Manifold Techniques Research Articles

Existence and multiplicity of positive solutions for Kirchhoff-Schrödinger-Poisson system with critical growth

This paper is concerned with the following Kirchhoff-Schrodinger-Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+ b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u + V(x)u+\phi u=\lambda g(x)u^{q-1}+h(x)u^{5}, &{} \ \ \text{ in } \mathbb {R}^{3},\\ -\Delta \phi = u^{2},~u>0,&{} \ \ \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$where $$a>0$$, $$b\ge 0$$, $$q\in [4,6)$$ and $$\lambda >0$$ is a parameter. Under some suitable conditions on V(x), g(x) and h(x), by using the Nehari manifold technique and the Ljusternik-Schnirelmann category theory, we relate the number of positive solutions with the topology of the global maximum set of h when $$\lambda $$ is small enough. Furthermore, with the aid of the Mountain Pass Theorem, we also obtain an existence result for $$\lambda $$ sufficiently large. Recent results from the literature are generally improved and extended.

Manifold-based isogeometric analysis basis functions with prescribed sharp features

We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed C0 continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential–geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in R3 is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in R2 so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to earlier constructions no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants in the partition of unity method. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish local polynomial approximants that are always C0 continuous across the tagged edges. The derived non-rational manifold-based basis functions correspond to the vertices of the mesh and may have an arbitrary number of creases and prescribed smoothness. This makes them particularly well suited for isogeometric analysis of Kirchhoff–Love thin shells with kinks, which require C1 continuous basis functions that are C0 continuous across the kinks. We demonstrate the convergence and utility of the new basis functions with linear and nonlinear beam, plate and shell examples.

Dynamical system analysis of Hessence scalar field in teleparallel gravity: Invariant manifold technique

This paper investigates the cosmological dynamics of the Hessence scalar field coupled with the dark matter in the background of the teleparallel gravity. We have assumed that the potential of the scalar field is exponential in nature whereas the [Formula: see text] appearing in the teleparallel theory has the form [Formula: see text]. The field equations of this system reduce to a nonlinear autonomous system and dynamical system analysis is then performed. Due to the nonlinearity and the existence of multiple zero eigenvalues, the traditional procedures of analysis break down. So some novel technique is required. One of the latest such techniques is the invariant manifold theory. By the application of this theory, one projects the variables linked with the zero eigenvalues onto the variables linked with the nonzero eigenvalues to compute the center manifolds and the reduced systems associated with the critical points. These reduced systems reflect the nature of the whole dynamical systems. They also have less dimension and are often simple in nature. Hence, it is possible to solve them directly. In this paper, we work exactly in this spirit and find the center manifolds and solve the corresponding reduced system for some of the critical points associated with the dynamical system. We discover some interesting results namely that there are certain bounds on the interaction term [Formula: see text] which asserts the stability of the systems. We also present various stability diagrams of the reduced systems. An asymptotic analysis is then done for the critical points at infinity. Finally, we discuss the cosmological interpretation of our results.

Existence and concentration of positive ground states for a 1-Laplacian problem in [formula omitted

This paper is concerned with the following quasilinear elliptic problem: −εΔ1u+V(x)u|u|=K(x)|u|p−1u, inRN,u∈BV(RN), where 0<p<1N−1, N≥2 and ε>0 is a small parameter. Under some mild conditions on the nonnegative functions V(x) and K(x), we establish the existence of a ground state solution uε for the above problem. Moreover, uε concentrates on the intersection set of global minimum points of V(x) and maximum points of K(x). The methods are based on the Nehari manifold technique and the Concentration-Compactness Principle of Lions.

Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

It is well known that a single nonlinear fractional Schrödinger equation with a potential V (x) may have a positive solution that is concentrated at the nondegenerate minimum point of V (x) when the positive parameter ε is sufficiently small [see G. Chen and Y. Zheng, Commun. Pure Appl. Anal. 13, 2359 (2014); J. Dávila et al., J. Differential Equations 256, 858 (2014); and M. M. Fall et al., Nonlinearity 28, 1937 (2015)]. In this work, we find multiple different positive solutions for a class of weakly coupled fractional Schrödinger systems with two potentials V1(x) and V2(x) that have some of the same minimum points; the positive solutions are concentrated at these minimum points. By using energy estimates, the Nehari manifold technique, and the Lusternik–Schnirelmann theory of critical points, we obtain some multiplicity results for a class of weakly coupled fractional Schrödinger systems. The existence and nonexistence of least energy positive solutions are also explored.

Dynamics Analysis in a Gierer–Meinhardt Reaction–Diffusion Model with Homogeneous Neumann Boundary Condition

A reaction–diffusion Gierer–Meinhardt system with homogeneous Neumann boundary condition on one-dimensional bounded spatial domain is considered in the present article. Local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are explored by analyzing in detail the associated eigenvalue problem. Moreover, properties of spatially homogeneous Hopf bifurcation are carried out by employing the normal form method and the center manifold technique for reaction–diffusion equations. Finally, numerical simulations are also provided in order to check the obtained theoretical conclusions.

Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary

Abstract We set out to obtain estimates of the Laplacian Spectrum of Riemannian manifolds with non-empty boundary. This was achieved using standard doubled manifold techniques. In simple terms, we pasted two copies of the same manifold along their common boundary thereby obtaining a Riemannian manifold with empty boundary and with a C 0 −metric. This made it possible to adapt some estimates of the spectrum dependent on the volume or genus of the manifold as calculated in recent years by several authors. In order to extend further estimates that depend on the curvature, it is necessary to regularize the metric of the doubled manifold so that the new metric is isometric to that of each copy and such that the curvature has a finite lower bound. Controlling the curvature in this way also makes estimates of topological invariants available.

The influence of air side and fuel side water addition on engine’s behaviour of a biofuel based compresion ingnition engine under oxygen enriched combustion

The effect of water injection at the air side and water addition at fuel side on engine?s performance of a Diesel engine was studied under oxygen enriched intake air using neat mahua oil as fuel. Initially experiments were carried out using neat mahua oil as fuel with different oxygen concentrations such as 21% (ambient), 22.4%, 23.8%, and 24.7% by volume at the air side. The optimal oxygen concentration was found based on the engine?s brake thermal efficiency. At the optimal oxygen concentration water injection was done on air side at 4% by mass and the experiments were repeated with neat mahua oil as fuel under oxygen enrichment mode. Finally, mahua oil emulsion was prepared using the same amount of water (i. e. 4%) and tested in the engine. A comparative study was made for the same amount of water (i. e. 4% as optimal) for water injection and neat mahua oil emulsion on engines behavior. Oxygen enrichment increased the brake thermal efficiency with all concentrations and reached the maximum value from 25.2% with ambient oxygen to a maximum of 30.6% at 23.8% of oxygen enrichment at the maximum brake mean effective pressure of 5.4 bar whereas it was 30.8% with neat diesel. The smoke, HC, and CO emissions were significantly reduced with oxygen enrichment. However, oxygen enrichment increased the NO emissions at all concentrations. Injection of water and emulsification techniques reduced the NO emissions considerably. Emulsification showed more reduction in NO emission than water injection for the same amount of water. It was concluded from the study that neat mahua oil could be effective used as fuel in compression ignition engines by combusting it under oxygen enriched condition. The optimal oxygen concentration of 23.8% could be recommended for the highest brake thermal efficiency. Injection of water at the intake manifold and emulsification techniques could solve the problem of higher NO emissions. The optimal amount of water that could be injected without affecting the engines power and brake thermal efficiency could be recommended as 4% by volume. Emulsification has the added advantage of further improvement in engine?s brake thermal efficiency.

A reduced-order model manifold technique for automated structural defects judging using the PGD with analytical validation

Automation is conquering new fields on a daily basis. Aiming for faster and more reliable products, industrials as well as researchers are oriented into automation. Non-destructive testing as well as defect quantification is not an exception. In fact, decisions with minimum allowable error are sought in real-time when facing any potential defect. In this work, we suggest a comprehensive method based on model order reduction techniques to judge if a structure shall be salvaged. The real-time decision is based on multidimensional parametric simulation, performed offline, using the Proper Generalized Decomposition (PGD). The PGD is a model order reduction technique that allows circumventing the curse of dimensionality by using domain decomposition. Therefore, the 6D simulation illustrated in this paper is performed within a few minutes on a standard laptop. Later on, a stress concentration manifold is built and used online for decision-making. The manifold is validated on a few selected solutions solved analytically using an analytical procedure. The aforementioned procedure is developed, in this paper, to calculate the tangential stress around circular holes of different sizes, in an infinite isotropic plate containing any number of holes and subjected to in-plane pressure loading at the tip of the infinite plate. The procedure is based on determining two Muskhelishvili complex potentials in terms of complex Fourier series, and applying the Schwartz alternating method repeatedly until the boundary conditions on the contour of every hole are satisfied.

Empirical Modeling of Zn/ZnO Nanoparticles Decorated/Conjugated with Fotolon (Chlorine e6) Based Photodynamic Therapy towards Liver Cancer Treatment

The current study is based on Zn/ZnO nanoparticles photodynamic therapy (PDT) mediated effects on healthy liver cells and cancerous cells. The synthesis of Zn/ZnO nanoparticles was accomplished using chemical and hydrothermal methods. The characterization of the synthesized nanoparticles was carried out using manifold techniques (e.g., transmission electron microscopy (TEM), X-ray diffraction (XRD), and energy dispersive X-ray spectroscopy (EDS)). In order to study the biotoxicity of the grown nanoparticles, they were applied individually and in conjunction with the third generation photosensitiser Fotolon (Chlorine e6) in the in vivo model of the normal liver of the Wister rat, and in the in vitro cancerous liver (HepG2) model both in the dark and under a variety of laser exposures (630 nm, Ultraviolet (UV) light). The localization of ZnO nanoparticles was observed by applying fluorescence spectroscopy on a 1 cm2 selected area of normal liver, whereas the in vitro cytotoxicity and reactive oxygen species (ROS) detection were carried out by calculating the loss in the cell viability of the hepatocellular model by applying a neutral red assay (NRA). Furthermore, a statistical analysis is carried out and it is ensured that the p value is less than 0.05. Thus, the current study has highlighted the potential for applying Zn/ZnO nanoparticles in photodynamic therapy that would lead to wider medical applications to improve the efficiency of cancer treatment and its biological aspect study.

Fractional Kirchhoff problem with critical indefinite nonlinearity

AbstractWe study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form urn:x-wiley:0025584X:media:mana201800044:mana201800044-math-0001where is a smooth bounded domain, and . Here M is the Kirchhoff coefficient and is the fractional critical Sobolev exponent. The parameter λ is positive and the is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results.

Halo orbits construction based on invariant manifold technique

The current investigation applied the invariant manifold technique to study the halo orbits around the libration points in circular restricted three-body problem. Two dominant (leading) directions were considered as dominant motions for the construction of spatial halo orbits, the third direction motion being the slave (following) motion. The dominant motions correspond to four-dimensional invariant manifolds in the phase space. The invariant nonlinear asymptotic relations (i.e. INARs) between the two dominant motions and the slave motion were established, enabling a transformation from the 3-DOF problem into a 2-DOF problem. Application of the INARs are also discussed. Such invariant nonlinear relations in polynomial expansion form could be used as: (I) approximate analytical solutions; (II) topological constraints to obtain more exact numerical solutions with differential correction. General findings in the current research revealed that the nonlinear asymptotic relations among the directions provided an alternative point of view to explore the overall dynamics of halo orbits around libration points with general rules. The effectiveness of the proposed method was also verified by numerical simulations.

The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms

In the present paper, we study the following singular Kirchhoff problem M∬R2N|u(x)−u(y)|2|x−y|N+2sdxdy(−Δ)su=λf(x)u−γ+g(x)u2s∗−1inΩ,u>0inΩ,u=0inRN∖Ω,where Ω⊂RN is an open bounded domain, dimension N>2s with s∈(0,1), 2s∗=2N∕(N−2s) is the fractional critical Sobolev exponent, parameter λ>0, exponent γ∈(0,1), M models a Kirchhoff coefficient, f∈L2s∗2s∗+γ−1(Ω) is a positive weight, while g∈L∞(Ω) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of λ. This approach allows us to avoid any restriction on the boundary of Ω.

Set-oriented design of interplanetary low-thrust trajectories using Earth Gravity Assist

A method by which to incorporate the Electric Delta-V Earth Gravity Assist (EDVEGA) scheme into a patched three-body approach to design an interplanetary transfer trajectory is presented in this paper. The EDVEGA scheme is a promising technique to reduce fuel consumption using Earth gravity assist and electric propulsion. In a patched three-body approach, the dimension of the problem that connects some trajectories is reduced by the use of special attainable sets. However, because of the singularity associated with Earth's center, it is difficult to connect an EDVEGA trajectory with a trajectory designed in the three-body problem. Therefore, this paper proposes a design method for an Earth–Mars transfer trajectory that combines the EDVEGA scheme with a patched three-body approach using Levi-Civita regularization and a special contour plot. Through comparison with conventional invariant manifold techniques, we demonstrate that fuel consumption is reduced by the proposed method.

Optimization of Fuel Consumption of a SI Engine Using Variable Valve Timing and Variable Length Intake Manifold Techniques

According to the world crisis about fuel consumption and environmental concerns regarding toxic emissions of internal combustion engines, the engines with higher efficiency and lower fuel consumption have been a topic of research in last decades. In this study, variable valve timing (VVT) and variable length intake manifold (VLIM) techniques are used to optimize the fuel consumption of an SI engine. At first, all components of engine are modeled in GT-POWER and a comparison with experimental results is performed to confirm the accuracy of the model. Then, the discrete-gird algorithm is employed to optimize the parameters in GT-POWER. The results obtained indicate that optimal valve timing and intake manifold length significantly reduces brake specific fuel consumption (BSFC).

A Robust Load Frequency Control Scheme for Power Systems Based on Second-Order Sliding Mode and Extended Disturbance Observer

This paper proposes a new robust load frequency control (LFC) scheme for multiarea power systems based on the second-order sliding mode control and an extended disturbance observer. First, a reduced-order model of the power system LFC is derived. In this model, the load variations and net exchange tie-line power deviations are combined as a lumped disturbance which can be estimated by the extended disturbance observer. Second, a novel sliding surface is designed with the new transformed state variables obtained from the estimated disturbance. The system dynamics can be indicated by sliding surface design using the eigenvalue assignment or the optimal sliding manifold technique. The sliding variable is driven to the sliding surface with a second-order sliding mode algorithm named supertwisting algorithm. The stability of the proposed LFC scheme and the extended disturbance observer is proved using Lyapunov method. The merits of the scheme include faster response speed, stronger robustness against disturbances arising from power system parameter errors, and unmodeled dynamics, and the full consideration of tie-line power flow scheduling variations. Finally, numerical simulations verify the effectiveness of the LFC scheme and reveal its advantages over the state of the arts.

Nonlinear stability in seismic waves

We analyse a passive system featuring a neutrally stable short wavelength mode. The system is modelled by the Nikolaevskiy equation relevant to elastic waves, reaction-diffusion systems and convection. After quickly falling onto a centre manifold the system then exhibits slow decay. Using the centre manifold technique, we deduce that the decay law is the inverse square root of time. The result is confirmed by direct computations of the system.

Planar periodic orbits’ construction around libration points with invariant manifold technique

This paper revisits the planar periodic motions around libration points in circular restricted three-body problem based on invariant manifold technique. The invariant manifold technique is applied to construct the nonlinear polynomial relations between ξ-direction and η-direction of a small celestial body during its periodic motion. Such direct nonlinear relations reduce the dimension of the dynamical system and facilitate convenient approximate analytical solutions. The nonlinear directional relations also provide terminal constraints for computing periodic motions. The method to construst periodic orbits proposed in this study presents a new point of view to explore the orbital dynamics. As an application in numerical simulations, nonlinear relations are adopted as topological terminal constraints to construct the periodic orbits with differential correction procedure. Numerical examples verify the validity of the proposed method for both collinear and triangular libration cases.

Gait variability and motor control in patients with knee osteoarthritis as measured by the uncontrolled manifold technique

Knee osteoarthritis (OA) causes pain, reduced muscular strength and stiffness of the affected joint. In response, the motor control mechanism is altered, potentially compromising stability during acts of daily living. Reduced walking stability can be quantified in terms of gait variability. This study therefore aimed to identify and quantify the effects of knee arthritis on gait variability. Fifty adults (25 males/25 females) with end-stage OA of the knee sufficiently symptomatic to require joint replacement, walked on a self-paced treadmill for 2min. A motion capture system was used to record 50 consecutive gait cycles from each patient. Kinematic variability of gait was analysed using the uncontrolled manifold technique (UCM). The position of the centre of mass (COM) was chosen as the task variable for the analysis. Results showed that our patient cohort were able to maintain a stable COM whilst walking, through adopting variable combinations of hip, knee and ankle kinematics. The greatest magnitudes of instability (based on the UCM ratios) occurred during initial contact and terminal stance. Active extension of the knee joint to approximately 5° is required during these gait cycle events, meaning that these gait events are highly quadriceps dependent. This study identified and quantified components of the gait cycle where patients with knee OA are most unstable. Employment of this technique could therefore allow specific personalised prescription for prehabilitation and rehabilitation.

Efficient Sparse Estimate of Sufficient Dimension Reduction in High Dimension

ABSTRACTIn this article, we propose a new efficient sparse estimate (ESE) in sufficient dimension reduction using distance covariance. Our method is model-free and does not need any kernel function or slicing selection. Moreover, it can naturally deal with multivariate response scenarios, making it appealing in a modified sequential algorithm that targets the large p small n problems. Compared with screening procedures that only use marginal utility, our method can extract more useful information from the data and is capable of determining the size of the selected submodel automatically while most of screening procedures cannot. Under mild conditions, based on manifold theories and techniques, it can be shown that our method would perform asymptotically as well as if the true irrelevant predictors were known, which is referred to as the oracle property. Extensive simulation studies and two real data examples demonstrate the effectiveness and efficiency of the proposed approach. It is remarkable that the analysis in cardiomyopathy microarray data reveals distinct and interesting findings. Supplemental materials for this article are available online.