised Hough method many such sets of times are chosen randomly from points on the arcs. The results are presented for example as peaks in an accumulator space for each variable. The method is demonstrated for a 0.18 m radius concrete pipe buried at a nominal I m depth in a road. Using data acquired at 600 MHz. frequency (around 0.16 m wavelength in soil) the estimated radius was 0.174 ± 0.059 m.

The problem of oscillation of a second order nonlinear difference system is considered and conditions for oscillation are derived by means of a complete analysis of nonoscillatory solutions. The case of systems with deviating arguments is also analyzed, and the results are shown to significantly generalize some existing ones.

The theory of semigroups of bounded linear operators in L1-like Banach spaces is employed to show the existence and uniqueness of a triple of non-negative functions representing the solution of a model of evolution of a population distributed in three stages of individuals, each of them being dependent on his own age and one stage also on mother age. The differential equations involved are linear and are coupled through linear boundary conditions. The detailed study of the spectrum of the linear, closed operator related to the system allows to obtain estimates for the asymptotic time behaviour of the solution; these results may be considered as a generalization of the Sharpe-Lotka theorem. Finally, the analytical structure of the solution is given.

We investigate two boundary value problems for the second order differential equation with $p$-Laplacian \[ (a(t)\Phi _{p}(x'))'=b(t)F(x), \quad t\in I=[0,\infty ), \] where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: \[ {\rm i)}\ x(0)=c>0, \ \lim _{t\rightarrow \infty }x(t)=0; \quad {\rm ii)}\ x'(0)=d

Continuous dependence of the threshold wave speed and of the travelling wave profiles for reaction-diffusion-convection equations $ u_t + h(u)u_x = (d(u)u_x)_x + f(u)$ is here studied with respect to the diffusion, reaction and convection terms.

Front propagation for the aggregation-diffusion-reaction equation v τ = [D(v)vx]x + f(v) is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation. © World Scientific Publishing Company.

A general method for solving boundary value problems associated to functional difference systems on the discrete half-line is presented and applied in studying the existence of positive unbounded solutions for a system of two coupled nonlinear difference equations. A further example, illustrating the method, completes the paper.

Je dokazan novy vysledek tykajici se oscilace nelinearniho diferencniho systemu dvou rovnic druheho radu, ktery kompletuje predchozi vysledky autoru. Dostava se takto kompletni charakterizace oscilace pro tuto tridu systemu.

The paper deals with the existence and properties of front propagation between the stationary states $0$ and $1$ of the reaction-diffusion-advection equation $$ v_\tau + h(v)v_x = \big( D(v) v_x \big)_x + G(v), $$ where $G$ is a bistable reaction term and $D$ is a strictly positive diffusive process. We show that the additional transport term $h$ can cause the disappearance of such wavefronts and prove that their existence depends both on the local behavior of $G$ and $h$ near the unstable equilibrium and on a suitable sign condition on $h$ in $[0,1]$. We also provide an estimate of the wave speed, which can be negative even if $\int_0^1 G(u) D(u) du>0$, unlike what happens to the mere reaction-diffusion dynamic occurring when $h\equiv 0$.

The optimal equilibrium harvesting policy is investigated for an age-dependent population of females whose life history consists of two stages termed eggs (or juveniles) and adults. Using a continuous-time linear model, we consider admissible harvesting policies in which a certain fraction of individuals of fixed ages is harvested per unit time in both stages to bring the population to an equilibrium level. Determination of the harvest rate that maximizes the sustainable yield, subject to a linear ecological or economic constraint, leads to a nonlinear, nonconvex optimization problem. The optimal policy is shown to consist of harvesting at most three ages. Thus, we say that the harvest is trimodal. In one stage, at most two ages are harvested, with the oldest being harvested completely; in the other, at most one age is harvested completely. In each stage, the age totally removed, if present, is older than the surplus age, which is the age at which the proportion of the expected number of eggs multiplied by the proportion of the expected number of adults first exceeds one. The three harvesting ages are dependent on the birth, maturation, and death rates and on the economic parameters of the problem. A simple algorithm to find the optimum harvesting strategy is described.

This paper analyses front propagation of the equation $u_\tau=[D(u)v_x]_x +f(v) \;\;\; \tau < 0, x \in \mathbb{R}$ where $f$ is a monostable (ie Fisher-type) nonlinear reaction term and $D(v)$ changes its sign once, from positive to negative values,in the interval $ v \in[0,1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $v$ at time $\tau$ and position $x$. The existence of infinitely many travelling wave solutions is proven. These fronts are parametrized by their wave speed and monotonically connect the stationary states u = 0 and v = 1. In the degenerate case, i.e. when D(0) and/or D(1) = 0, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.