Degenerate Problems Research Articles

A nonconventional stability approach for a nonlinear Crank–Nicolson method solving degenerate Kawarada problems

Conventionally, the numerical stability of finite difference approximations of nonlinear Kawarada problems is shown only via frozen source terms, that is, ignoring potential jeopardization from quenching nonlinearities. The approach leaves an inadequacy behind even in the sense of localized stability analysis. This paper provides a much improved analysis of the numerical stability without freezing nonlinear source terms of the underlying equations. The strategy implemented can be extended for similar endeavors to higher dimensional cases. Simulation experiments are included.

Systems of fully nonlinear degenerate elliptic obstacle problems with Dirichlet boundary conditions

In this paper, we prove existence and uniqueness of viscosity solutions to the following system: For $$ i\in \left\{ 1,2,\dots ,m\right\} $$ $$\begin{aligned}{} & {} \min \biggl \{ F\bigl ( y,x,u_{i}(y,x),D u_{i}(y,x),D^2 u_{i}(y,x)\bigl ), u_{i}(y,x)-\max _{j\ne i}\bigl ( u_{j}(y,x)-c_{ij}(y,x)\bigl )\biggl \}\\{} & {} =0, \left( y,x \right) \in \Omega _{L}\\{} & {} u_{i}(0,x)=g_{i}(x), x\in \bar{\Omega },\ u_i(y,x)=f_i(y,x), (y,x)\in (0,L)\times \partial {\Omega } \end{aligned}$$ where $$ \Omega \subset \mathbb {R}^n $$ is a bounded domain, $$ \Omega _{L}:=(0,L)\times \Omega $$ and $$ F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}$$ is a general second-order partial differential operator which covers even the fully nonlinear case. (We will call a second-order partial differential operator $$F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}$$ fully nonlinear if and only if, it has the following form $$\begin{aligned} F \left( y,x,u,D_x u,D_{xx}^2 u\right) :=\sum _{|\alpha |=2}\alpha _{\alpha }\left( y,x,u,D_x u,D_{xx}^2 u \right) D^{\alpha }u(y,x)+\alpha _{0}\left( y,x,u,D_x u \right) \end{aligned}$$ with the restriction that at least one of the functional coefficients $$ \alpha _{\alpha },\ |\alpha |=2, $$ contains a partial derivative term of second order.) Moreover, F belongs to an appropriate subclass of degenerate elliptic operators. Regarding uniqueness, we establish a comparison principle for viscosity sub and supersolutions of the Dirichlet problem. This system appears among others in the theory of the so-called optimal switching problems on bounded domains.

Families of hybridized interior penalty discontinuous Galerkin methods for locally degenerate advection-diffusion-reaction problems

We analyze families of primal high-order hybridized discontinuous Galerkin (HDG) methods for solving degenerate second-order elliptic problems. One major problem regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the whole domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptical in the rest. This problem is thus quite delicate to solve since the exact solution can be discontinuous at interfaces separating the elliptic and hyperbolic parts. The proposed hybridized interior penalty DG (H-IP) method is developed in a unified and compact fashion. It can handle pure-diffusive or -advective regimes as well as intermediate regimes that combine these two mechanisms for a wide range of Péclet numbers, including the tricky case of local evanescent diffusivity. To this end, an adaptive stabilization strategy based on the addition of jump-penalty terms is considered. An upwind-based scheme using a Lax–Friedrichs correction is favored for the hyperbolic region, and a Scharfetter–Gummel-based technique is preferred for the elliptic one. The well-posedness of the H-IP method is also briefly discussed by analyzing the strong consistency and the discrete coercivity condition in a self-adaptive energy-norm that is regime dependent. Extensive numerical experiments are performed to verify the model’s robustness for all the abovementioned regimes.

A computational building block approach towards multiscale architected materials analysis and design with application to hierarchical metal metamaterials

In this study we report a computational approach towards multiscale architected materials analysis and design. A particular challenge in modeling and simulation of materials, and especially the development of hierarchical design approaches, has been to identify ways by which complex multi-level material structures can be effectively modeled. One way to achieve this is to use coarse-graining approaches, where physical relationships can be effectively described with reduced dimensionality. In this paper we report an integrated deep neural network architecture that first learns coarse-grained representations of complex hierarchical microstructure data via a discrete variational autoencoder and then utilizes an attention-based diffusion model solve both forward and inverse problems, including a capacity to solve degenerate design problems. As an application, we demonstrate the method in the analysis and design of hierarchical highly porous metamaterials within the context of nonlinear stress–strain responses to compressive deformation. We validate the mechanical behavior and mechanisms of deformation using embedded-atom molecular dynamics simulations carried out for copper and nickel, showing good agreement with the design objectives.

Dissecting a vegetable landrace: Components of variation in Spanish ‘Moruno’ tomatoes as a case studio

The variability of landraces is dissected and related to farmers' practices through surveys. Spanish 'Moruno' tomatoes, recognised for their excellent flavour, were selected as a case study. For this purpose, 30 populations were characterised in a four-year program. Higher intra-population than inter-population variability was found. Variability is generally reduced in the traits used by farmers as selection criteria. Farmers rarely used flavour as a selection criterion, but it is one of the main characteristics used by them to define the landrace. Seed exchange and growing different landraces simultaneously are commonplace, and outcrossing might occur, thus justifying in part the seed degeneration problems experienced by some farmers. At the same time, farmers select seeds for the next year on a per-fruit basis rather than a per-plant basis, justifying the maintenance of a high level of variability. In such cases, high pressure would be applied to key morphological traits but not to flavour. Accordingly, the sugars, acids and volatiles profiles related to flavour in the landraces' populations are highly variable, though the environment also exercises a high effect. It is necessary to make population selections to offer farmers materials combining the best organoleptic perception and a high stable yield. This would reduce the gap between the price premium received by farmers and the expected fair price. It would also be necessary to develop consumer information campaigns to exploit their willingness to pay for the extra value offered by landraces. Only then long-term on-farm conservation would be economically feasible. Although achieving a trade-off between yield and flavour is difficult, it is possible to identify populations that reach a compromise between them. In germplasm banks, it is impossible to evaluate all the materials in the same year. The use of hybrid controls, with no genetic variation, is helpful in considering the environmental effects. Still, genotype x environment interactions are evident, and using selected control populations of landraces is necessary to evaluate possible performances closer to the type of materials being evaluated.

DEVELOPMENT OF METHODS OF BREEDING WORK WITH INSECTS OF THE SPECIES HERMETIA ILLUCENS

Background. The insect Hermetia illucens (black soldier fly), is widely used for the disposal of food and agricultural waste. This is due to the omnivorous nature of the fly in relation to protein and carbohydrate foods and a fast reproductive cycle. The big problem of industrial cultivation of this insect is the degeneration of populations of the black soldier fly. When bred in captivity, due to the relatively small number of population, the degradation of key parameters, such as viability, fertility, productivity, etc. of H. illucens culture occurs. It is noted that during the first five generations, despite polygamy during crossing, the effect of harmful recessive alleles in a homozygous state manifests itself. Currently, the problem of degeneration of flies associated with inbred depression is solved by mass cultivation of large outbred populations in cages, with periodic reintroduction of producers from natural populations.
 Purpose. Development of a method for crossing lines of Hermetia illucens flies for breeding, determining patterns of population development, as well as increasing their productivity and adaptive potential.
 Materials and Methods. In selection work with the lines of the insect species H. illucens, individual crossing was used. Individual selections were made by selecting larvae and pupae by size and body weight. The crosses were carried out according to the scheme: one male and one female were placed in plastic containers while maintaining the required temperature and humidity. Eggs obtained from this pair of flies were placed on a nutrient substrate in a separate container, hatched larvae were cultured in isolation in the same container. During the growth process, the largest larvae and pupae from this population were selected. After the formation of adult individuals (images) of flies, they were examined, their sex was determined and again one male and one female were placed in a separate container. Thus, inbred lines of isofemale (descendants of one female) flies were obtained.
 Results. We developed the method of individual crossing made it possible to trace the development of populations of flies obtained from one female and one male, to consider the patterns and changes in the population. The tendency of the degeneration of the fly population in inbred crosses was determined, in each subsequent generation there were fewer fertilized eggs and larvae. This is due to the fact that recessive lethal genes become homozygous and manifest their effect. Subsequent selections in each generation of the largest and most active larvae will create a population devoid of most lethal and harmful genes.
 Conclusions. As a result of the research work carried out, it was possible to develop a method of individual crossing of males and females of the insect Hermetia illucens. It was determined that the time from oviposition onset to the maximum number of larvae occurs on 3-4th day after egg laying. With the crossing of close relatives, the tendency of the degeneration of the fly population was determined, in each subsequent generation there were fewer fertilized eggs and larvae. This is due to the manifestation of recessive lethals. Subsequent selections in each generation of the largest and most active larvae will create a population devoid of most lethal and harmful genes. Populations of inbred flies have stable characteristics for this species and, despite a decrease in the number of fertilized eggs in clutches, as well as a smaller number of larvae obtained from them, they have normal distribution of size and body weight, which allows selection for economically valuable traits.

A Dynamic Analysis of the Twin-Deficit Hypothesis: the Case of a Developing Country

The main economic challenge is rising aggregate demand, which leaves the economy short on resources and leads to expanding fiscal and external account deficits. The current study uses autoregressive distributed lag (ARDL) model to evaluate the twin deficit hypothesis in the context of Pakistan in an effort to find an answer to this question. The study uses augmented ARDL, popularized by McNown et al. (Appl Econ 50:1509–1521, 2018) and Sam et al. (Econ Model 80:130–141, 2019), to address the degenerate problems that might arise while applying the ARDL approach. Two separate models were estimated, one with the current account balance as the dependent variable and the other with the balance of trade. In the long run, both models confirm the conventional interpretation of twin deficit hypothesis in Pakistan, with the causality running only from the fiscal deficit to the balance of trade. Other control variables in both models are crucial in understanding the current account balance and balance of trade. According to models, an increase in the exchange rate, as measured by the log of the nominal effective exchange rate, improves both current account and trade balance, verifying the elasticity approach in the long run. The openness of the economy is found to worsen current account balance, and the result is statistically significant. Contrarily, openness has been improved trade balance, but the result is statistically insignificant. To control a large and persistent external deficit, the government has to reduce its fiscal deficit, and such a strategy would be successful when monetary policy is accommodative.

Self-rated eyesight among healthy older Australians: Baseline results of the ASPREE Longitudinal Study of Older Persons.

We aimed to describe the self-reported level of eyesight amongst a cohort of relatively healthy older Australian adults, and to investigate associations between poorer self-rated eyesight and demographic, health, and functional characteristics METHODS: The ASPirin in Reducing Events in the Elderly (ASPREE) Longitudinal Study of Older Persons (ALSOP) study was embedded in a multisite trial which recruited independently living Australians from general practices (2010-2014). Self-rated eyesight was recorded on a paper-based questionnaire as Excellent, Good, Fair, Poor, Very poor, or Completely blind at the baseline study wave RESULTS: Data from 14 592 participants (aged 70-95 years, 54.61% female) were included in this cross-sectional analysis. Eighty percent of participants reported excellent or good eyesight (n = 11 677). People with complete blindness were precluded from enrolling but 299 participants (2.0%) reported poor or very poor eyesight, and 2616 rated their eyesight as fair (17.9%). Lower levels of eyesight were associated with being older, female, fewer years of formal education, a primary language other than English, smoking, and self-reported macular degeneration, glaucoma, retinopathy, cataracts, and hearing problems (each p ≤ 0.021). People with lower levels of eyesight had a higher number of falls, frailty characteristics, and depressive symptoms, and lower mental and physical health functioning scores (each p < 0.001) CONCLUSIONS: Whilst most of these healthy older Australians reported good or excellent eyesight, a notable minority reported poor or very poor eyesight, and this was associated with a range of poorer health measures. These findings support the need for additional resources to prevent vision loss and associated sequelae.

A Degenerate Primer Design Method Based on Ant Colony Optimization

For the existed biological degenerate primer design problem, a solution method based on an ant colony optimization algorithm is proposed, in which an optimization model considers primer coverage as well as degeneracy while allowing for a small number of base mismatches. Using the construction graph model, the algorithm makes full use of the dynamic change information during the iterative optimal path optimization process, strengthens the explored new path pheromone, and selects the primers that meet the constraints. The results of DNA template sequence matching experiments show that the degenerate primers designed by the proposed algorithm outperform existing approaches.

Meshless generalized finite difference method with a domain-selection method for solving degenerate boundary problems

This paper is dedicated to presenting a combined Generalized Finite Difference Method (GFDM) and Domain-Selection Method (DSM) approach for solving degenerate boundary problems which are usually encountered in engineering simulations. Compared with traditional Domain-Decomposition Method (DDM), the most critical advantage of the present GFDM-DSM is that it is entirely free from matching continuous boundary conditions for different parts of the fluid domain of interest, which is sometimes hard to perform for specific situations. Instead, the only prerequisite of the present GFDM-DSM is to identify which sub-domain each computational node belongs to, and then a linear algebraic system can be assembled according to the pattern of domain-selection. Four benchmarks are performed to validate the convergence and accuracy of the present GFDM-DSM. It is shown that the present GFDM-DSM holds satisfactory accuracy for solving degenerate boundary problems. In particular, the present DSM is a flexible strategy since it can be also combined with other localized domain-type meshless methods in practical applications.

Proximal Stabilized Interior Point Methods and Low-Frequency-Update Preconditioning Techniques

In this work, in the context of Linear and convex Quadratic Programming, we consider Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is strongly supported by theoretical results concerning convergence and the rate of convergence, and can handle degenerate problems. Moreover, in the second part of this work, we analyse the interactions between the regularization parameters and the computational footprint of the linear algebra routines used to solve the Newton linear systems. In particular, when these systems are solved using an iterative Krylov method, we are able to show—using a new rearrangement of the Schur complement which exploits regularization—that general purposes preconditioners remain attractive for a series of subsequent IPM iterations. Indeed, if on the one hand a series of theoretical results underpin the fact that the approach here presented allows a better re-use of such computed preconditioners, on the other, we show experimentally that such (re)computations are needed only in a fraction of the total IPM iterations. The resulting regularized second order methods, for which low-frequency-update of the preconditioners are allowed, pave the path for an alternative class of second order methods characterized by reduced computational effort.

The modern view of the state of the problem of age-macular degeneration, its connection with genetics

The modern view of the state of the problem of age-macular degeneration, its connection with genetics

On the rate of convergence of a numerical scheme for Fractional conservation laws with noise

Abstract We consider a semidiscrete finite volume scheme for a degenerate fractional conservation law driven by a cylindrical Wiener process. Making use of the bounded variation (BV) estimates, and a clever adaptation of classical Kružkov theory, we provide estimates on the rate of convergence for approximate solutions to degenerate fractional problems. The main difficulty stems from the degenerate fractional operator and requires a significant departure from the existing strategy to establish Kato’s type of inequality. Indeed, recasting the mathematical framework recently developed in Bhauryal et al. (2021, J. Differential Equations, 284, 433–521), we establish such Kato’s type of inequality for a finite volume scheme. Finally, as an application of this theory, we demonstrate numerical convergence rates.

Biomechanical comparisons of dynamic fixators with rod-rod and screw-spacer joints on lumbar hybrid fixation

BackgroundHybrid fixators with quite different joint design concepts have been widely to suppress adjacent segment degeneration problems. The kinematic and kinetic responses of the adjacent and transition segments and contact behaviors at the bone-screw interfaces served as the objective of this study. MethodsThe moderately degenerated L4/L5 and mildly degenerative L3/L4 segments were respectively immobilized by a static fixator and further bridged by the rod-rod (Isobar) and screw-spacer (Dynesys) fixator. The joint stiffness and mobility of the rod-rod system and the cable pretension of the screw-spacer system were systematically varied. FindingsThe flexion of the screw-spacer system provided higher mobility to the transition segment, reducing adjacent-segment problems. The cable pretension had a minor effect on the construct behavior. However, due to limited joint mobility, the rod-rod system showed higher constraints to the transition segment and induced more adjacent-segment compensations. The increased mobility of the rod-rod joint caused it to behave as a more dynamic fixator that increased adjacent-segment compensations at the transition segment. Comparatively, increasing the joint mobility showed more significant effects on the construct behaviors than decreasing the joint stiffness. Furthermore, increased constraint by the rod-rod joint induced higher stress and risk of loosening at the bone-screw interfaces InterpretationIf the protection of the transition segment is the major concern, the rod-rod system can be used to constrain the intervertebral motion and share the higher loads through the fixator. Otherwise, the screw-spacer system is recommended in situations where higher loads onto the transition disc are allowable.

Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order

We study the long-time behavior of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model, for instance, the dynamic behavior of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power law or Ellis law for the fluid viscosity. In all three cases, positive constants (i.e., positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behavior of solutions with respect to the -norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state converge to equilibrium at rate for some . Finally, in the case of an Ellis fluid, steady states are exponentially stable in .

The degenerate hyperbolic problem for the two‐dimensional nonlinear wave system

This paper constructs a local classical solution of degenerate hyperbolic problem for the two-dimensional nonlinear wave system. To deal with the parabolic degeneracy, we introduce a new set of coordinates to transform the nonlinear wave system to a new system that has explicitly singularity-regularity structures. By constructing a weighted metric space, we establish the existence of solution for the new system. Returning the solution to the original variables, we obtain the existence of the classical solution for the nonlinear wave system with degenerate boundary value problem.

Optimal entry decision of unemployment insurance under partial information

The aim of this paper is to study the optimal time for the individual to join an unemployment insurance scheme which is intended to protect workers against the consequences of job loss and to encourage the unemployed workers to find a new job as early as possible. The wage dynamic is described by a geometric Brownian motion model under drift uncertainty and the problem is a kind of two-dimensional degenerate optimal stopping problems which is hard to analyze. The optimal time of decision for the workers is given by the first time at which the wage process hits the free boundary which therefore plays a key role in solving the problem. This paper analyzes the monotonicity and continuity of the free boundary and derives a nonlinear integral equation for it. For a particular case the closed-form formula of free boundary is obtained and for the general case the free boundary is solved by the numerical solution of the nonlinear integral equation. The key in the analysis is to convert the degenerate problem into the non-degenerate one using the probability approach.

Strong 2-degenerate graph embeddings

A result of Friedman and Pippenger gives a sufficient condition on the expansion properties of a graph to contain all small trees with bounded maximum degree. In this paper, we extend the tree embedding problem of expanding graphs to the degenerate graph embedding problem, and give a sufficient condition for a graph to contain strong [Formula: see text]-degenerate subgraphs with bounded maximum codegree.

A global result for a degenerate quasilinear eigenvalue problem with discontinuous nonlinearities

A global result for a degenerate quasilinear eigenvalue problem with discontinuous nonlinearities

The Heston stochastic volatility model has a boundary trace at zero volatility

We establish boundary regularity results in Hölder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) mathbb {H} = mathbb {R}times (0,infty )subset mathbb {R}^2. Starting with nonsmooth initial data u_0in H, we take advantage of smoothing properties of the parabolic semigroup textrm{e}^{-t{mathcal {A}}}:,Hrightarrow H, tin mathbb {R}_+, generated by the Heston model, to derive the smoothness of the solution u(t) = textrm{e}^{-t{mathcal {A}}} u_0 for all t>0. The existence and uniqueness of a weak solution is obtained in a Hilbert space H = L^2(mathbb {H};mathfrak {w}) with very weak growth restrictions at infinity and on the boundary partial mathbb {H}= mathbb {R}times { 0}subset mathbb {R}^2 of the half-plane mathbb {H}. We investigate the influence of the boundary behavior of the initial data u_0in H on the boundary behavior of u(t) for t>0.