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In this paper, we prove the existence of at least three weak solutions for a doubly eigenvalue elliptic systems involving the p-biharmonic equation with Hardy potential of Kirchhoff type with Navier boundary condition. More precisely, by using variational methods and three critical points theorem due to B. Ricceri, we establish multiplicity results on the existence of weak solutions for such problems where the re-action term is a nonlinearity function f which satisfies in the some convenient growth conditions. Indeed, using a consequence of the critical point theorem due to Ricceri, which in it the coercivity of the energy Euler functional was required and is important, we attempt the existence of multiplicity solutions for our problem under algebraic conditions on the nonlinear parts. We also give an explicit example to illustrate the obtained result. [ABSTRACT FROM AUTHOR]

We investigate the concavity region of solutions to the Poisson equation on planar-bounded domains, by characterizing the nodal set D of the Hessian determinant of the solutions. In the case of solutions to the homogeneous Dirichlet problem on convex polygons, we prove that D is a smooth curve that touches every side of the polygon in exactly one point. [ABSTRACT FROM AUTHOR]

We investigate solutions for nonlinear operator equations and obtain some abstract existence results by linking methods. Some well-known theorems about periodic solutions for second-order Hamiltonian systems by M. Schechter are special cases of these results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the finite-dimensional approximation of quasi-static evolutions of critical points of the phase-field model of brittle fracture. In a space discretized setting, we first discuss an alternating minimization scheme which, together with the usual time-discretization procedure, allows us to construct such finite-dimensional evolutions. Then, passing to the limit as the space discretization becomes finer and finer, we prove that any limit of a sequence of finite-dimensional evolutions is itself a quasi-static evolution of the phase-field model of fracture. Our proof shows for the first time the consistency of a numerical scheme for evolutions of fractures along critical points. [ABSTRACT FROM AUTHOR]

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In this paper, we study the class of simple systems on R induced by homeomorphisms having finitely many non-ordinary points. We char- acterize the family of homeomorphisms on R having finitely many non- ordinary points upto (order) conjugacy. For x, y ∊ R, we say x ~ y on a dynamical system (R; f) if x and y have same dynamical properties, which is an equivalence relation. Said precisely, x ~ y if there exists an increasing homeomorphism h : R → R such that h ○ f = f ○ h and h(x) = y. An element x ∊ R is ordinary in (R; f) if its equivalence class [x] is a neighbourhood of it. [ABSTRACT FROM AUTHOR]

In a recent paper, we deduced a new energy functional for pure traction problems in elasticity, as the variational limit of nonlinear elastic energy functional related to a material body subject to an equilibrated force field: a kind of Gamma limit with respect to the weak convergence of strains, when a suitable small parameter tends to zero. This functional exhibits a gap that makes it different from the classical linear elasticity functional. Nevertheless, a suitable compatibility condition on the force field ensures coincidence of related minima and minimizers. Here, we show some relevant properties of the new functional and prove stronger convergence of minimizing sequences for suitable choices of nonlinear elastic energies. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of an iterative method based on the Ermakov-Kalitkin class of iterative schemes for solving nonlinear equations. As it was proven in "A new family of iterative methods widening areas of convergence, Appl. Math. Comput.", this family has the property of getting good estimations of the solution when Newton's method fails. Moreover, the set of converging starting points for several non-polynomial test functions was plotted and they showed to be wider in the case of proposed methods than in Newton's case, for small values of the parameter. Now, we make a complex dynamical analysis of this parametric class in order to justify the stability properties of this family. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we propose a parameter‐free algorithm to calculate ε, a parameter of small quantity initially introduced into the nonlinear weights of weighted essentially nonoscillatory (WENO) scheme to avoid denominator becoming zero. The new algorithm, based on local smoothness indicators of fifth‐order weighted compact nonlinear scheme (WCNS), is designed in a manner to adaptively increase ε in smooth areas to reduce numerical dissipation and obtain high‐order accuracy, and decrease ε in discontinuous areas to increase numerical dissipation and suppress spurious numerical oscillations. We discuss the relation between critical points and discontinuities and illustrate that, when large gradient areas caused by high‐order critical points are not well resolved with sufficiently small grid spacing, numerical oscillations arise. The new algorithm treats high‐order critical points as discontinuities to suppress numerical oscillations. Canonical numerical tests are carried out, and computational results indicate that the new adaptive algorithm can help improve resolution of small scale flow structures, suppress numerical oscillations near discontinuities, and lessen susceptibility to flux functions and interpolation variables for fifth‐order WCNS. The new adaptive algorithm can be conveniently generalized to WENO/WCNS with different orders. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple solutions for the boundary value problem where Ω is a bounded domain with smooth boundary in are Carathéodory functions and is the Grushin operator. This result is a generalization of that of Rabinowitz and of Luyen and Tri. [ABSTRACT FROM AUTHOR]

In this paper we obtain results on existence of standing waves in Discrete Nonlinear Shrödinger equation (DNLS) with saturable nonlinearity on a two-dimensional lattice. We consider two types of solutions: with periodic amplitude and vanishing at infinity (localized solution). Sufficient conditions for the existence of such solutions are obtained with the aid of Nehari manifold and periodic approximations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish the existence of two non-zero solutions for a mixed boundary value problem with the Sturm–Liouville equation. The approach is based on a recent two critical point theorem. [ABSTRACT FROM AUTHOR]

Abstract In this work we study the existence of solutions to the critical Brezis–Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet–Neumann boundary conditions, i.e., { (− Δ) s u = λ u + u 2 s ⁎ − 1 , u > 0 in Ω , u = 0 on Σ D , ∂ u ∂ ν = 0 on Σ N , where Ω ⊂ R N is a regular bounded domain, 1 2 < s < 1 , 2 s ⁎ is the critical fractional Sobolev exponent, 0 ≤ λ ∈ R , ν is the outwards normal to ∂Ω, Σ D , Σ N are smooth (N − 1) -dimensional submanifolds of ∂Ω such that Σ D ∪ Σ N = ∂ Ω , Σ D ∩ Σ N = ∅ , and Σ D ∩ Σ ‾ N = Γ is a smooth (N − 2) -dimensional submanifold of ∂Ω. [ABSTRACT FROM AUTHOR]

Depression is a common mental illness and research has focused on late childhood and adolescence in an attempt to prevent or reduce later psychopathology and/or social impairments. It is important to establish and study population-averaged trajectories of depressive symptoms across adolescence as this could characterise specific changes in populations and help identify critical points to intervene with treatment. Multilevel growth-curve models were used to explore adolescent trajectories of depressive symptoms in 9301 individuals (57% female) from the Avon Longitudinal Study of Parents and Children, a UK based pregnancy cohort. Trajectories of depressive symptoms were constructed for males and females using the short mood and feelings questionnaire over 8 occasions, between 10 and 22 years old. Critical points of development such as age of peak velocity for depressive symptoms (the age at which depressive symptoms increase most rapidly) and the age of maximum depressive symptoms were also derived. The results suggested that from similar initial levels of depressive symptoms at age 11, females on average experienced steeper increases in depressive symptoms than males over their teenage and adolescent years until around the age of 20 when levels of depressive symptoms plateaued and started to decrease for both sexes. Females on average also had an earlier age of peak velocity of depressive symptoms that occurred at 13.5 years, compared to males who on average had an age of peak velocity at 16 years old. Evidence was less clear for a difference between the ages of maximum depressive symptoms which were on average 19.6 years for females and 20.4 for males. Identifying critical periods for different population subgroups may provide useful knowledge for treating and preventing depression and could be tailored to be time specific for certain groups. Possible explanations and recommendations are discussed. [ABSTRACT FROM AUTHOR]

We are concerned with the nonlinear elliptic equation with Dirichlet boundary condition: , u>0 in Ω, u=0 on where Ω is a smooth bounded domain in and is a -positive function in . Under some suitable conditions on and using dynamical method involving a partial description of the lack of compactness of the associated variational problem, we prove an existence result. [ABSTRACT FROM AUTHOR]

Abstract The purpose of the article is to provide sufficient conditions for having a positive weak solution for a fourth-order semilinear elliptic equation in a compact Riemannian manifold, also we furnish the Palais–Smale property for a suitable functional. [ABSTRACT FROM AUTHOR]

Optical and crystalline structure properties of Ga2Se3 crystals were analyzed utilizing ellipsometry and x-ray diffraction (XRD) experiments, respectively. Components of the complex dielectric function (ε = ε1 + iε2) and refractive index (N = n + ik) of Ga2Se3 crystals were spectrally plotted from ellipsometric measurements conducted from 1.2 eV to 6.2 eV at 300 K. From the analyses of second-energy derivatives of ε1 and ε2, interband transition energies (critical points) were determined. Absorption coefficient-photon energy dependency allowed us to achieve a band gap energy of 2.02 eV. Wemple and DiDomenico single effective oscillator and Spitzer-Fan models were accomplished and various optical parameters of the crystal were reported in the present work. [ABSTRACT FROM AUTHOR]

The Leray-Lions operators are versatile enough to be particularized to various elliptic operators, so they receive a lot of attention. This paper introduces to the mathematical literature Leray-Lions type operators that are appropriate for the study of the variable exponent problems of higher order. We establish some properties concerning these general operators and then we apply them to a fourth order problem with variable exponents. [ABSTRACT FROM AUTHOR]

One of the current tasks of disaster management is to effectively counter toxic accidents on traffic communications. The paper demonstrates the procedure of the use of geographic data and knowledge with GIS technology for the operational mitigation of accident impacts on the traffic communication with leakage of toxic substance. A simulated leakage of toxic liquid substance on a highway in the Czech Republic was chosen as an example. The process is divided into two units. In the first preparatory block, data on soils and the geological environment are analysed and purpose oriented pre-processed. The data layer generally describes the expected movement of pollutants, e.g. predominant surface runoff, or predominant infiltration and/or a balanced combination of both of them. In the second operational unit, a location of the accident is precisely identified and the estimation of possible routes of pollutant runoff is performed with respect to the current status of the territory. Key points on these routes are identified with the aim to select mitigation measures and optimum access routes modelled for intervention techniques to reach key points in order to prevent contamination of water bodies. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear differential problem involving the p-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results. [ABSTRACT FROM AUTHOR]

Second-order linear Hamiltonian systems at resonance with periodic nonlinearity is investigated. An existence result of solutions for such systems is obtained by means of variational methods, saddle point theorem, and an index theory for second-order linear Hamiltonian systems. Meanwhile, two examples and two extensions are presented. [ABSTRACT FROM AUTHOR]

Phase equilibria in the binary system ZnSO4–H2O at temperatures of 240–450°C and pressures to 34 MPa were studied by various methods (visual observation of phase transformations in water–salt systems of various compositions (30–60 wt%) in thick-walled quartz ampules (3 mm i.d.), taking samples of solutions and their chemical analysis, and autoclave measurements of P–V–T parameters in phase transitions). It was shown that the decrease in the ZnSO4 solubility with increasing temperature, which is observed to 350–370°C, gives way to its abrupt increase with further heating; i.e., the negative temperature coefficient of solubility, which is characteristic of type 2 systems, changes to a positive one, as in type 1 systems. The ZnSO4–H2O system is of type 1 complicated by phase separation of saturated (and unsaturated) solutions, which leads to the sharp increase in the salt solubility above 375°C. [ABSTRACT FROM AUTHOR]

Let pn be a random, degree n polynomial whose roots are chosen independently according to the probability measure μ on the complex plane. For a deterministic point ξ lying outside the support of μ, we show that almost surely the polynomial qn(z) := pn(z)(z − ξ) has a critical point at distance O(1/n) from ξ. In other words, conditioning the random polynomials pn to have a root at ξ almost surely forces a critical point near ξ. More generally, we prove an analogous result for the critical points of qn(z) := pn(z)(z − ξ1) · · · (z−ξk), where ξ1, . . ., ξk are deterministic. In addition, when k = o(n), we show that the empirical distribution constructed from the critical points of qn converges to μ in probability as the degree tends to infinity, extending a recent result of Kabluchko. [ABSTRACT FROM AUTHOR]

Abstract The pressure stability field of the Mg-perovskite phase was investigated by characterizing the evolution of the electron arrangement in the crystal. Ab initio calculations of the perovskite structures in the range 0–185 GPa were performed at the HF/DFT (Hartree-Fock/Density Functional Theory) exchange–correlation terms level. The electron densities, calculated throughout the ab-initio wave functions, were analysed by means of the Bader's theory, coupled with Thom's catastrophe theory. To the best of our knowledge the approach is used for the first time. The topological results show the occurrence of two topological anomalies at P~20 GPa and P~110 GPa which delineate the pressure range where Mg-perovskite is stable. The paper accomplishes the twofold objectives of providing a contribution in shading light into the behaviour of the dominant component of the Earth's lower mantle across the D" layer and of proposing a novel approach in predicting the stability of a compound at extreme conditions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with 2p-order Hamiltonian systems with impulsive effects. We investigate the variational structure associated to this system. In addition, we obtain some results of multiple solutions for asymptotically linear 2p-order Hamiltonian systems via variational methods and critical point theorems. Meanwhile, some examples are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

Future singularities arising in a family of models for the expanding universe, characterized by sharing a convenient parametrization of the energy budget in terms of the deceleration parameter, are classified. Finite-time future singularities are known to appear in many cosmological scenarios, in particular, in the presence of viscosity or nongravitational interactions, the last being known to be able to suppress or just change in some cases the type of the cosmological singularity. Here, a family of models with a parametrization of the energy budget in terms of the deceleration parameter are studied in the light of Gaussian processes using reconstructed data from 4 0 -value H (z) datasets. Eventually, the form of the possible nongravitational interaction between dark energy and dark matter is constructed from these smoothed H (z) data. Using phase space analysis, it is shown that a noninteracting model with dark energy ω de = ω 0 + ω 1 q (q being the deceleration parameter) may evolve, after starting from a matter-dominated unstable state, into a de Sitter universe (the solution being in fact a stable node). Moreover, for a model with interaction term Q = 3 H b ρ dm (b is a parameter and H , the Hubble constant) three stable critical points are obtained, which may have important astrophysical implications. In addition, part of the paper is devoted to a general discussion of the finite-time future singularities obtained from direct numerical integration of the field equations, since they appear in many cosmological scenarios and could be useful for future extended studies of the models here introduced. Numerical solutions for the new models, produce finite-time future singularities of Type I or Type III, or an ω -singularity, provided general relativity describes the background dynamics. [ABSTRACT FROM AUTHOR]

The present study is aimed at solving the problem of in situ thermometry of lowtemperature processes of molecular beam epitaxy of indium antimonide. A spectral ellipsometric method for measuring the temperature of InSb epitaxial layers is proposed. The method is based on the temperature dependence of the energy positions of the critical points. The spectra of ellipsometric parameters of the material in the temperature range from 25 to 270 °C are measured. The analysis of these spectra shows that the most temperature-sensitive parameters are the spectral positions of the peaks of the ellipsometric parameter, which are manifested near the critical points E1 and E1 + Δ1. It is found that the dependences of the peak positions on temperature in the above-mentioned temperature range are linear functions with the slope factors of 0.21 and 0.10 nm/°C, respectively. These factors determine the sensitivity of the method and ensure the temperature measurement accuracy within 2–3 °C. [ABSTRACT FROM AUTHOR]

Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result. [ABSTRACT FROM AUTHOR]

In this paper, we study some type of equations which may model the behavior of species inhabiting in some habitat. For our purpose, using a priori bounded techniques, we obtain a positive solution to a family of non-local partial differential equations with non-homogeneous boundary conditions. [ABSTRACT FROM AUTHOR]

Visual exploration of flow fields is important for studying dynamic systems. We introduce semantic flow graph (SFG), a novel graph representation and interaction framework that enables users to explore the relationships among key objects (i.e., field lines, features, and spatiotemporal regions) of both steady and unsteady flow fields. The objects and their relationships are organized as a heterogeneous graph. We assign each object a set of attributes, based on which a semantic abstraction of the heterogeneous graph is generated. This semantic abstraction is SFG. We design a suite of operations to explore the underlying flow fields based on this graph representation and abstraction mechanism. Users can flexibly reconfigure SFG to examine the relationships among groups of objects at different abstraction levels. Three linked views are developed to display SFG, its node split criteria and history, and the objects in the spatial volume. For simplicity, we introduce SFG construction and exploration for steady flow fields with critical points being the only features. Then we demonstrate that SFG can be naturally extended to deal with unsteady flow fields and multiple types of features. We experiment with multiple data sets and conduct an expert evaluation to demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

Responses defined at critical points are particularly important for reactor safety analyses and licensing (e.g., the maximum fuel and/or clad temperature). The novel mathematical framework of the first-order comprehensive adjoint sensitivity analysis methodology for critical points (1st-CASAM-CP) is applied in this work to develop a reactor safety thermal-hydraulics benchmark model which admits exact closed-form expressions for the adjoint functions and for the first-order sensitivities of responses defined at critical points (maxima, minima, saddle points) in physical systems characterized by imprecisely known parameters, external and internal boundaries. This benchmark model is designed for verifying the capabilities and accuracies of computational tools for modeling numerically thermal-hydraulics systems. The unique and extensive capabilities of the 1st-CASAM-CP methodology are demonstrated in this work by considering two responses of paramount importance in reactor safety, namely, (i) the maximum rod surface temperature, which occurs at the imprecisely known interface between the subsystem that models the heat conduction inside the heated rod and the subsystem modeling the heat convection process surrounding the rod; and (ii) the maximum temperature inside the heated rod, which has a critical point with two components, one located at a precisely known boundary of the subsystem that models the heat conduction inside the heated rod, while the other component depends on an imprecisely known boundary (i.e., the rod length). The exact analytical expressions developed in this work for the sensitivities of the maximum internal rod temperature and maximum rod surface temperature, as well as for the sensitivities of the locations where these respective maxima occur, provide exact benchmarks for verifying the accuracy of thermal-hydraulics computational tools. The sensitivities of such responses and of their critical points with respect to model parameters enable the quantification of uncertainties induced by uncertainties stemming from the system’s parameters and boundaries in the respective responses and their underlying critical points.

This work presents the novel first-order comprehensive adjoint sensitivity analysis methodology for critical points (1st-CASAM-CP), which enables the exact and efficient computation of the first-order sensitivities of responses defined at critical points (maxima, minima, saddle points) of coupled nonlinear models of physical systems characterized by imprecisely known parameters underlying the models, boundaries, and interfaces between the coupled systems. Responses defined at critical points are important in many applications, including system optimization, safety analyses and licensing. For the design and licensing of nuclear reactors, such essentially important responses include the maximum temperatures of the fuel and cladding in hot channels. The 1st-CASAM-CP presented in this work makes it possible to determine, using a single large-scale “adjoint” computation, the first-order sensitivities of the magnitude of a response defined at a critical point of a function in the phase-space of the systems’ independent variables. In addition, the 1st-CASAM-CP enables the computation of the sensitivities of the location in phase-space of the critical point at which the respective response is located: one “adjoint” computation is required for each component of the respective critical point in the phase-space of independent variables. By enabling the exact and efficient computation of the sensitivities of responses and of their critical locations to imprecisely known model parameters, boundaries, and interfaces, the 1st-CASAM-CP significantly extends the practicality of analyzing crucially important responses for large-scale systems involving many uncertain parameters, interfaces, and boundaries.

In this paper, we prove a bifurcation phenomenon in a two-phase, singularly perturbed, free boundary problem of phase transition. We show that the uniqueness of the solution for the two-phase problem breaks down as the boundary data decreases through a threshold value. For boundary values below the threshold, there are at least three solutions, namely, the harmonic solution which is treated as a trivial solution in the absence of a free boundary, a nontrivial minimizer of the functional under consideration, and a third solution of the mountain-pass type. We classify these solutions according to the stability through evolution. The evolution with initial data near a stable solution, such as the trivial harmonic solution or a minimizer of the functional, converges to the stable solution. On the other hand, the evolution deviates away from a non-minimal solution of the free boundary problem. [ABSTRACT FROM AUTHOR]

We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a \(p(x)\)-biharmonic operator \[\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(p\in C(\overline{\Omega})\), \(\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)\) is the \(p(x)\)-biharmonic operator, and \(\lambda\gt 0\) is a parameter. We establish sufficient conditions under which there exists a positive number \(\lambda^{*}\) such that the above problem has at least two nontrivial weak solutions for each \(\lambda\gt\lambda^{*}\). Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces \(L^{p(x)}(\Omega)\) and \(W^{k,p(x)}(\Omega)\).

We present a qualitative analysis of chiral cosmological model (CCM) dynamics with two scalar fields in the spatially flat Friedman–Robertson–Walker Universe. The asymptotic behavior of chiral models is investigated based on the characteristics of the critical points of the selfinteraction potential and zeros of the metric components of the chiral space. The classification of critical points of CCMs is proposed. The role of zeros of the metric components of the chiral space in the asymptotic dynamics is analysed. It is shown that such zeros lead to new critical points of the corresponding dynamical systems. Examples of models with different types of zeros of metric components are represented.

Retrofitting of industrial heat recovery systems can contribute significantly to meeting energy efficiency targets for industrial plants. One issue to consider when screening retrofit design proposals is that industrial heat recovery systems must be able to handle variations, e.g., in inlet temperatures or heat capacity flow rates, in such a way that operational targets are reached. Consequently, there is a need for systematic retrofitting methodologies that are applicable to multi-period heat exchanger networks (HENs). In this study, a framework was developed to achieve flexible and cost-efficient retrofit measures of (industrial) HENs. The main idea is to split the retrofitting processes into several sub-steps. This splitting allows well-proven (single period) retrofit methodologies to be used to generate different design proposals, which are collected in a superstructure. By means of structural feasibility assessment, structurally infeasible design proposals can be discarded from further analysis, yielding a reduced superstructure. Additionally, critical point analysis is applied to identify those operating points within the uncertainty span that determine necessary overdesign of heat exchangers. In the final step, the most cost-efficient design proposal within the reduced superstructure is identified. The proposed framework was applied to a HEN retrofit case study to illustrate the proposed framework.

Some existence theorems are obtained for infinitely many periodic solutions of ordinary p-Laplacian systems by minimax methods in critical point theory.