Continuity Assumption Research Articles

The classical obstacle problem with Hölder continuous coefficients

Weiss’ and Monneau’s type quasi-monotonicity formulas are established for quadratic energies having matrix of coefficients which are Dini, double-Dini continuous, respectively. Free boundary regularity for the corresponding classical obstacle problems under Hölder continuity assumptions is then deduced.

On [H^{1}]^{3 times 3}, [H(text {curl})]^3 and H(text {sym Curl}) finite elements for matrix-valued Curl problems

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [{{,mathrm{textit{H}^1},}}]^{3times 3}, [ H (textrm{curl}{})]^3 and H (textrm{sym}, textrm{Curl}{}) for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [ H (textrm{curl}{})]^3 in place of the classical [{{,mathrm{textit{H}^1},}}]^{3 times 3}. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H (textrm{sym}, textrm{Curl}{}). The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory.

A Novel 2-D Phase Unwrapping Method Based on PUMA for Aerospace Topographic Mapping Systems

Two-dimensional phase unwrapping (PU) is the key step of interferometric synthetic aperture radar technology. The accuracy of the PU results directly affects the quality of the final product. Due to the limitation of the phase continuity assumption, the results obtained by the single-baseline PU in large gradient regions are not ideal. To solve this problem, this article proposes an improved PU max-flow/min-cut algorithm (PUMA) which can efficiently deal with the phase of large gradient regions. First, we construct a quasiconvex function which is the combination of logarithmic and quadratic functions as the potential function in the energy function model of the PUMA method. Then, we use the gradient information of the external digital elevation model to assist in setting the weight of the potential function. Finally, the unwrapped phase is obtained by minimizing the new energy function model. The experimental results on the TanDEM-X dataset show that the improved approach in this article can achieve higher accuracy PU results in the region of large gradient variation than the existing PU methods.

Bregman proximal point type algorithms for quasiconvex minimization

We discuss a Bregman proximal point type algorithm for dealing with quasiconvex minimization. In particular, we prove that the Bregman proximal point type algorithm converges to a minimal point for the minimization problem of a certain class of quasiconvex functions without neither differentiability nor Lipschitz continuity assumptions, this class of nonconvex functions is known as strongly quasiconvex functions and, as a consequence, we revisited the general case of quasiconvex functions.

Fast relaxed inertial Tseng’s method-based algorithm for solving variational inequality and fixed point problems in Hilbert spaces

Motivated and inspired by the works of Ceng et al. (2010) and Yao and Postolache (2012), we first study a relaxed inertial Tseng’s method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of fixed points of an κ-demicontractive mapping in real Hilbert spaces. The strong convergence of the algorithm is proved with conditions weaker than the conditions of other methods studied in the literature. Next, we also obtain an R-linear convergence rate of relaxed inertial Tseng’s method under strong pseudomonotonicity and Lipschitz continuity assumptions of the variational inequality mapping. As far as we know, these results have not been considered before in the literature. Finally, some numerical examples illustrate the effectiveness of our algorithms.

Three-dimensional multi-field digital twin technology for proton exchange membrane fuel cells

In times of the commercialization process of proton exchange membrane fuel cells (PEMFCs), a full knowledge of in-situ state in PEMFCs is of critical significance to the in-situ operational process and the evaluation of material stage and potential damage. The conventional experimental observation and in-situ prediction models can only obtain very limited information while the computational fluid dynamics approach takes too long time to get the detailed information. To reach a full knowledge of PEMFC real-time state, a novel 3D multi-physics digital twin model for PEMFCs is proposed based on the proper orthogonal decomposition (POD) method. In the model, firstly, for one kind of PEMFC, 139 ex-situ snapshots are designed and simulated based on the three-dimensional two-phase non-isothermal numerical model with the assumption of liquid pressure continuity in the whole membrane electrode assembly. Then the modes of each field in snapshots are extracted by singular value decomposition method using Jacobi algorithm. Finally, the coefficients in the POD prediction equation are obtained by using the multivariate adaptive regression splines. The digital twin results of voltage, temperature, membrane water content and liquid water saturation fields are exhibited and analyzed. Results suggest that for the studied PEMFC, the digital twin technique can capture the global values and the local distribution characteristics of each above physical fields well in 0.913 s. The mean global deviations of the above four fields of 20 groups of random conditions within wide current density and operational condition ranges are 5.7 %, 1.3 %, 8.9 % and 12.0 % respectively. Even though the practical results can only be applied for the studied PEMFC, the proposed methodology has its general application range.

Other-regarding behaviour in fuzzy non-cooperative games: Existence of altruistic equilibria

We study fuzzy non-cooperative games where the players care about the others. Two types of other-regarding behaviour of players are investigated. First, we assume that the players are full altruistic. So, we introduce the full altruistic fuzzy equilibrium and give necessary and sufficient conditions for the existence of such an equilibrium. Then, we consider players that subordinate the other-regarding behaviour to own interest to be preserved. In this case, each player has a fuzzy target and supports the opponents choosing strategies which give improvements with respect to the target. We introduce the partial altruistic fuzzy equilibrium and obtain the existence of such an equilibrium under weak continuity and convexity assumptions.

Fixed-time stability of projection neurodynamic network for solving pseudomonotone variational inequalities

In this paper, we propose a novel modified neurodynamic network (MNN) based on projection to solve pseudomonotone variational inequalities. Under the strong pseudo-monotonicity and Lipschitz continuous assumptions, the MNN admits a unique solution. The relations between the MNN and the corresponding neurodynamic network are obtained. Moreover, we establish the globally fixed-time stability of the MNN. The convergence time of the MNN is not depended on the initial conditions and uniformly bounded. Numerical examples are also reported to show the effectiveness and superiority of the MNN.

Equilibrium non-existence in generalized games

A generalized game is a strategic situation in which agents' behavior restricts their opponents' available action choices, giving rise to interdependencies in terms of what strategy profiles remain mutually feasible. This paper proposes a novel example of a simple jointly convex generalized game in which the well-known convexity, compactness, continuity, and concavity assumptions are satisfied, but no Nash equilibrium exists. The essence of this contribution lies in answering a question left open by Banks and Duggan (2004): whether the supplemental condition of lower hemicontinuity of feasibility correspondences can be dropped from these authors' equilibrium-existence theorem. It cannot.

Sequential Multi-Hypothesis Testing in Multi-Armed Bandit Problems: An Approach for Asymptotic Optimality

We consider a multi-hypothesis testing problem involving a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -armed bandit. Each arm’s signal follows a distribution from a vector exponential family. The actual parameters of the arms are unknown to the decision maker. The decision maker incurs a delay cost for delay until a decision and a switching cost whenever he switches from one arm to another. His goal is to minimise the overall cost until a decision is reached on the true hypothesis. Of interest are policies that satisfy a given constraint on the probability of false detection. This is a sequential decision making problem where the decision maker gets only a limited view of the true state of nature at each stage, but can control his view by choosing the arm to observe at each stage. An information-theoretic lower bound on the total cost (expected time for a reliable decision plus total switching cost) is first identified, and a variation on a sequential policy based on the generalised likelihood ratio statistic is then studied. Due to the vector exponential family assumption, the signal processing at each stage is simple; the associated conjugate prior distribution on the unknown model parameters enables easy updates of the posterior distribution. The proposed policy, with a suitable threshold for stopping, is shown to satisfy the given constraint on the probability of false detection. Under a continuous selection assumption, the policy is also shown to be asymptotically optimal in terms of the total cost among all policies that satisfy the constraint on the probability of false detection.

A characterisation of the continuum Gaussian free field in arbitrary dimensions

We prove that under certain mild moment and continuity assumptions, the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

Study of the SH-wave propagation in an MEFR layer bounded by heterogeneous viscoelastic layer and elastic half-space

PurposeThe purpose of this study is stated regarding the impact of the horizontally polarized shear wave vibration on a composite medium in the terms of phase and damped velocity.Design/methodology/approachThe assumed composite is composed of magneto-elastic fiber-reinforced (MEFR) layer constrained between heterogeneous viscoelastic layer and heterogeneous elastic half-space. The considered heterogeneity is associated with the directional rigidity and mass density in the uppermost layer and half-space of quadratic and trigonometric types, respectively. The coupled field equations related to the respective medium are solved analytically by employing the method of separation of variables.FindingsThe dispersion relation of the stated problem is secured by using the continuity assumptions, imposed at the stress-free surface and the interfaces of the expressed medium. The adopted numerical examples are used to compute the dispersion relation and plot the graphs between phase/damped velocity and wave number. Parametric studies on the phase and damped velocity yield five main conclusions: (1) Phase velocity decreases with increasing value of wave number and damped velocity increases up to a certain number and then starts falling simultaneously with increasing magnitude of wave number while keeping the rest parametric values fixed. (2) The presence of heterogeneity in the upper layer enhances the phase velocity and diminishes the damped velocity, but the presence of heterogeneity in the half-space enhances both the phase and damped velocity. (3) The appearance of reinforced parameters enhances the phase velocity for the considered crystalline graphite material and diminishes the phase velocity for the rest materials (carbon fiber-epoxy resin and steel) of the MEFR layer. Similarly, damped velocity decreases for the assumed crystalline graphite material of the MEFR layer and increases for the rest materials of the MEFR layer. (4) The induced dissipation factor due to viscoelastic property shows reversal decreasing and increasing effect on phase and damped velocity of SH-wave. (5) Ascending values of the angle at which the wave crosses the magnetic field increase the phase velocity and decrease the damped velocity for all the considered MEFR examples.Originality/valueTill date, the mathematical modeling as well as vibrational analysis of wave propagation through the composite structure consisting of MEFR layer constrained between viscoelastic media and elastic half-space under the effect of different varying properties with depth remains a new challenging issue for the researchers around the globe. The current analysis is an approach to move ahead in the era of wave propagation in different realistic models based on their parametric studies. Also, these studies are very helpful to find their applications in the field of mechanical, construction, aerospace, automobile, biomedical, marine, manufacturing industries and many branches of science and technology where magnetic fields induced in elastic deformation occur.

Time‐dependent neural networks with activation functions violating the standard Lipschitz condition

A Hopfield neural network system with discrete (or variable delays) is considered in this paper. The standard assumption of Lipschitz continuity of the activation functions is dropped partially. Having in mind that this condition is needed not only for the uniqueness of solutions but also for the stability of the system, the present work improves the existing ones in the literature. The other feature here, which is in fact the main one, is the treatment of time‐dependent activation functions. The time‐independent case has been discussed by one of the authors in Tatar (2020). Unfortunately, it is not applicable to the present situation. Indeed, when applied, it will require a uniform boundedness condition. To overcome this difficulty, we provide here a new argument in addition to the introduction of some functionals.

Investigation on Interlayer Behaviors of a Double-Layered Heterogeneous Asphalt Pavement Structure for Steel Bridge Deck

The EAC+SMA composite structure, made from epoxy asphalt concrete (EAC) with stone matrix asphalt (SMA) concrete on top, is a double-layered heterogeneous structure applied to steel bridge deck pavement (SBDP) in recent years. The interlayer bonding performance is key to the long-life application of EAC+SMA. In this study, the objective is to investigate the interlayer bonding performance of EAC+SMA composite structure to identify its best design and construction strategy. To achieve this goal, a series of pull-off tests and interlayer shear tests under different test conditions were performed on four commonly used EAC+SMA composite structures. The effects of design parameters, the construction conditions, and the application environment of EAC+SMA structure on its interlayer bonding performance were evaluated. The results showed that the EAC+SMA composite structure with SMA13 (SMA mixture with the nominal maximum aggregate size of 13.2 mm) as the wearing layer material and epoxy resin–based bonding material possessed the best interlayer bonding performance among the four composite structures. It is suggested that the bonding layer and SMA layer should be constructed after the Marshall stability of the EAC layer reaches 15 kN, and the construction interface should be clean to achieve the best interlayer bonding performance for the EAC+SMA composite structure. In addition, the mixed-mode cohesive zone model was developed to characterize the interlayer bonding behaviors in the mechanical analysis of EAC+SMA composite structures, and the proposed numerical characterization method could be used to obtain more accurate mechanical results for the structure design compared with the traditional numerical characterization method under the interlayer continuous assumption.

Simplified Option Price Derivations

Previous academic research reveals that mean-variance asset pricing (MVAP) models such as the single-period capital asset pricing model (CAPM) fail to produce rational European option prices. This article shows two adaptations of MVAP models that may be used to value derivatives with nonlinear payouts. The first removes static option arbitrage in investors’ optimized aggregate portfolio selection. The second linearizes the pricing kernel, using a static version of the self-financing condition applied in dynamic option modeling. Both adaptations produce risk-neutral derivative prices in equilibrium for all finite-moment probability distributions of underlying asset prices with compact support. The derivation does not require stochastic calculus, frictionless continuous trading assumptions, or the solution of differential equations. The resulting model is a hybrid of equilibrium and arbitrage techniques that rationally values assets and derivatives.

Reflected BSDEs with two completely separated barriers and regulated trajectories in general filtration

Abstract In this paper, we study doubly reflected backward stochastic differential equations defined on probability spaces equipped with filtration satisfying only the usual assumptions of right continuity and completeness in the case where the barriers L and U are not necessarily right continuous. We suppose that the barriers L and U and their left limits are completely separated and we show existence and uniqueness of the solution.

Principal eigenvalue, maximum principles and linear stability for nonlocal diffusion equations in metric measure spaces

We study principal eigenvalues and maximum principles for stationary nonlocal operators in spaces of integrable functions defined on general metric measure spaces under minimal assumptions on the kernels. Several characterizations of the principal eigenvalue are given as well as several conditions guaranteeing existence. Characterization of the (strong) maximum principle is also given. For evolution problems we prove the strong maximum principle and characterize stability in terms of the sign of the principal eigenvalue. We recover, extend and improve all previously known results, obtained for smooth open sets in euclidean space under continuity assumptions on the data.

Global higher integrability for minimisers of convex obstacle problems with (p,q)-growth

We prove global W^{1,q}(Omega ,{mathbb {R}}^N)-regularity for minimisers of {{mathscr {F}}(u)=int _Omega F(x,Du),{mathrm{d}}x} satisfying uge psi for a given Sobolev obstacle psi . W^{1,q}(Omega ,{mathbb {R}}^N) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform alpha -Hölder continuity assumption in x and natural (p, q)-growth conditions in z with q<frac{(n+alpha )p}{n}. In the autonomous case Fequiv F(z) we can improve the gap to q<min left( frac{np}{n-1},p+1right) , a result new even in the unconstrained case.

ELUCIDATING THE FRACTAL NATURE OF POWDER BED IN SELECTIVE LASER MELTING OF METALLIC COMPONENTS

In this work, the fractal nature of Selective Laser Melting (SLM) additive manufacturing process (AM) is elucidated. Fractal dimension and lacunarity of metallic powders are calculated from Scanning Electron Microscopy (SEM) images adapted from literature. The complexity and homogeneity of the textures of the powder beds are also studied through the comparison of fractal dimension and lacunarity. It is found that better densification results are obtained when the powder bed’s fractal dimension is closer to the golden mean number of 1.618. Furthermore, this finding is extended to expressions for predicting the component’s bulk density produced via SLM by setting the [Formula: see text] exponent equal to the golden mean value and finding the proportionality constant, [Formula: see text], using a nonlinear least squares method. The proposed approach works well since theoretical prediction and experimental data compare well with root-mean-square-error (RMSE) values that do not exceed [Formula: see text]. This work sheds new light on enhancing additive manufacturing technologies considering the fractal nature of SLM since its process mathematical models are constructed around Euclidian space-time with continuous smooth assumptions that should be adapted to include the fractal nature of the manufacturing process aiming to improve their precision. The underlying interweaving of SLM, as a fractal process, and the golden mean number is revealed.

A modified projection neural network with fixed-time convergence

In this study, we propose a modified projection neural network (PNN) with fixed-time convergence to solve the nonlinear projection equations. Under the assumptions of Lipschitz continuity and strict monotonicity, the existence of the solution and the stability in the Lyapunov sense of the proposed modified PNN are proved, which guarantee the convergence in fixed time. The convergence time of the proposed PNN has an upper bound which is independent ofarbitrary initial conditions. Compared with other existing PNN,the presented modified PNN can also solve the non-smooth, nonlinear, and constrained convex optimization problems. In the final step, the numerical simulations demonstrate that the presented modified PNN has a faster convergence rate and gets a more precise solution than the existing methods.