Separate Solutions Research Articles

Aqueous 2-Ethyl-4-methylimidazole Solution for Efficient CO2 Separation and Purification

Carbon capture and storage (CCS) technology is considered as one of the most effective short-term solutions in reducing atmospheric CO2 concentrations. A key of CCS technology is to seek the absorbent with low cost, fast absorption rate, and high stability. In this study, we show that 2-ethyl-4-methylimidazole is particularly suitable for efficient CO2 capture. The aqueous solution of 2-ethyl-4-methylimidazole displays a maximum CO2 molar absorption capacity of 1.0 mol∙mol−1 and the absorbed CO2 can be completely released through heating the solution at a relatively low temperature (<100 °C). Stability tests show that the aqueous system is quite stable, with less than 10% loss of the molar absorption capacity after eight absorption–desorption cycles. Time-related in-situ attenuated total reflection infrared absorption spectroscopy and 13C nuclear magnetic resonance spectroscopy studies reveal that the intermediates are HCO3− and H2CO3 in the process of CO2 absorption–desorption. These intermediates are easily decomposed, which are responsible for the low CO2 desorption temperature and high desorption efficiency of the system. Moreover, the aqueous solution of 2-ethyl-4-methylimidazole is able to separate and purify CO2 from flue gas and even ambient air. Consequently, 2-ethyl-4-methylimidazole is a promising low-cost CO2 absorbent for industrial implementation.

Investigation of Performance of Nano Silica Cement Additive on Sulphate Attack In Geothermal Wells

Purpose: This research was intended to evaluate Nano silica as an additive to improve the sulphate resistance of cement used in geothermal wells.
 Design/ Methodology/ Approach: Sulphate resistance was determined by measuring the longitudinal change in cement cube specimens that were cured in sodium sulphate solution for 21 days. Cube specimens with varied concentrations of Nano silica (0%, 0.3%, 0.6%, 0.9% and 1.2%) were used in the study. Five separate solutions were maintained at 23℃, 40℃, 65℃, 70℃ and 80℃ for 21 days. Final length measurements were taken and compared as a percentage of initial length measurements.
 Findings: Beyond 65℃, the sulphate resistance of cement improved for each percentage concentration of Nano silica replacement. Control specimens with 0% Nano silica had the most inferior performance at all temperatures. Higher concentrations of 1.2% and 0.6% Nano silica replacement gave the most resistance between 23℃ and 65℃. A lower concentration of 0.3% proved more suitable between 65℃ and 80℃.
 Research limitation: The results of the experiment indicate performance in low-temperature geothermal wells.
 Practical implications: The application of this additive can improve the durability and strength of the cement, reducing the potential for degradation due to exposure to sulphates. This can lead to a longer lifespan of the geothermal well, reducing maintenance costs and increasing its overall efficiency.
 Social implications: Improved cement designs that create longer-lasting cement sheaths can be developed from this research, thereby fostering geothermal energy development. The option to replace certain volumes of cement with Nano silica contributes to a reduction of the carbon footprint by minimising the demand for, and therefore the production of cement.  
 Originality/ Value/ Novelty: Previous research that tested cement at high temperatures analysed mechanical resistance. This research examined the sulphate resistance of cement at high temperatures.

The approach to supply chain cooperation in the implementation of sustainable development initiatives and company's economic performance

Research background: The idea of sustainable development, in the face of the challenges encountered by contemporary society, is gaining increasing popularity. Currently, it recognizes the substantial role that companies play in its successful implementation. Initiatives in the field of sustainable development may be undertaken by companies independently as part of their own activities, or together with entities forming the supply chain as an element of sustainable supply chain management. Purpose of the article: Identification of groups of companies that are characterised by a different approach to cooperation in the field of sustainable development in the supply chain. Methods: The quantitative research was conducted in September 2020 with the use of the CATI (Computer-Assisted Telephone Interview) technique and a standardised survey questionnaire. A total of 500 randomly selected companies located in Poland participated in this study. The respondents were representatives of top management of the companies. In order to identify various groups of companies, a cluster analysis was performed using the k-means method in SPSS. Findings & value added: Based on the literature analysis, 3 areas of sustainable development have been identified, in which companies can become involved ? green design, sustainable operations, and reverse logistics & waste management. For each of the 3 areas, 3 clusters of companies were identified: companies that are not involved in sustainable development at all (1), companies that carry out most of the sustainable development initiatives independently (2), companies that carry out most of the sustainable development initiatives jointly with supply chain partners (3). The article also shows that the companies in different cluster differ in terms of perceived economic benefits achieved thanks to the implementation of sustainable development initiatives. This may suggest the need to develop separate sustainability solutions for such groups of companies in the future.

Merging Renewable Carbon-Based Materials and Emerging Separation Concepts to Attain Relevant Purification Applications in a Circular Economy

The globe demands a more efficient approach to provide materials and products for the economic and living needs of a modern society. Separation processes have an important role in order to achieve the recoveries and purities required in terms of quality specifications. There are a variety of porous materials taking the role of separation agents which work effectively in many technological industrial applications based on adsorption and membrane processes. The separation agents derived from renewable materials represent a desirable alternative to face the sustainability challenges in industrial settings. In this sense, the biomass residues which are abundant in agroindustrial regions represent an opportunity to tackle the needs for sustainable separation processes. Innovation plays a key role in order to combine renewable materials and emerging separation concepts to find compelling solutions for the needs of water and gas treatment purifications. These solutions are achieved more effectively through the joint research and innovation effort between materials and process aspects. This commentary remarks the fruitful opportunity existing for separation technology to take advantage of biomass residues to be converted into tailor-made separation agents. It focuses on novel separation applications where the biocarbon electroactive properties play a significant role. These technological approaches for biocarbon are beyond its most common use as an activated carbon adsorbent. Recent research contributions providing original solutions on this aspect are discussed here mainly for gas separation of CO2 and volatile organic compounds, freshwater recovery, and electrolyte separation among others. Innovative separation solutions including tailor-made biocarbons are expected to grow at a fast pace in the coming years.

Speciation of Fe(II) and Fe(III) in Chinese Yellow Rice Wine Using Direct Immersion Dual-Drop Microextraction Coupled with Graphite Furnace Atomic Absorption Spectrometry (GFAAS)

A new direct immersion dual-drop microextraction (DIDDME) procedure is described for Fe(III) and Fe(II) speciation followed by detection with graphite furnace atomic absorption spectrometry. This method is based upon the selective reaction of 1-(2-pyridylazo)-2-naphthol (PAN) and N-benzoyl-N-phenylhydroxylamine (BPHA) with Fe(II) and Fe(III) to form complexes at pH 1.5. Two solvent drops that contain PAN and BPHA were concurrently immersed in a sample solution for simultaneous separation of Fe(III) and Fe(II), respectively. This method eliminated time-consuming and labor-intensive oxidation of Fe(II) or reduction of Fe(III), which may provide incomplete conversion and contamination. Using the optimal conditions, the detection limits were 0.067 ng mL−1 and 0.054 ng mL−1 for Fe(II) and Fe(III) with relative standard deviations of 6.0% and 5.1%, respectively. Enrichment factors of 300-fold were obtained. The procedure was successfully utilized for the determination of Fe(II) and Fe(III) in Chinese yellow wine. To evaluate accuracy of this method, a rice certified reference material was analyzed and the result was in good agreement with the certified value.

Direct separation approach and multi-valued localized excitation for (M+N)-dimensional nonlinear system

In this paper, we propose a new variable separation method that does not need the Hirota’s bilinear form and directly gives the analytic form of the solution u instead of its potential uy. This new method covers the N-soliton solution obtained by the Hirota’s direct method and the multi-valued solution of the multi-linear variable separation approach(MLVSA), and is applicable to the (M+N)-dimensional nonlinear model. We would like to call it the “direct separation approach” (DSA). Taking the extended (3+1)-dimensional Kadomtsev–Petviashvili equation as an example, we introduce an entirely fresh variable separation ansatz and substitute it into the equation, resulting in a new variable separation solution. With the help of this variable separation solution, we construct two types of N-soliton solutions via the single-valued functions. Then, the rogue waves and four typical folded solitary waves are obtained by using the multi-valued functions. In addition, we study the head-on and chase-after collision between two, three, and four folded solitary waves and systematically analyze their dynamic behaviors, such as the phase shifts and their difference values of the interaction. Especially, we detect a multi-directional movement in the collision between four folded solitary waves, which significantly differs from the others. The obtained results may help us simulate complex folded appearance in real life and better understand the wave motion of the solitary waves.

Study on the Extraction Method of Microplastic System in Textile Wastewater

Microplastic pollution has become a global environmental problem. Textile microplastics are an important component of microplastic pollution, but little is known about their contamination in the industrial environment. The lack of standardized methods for detecting and quantifying textile microplastics is a major obstacle to determining the risks they pose to the natural environment. This study systematically examines the pretreatment options for the extraction of microplastics from printing and dyeing wastewater. The effectiveness of potassium hydroxide, nitric acid-hydrogen peroxide mixed solution, hydrogen peroxide, and Fenton's reagent for the removal of organic matter from textile wastewater is compared. Three textile microplastics, polyethylene terephthalate, polyamide, and polyurethane, are studied. The effects of the digestion treatment on the physicochemical properties of textile microplastics are characterized. The separation efficiency of sodium chloride, zinc chloride, sodium bromide, sodium iodide, and sodium chloride-sodium iodide mixed solution on the textile microplastics is tested. The results showed that Fenton's reagent achieved a 78% removal rate of organic matter from printing and dyeing wastewater. Meanwhile, it has less of an effect on the physicochemical properties of textile microplastics after digestion and is the best reagent for digestion. The zinc chloride solution achieved a 90% recovery for separating textile microplastics with good reproducibility. It does not affect the subsequent characterization analysis after separation and is the best solution for density separation.

Fabrication of high-strength graphene oxide/carbon fiber nanocomposite membranes for hydrogen separation applications

Graphene oxide (GO) is a promising nanosheet for developing next-generation thin-membrane technology to enhance gas permeability and selectivity performance based on the interlayer spacing principle. This study aimed to strengthen GO membranes for enhancing gas permeability and selectivity using non-woven carbon fiber (CF) fabric. The GO/CF membranes were synthesized with 1 wt% of GO and different CF orientations in the matrices with various GO layer thicknesses (200–800 µm). The resulting membranes were characterized for their surface and cross-sectional morphology, pore sizes, mechanical, chemical, and thermal analysis. The SEM results showed successful deposition of GO flakes onto the CF surface with high cohesion with the fibers, accompanied by some deviations in the GO edges, allowing gases to pass through the membranes. In addition, thinner membranes had higher Young's modulus (56 MPa) and surface roughness in the ranges of 84–184 nm. While the gas permeability measurements revealed that the thinner membranes (200 µm) had higher gas permeability in the range 7809214 Barrer (CO2) and 23154004 Barrer (H2) with an increase of 68–92% compared to the thicker membranes (600 and 800 µm). Moreover, thinner membranes showed higher selectivities about 5.2 (H2/CO2), 4.3 (H2/N2) and 3.6 (H2/CH4). Overall, these results indicate that GO/CF membranes are a promising solution for hydrogen separation due to their high strength and ease of fabrication.

Calculation of molecular g-tensors by sampling spin orientations of generalised Hartree-Fock states

The variational inclusion of spin–orbit coupling in self-consistent field (SCF) calculations affords single-determinant wave functions that completely break spin symmetry. The individual components of the molecular g-tensor are commonly obtained from separate SCF solutions that align the magnetic moment along one of the three principal tensor axes. However, this strategy raises the question if energy differences between solutions are relevant, or how convergence is achieved if the principal axis system is not determined by molecular symmetry. The present work resolves these issues by a simple two-step procedure akin to the generator coordinate method (GCM). First, the orientation of the magnetic moment is constrained in generalised Hartree-Fock (GHF) calculations. Then, non-orthogonal configuration interaction between GHF determinants yields a Kramers doublet for the calculation of the complete g-tensor. Alternatively, diagonalisation in a basis spanned by spin rotations of a single GHF determinant results in qualitatively correct g-tensors by eliminating errors related to spin contamination. For small first-row molecules, these approaches are evaluated against experimental data and full configuration interaction results. It is further demonstrated for two fictitious tetrahedral and species that a GCM strategy can describe the spin–orbit splitting of orbitally-degenerate ground states, which may lead to negative g-values.

Lattice Boltzmann method for solute transport in dual-permeability media

Dual-permeability models are currently under active investigation for predicting the flow and transport of pollutant concentration in porous media. The basic idea of these models is usually to separate the medium into two distinct flow regions with separate hydraulic and solute movement properties. Most of the dual-permeability approaches presented to date assume the reaction terms to be independent of space. For local reaction processes in soils, however, the depth effects are important and should be incorporated into the dual-permeability models. To reveal the transport phenomenon in such case, the best way is to obtain an exact solution with a suitable analytical approach, but it may be hard or even impossible for this general scenario. As such, this paper seeks an alternative method, and propose a lattice Boltzmann method for solute transport in dual-permeability media. The Chapman–Enskog analysis shows that the present model can recover the governing equations correctly. The capability and accuracy of this model is first verified by two problems: the non-reactive mass transfer in dual-permeability porous media, and the solute transport in a depth-dependent soil. With this model, we further studied the contaminant transport in dual-permeability soil with depth-dependent reaction rates. The numerical results show that when the exchange coefficient is relatively smaller, there are usually two peaks for the breakthrough curves (BTCs), whereas it degenerates to a single peak as the exchange coefficient increases. In addition, we note that the distributions of the concentration in different domains are nearly the same for a larger exchange coefficient. As for the pore water velocity, it is found that when the difference in the pore water velocity in the two domains is significant, the BTC exhibits a bimodal distribution. However, there is only one peak for the BTCs when the above difference is insignificant. Further, as compared with the constant-reaction rate cases, we observe that the BTCs obtained for the media with depth-dependent rates lie in the intermediate zone defined by the above two extremes. Finally, we adopt the current approach to simulate the solute transport in an Andisol and observe that the difference between the numerical and experimental results is insignificant, indicating our approach is capable in modeling solute transport in porous media.

Trends of Technological Development of Platinum Group Metal Recycling: Solubilization and Physical Concentration Processes

Spent automobile catalysts are the most important secondary resource for platinum group metals (PGMs), and their recycling is essential not only to ensure a steady supply of PGMs but also to preserve the natural environment. However, several factors limit the efficiency of PGM recovery in current recycling processes. Specifically, PGMs represent a small proportion of the entire catalyst and are difficult to dissolve in aqueous solutions for subsequent separation. In recent decades, multiple technologies have been developed to make the recycling of PGMs more efficient and environmentally-friendly. Herein, we introduce the typical industrial processes for recovering PGMs from spent catalysts, along with established and emerging trends in the technological development of PGM recycling. Furthermore, we review novel recycling techniques for converting PGMs in spent catalysts into more soluble states and/or physically concentrating PGMs from spent catalysts.

Edge computing on TPU for brain implant signal analysis

The ever-increasing number of recording sites of silicon-based probes imposes a great challenge for detecting and evaluating single-unit activities in an accurate and efficient manner. Currently separate solutions are available for high precision offline evaluation and separate solutions for embedded systems where computational resources are more limited.We propose a deep learning-based spike sorting system, that utilizes both unsupervised and supervised paradigms to learn a general feature embedding space and detect neural activity in raw data as well as predict the feature vectors for sorting. The unsupervised component uses contrastive learning to extract features from individual waveforms, while the supervised component is based on the MobileNetV2 architecture. One of the key advantages of our system is that it can be trained on multiple, diverse datasets simultaneously, resulting in greater generalizability than previous deep learning-based models.We demonstrate that the proposed model does not only reaches the accuracy of current state-of-art offline spike sorting methods but has the unique potential to run on edge Tensor Processing Units (TPUs), specialized chips designed for artificial intelligence and edge computing. We compare our model performance with state of art solutions on paired datasets as well as on hybrid recordings as well. The herein demonstrated system paves the way to the integration of deep learning-based spike sorting algorithms into wearable electronic devices, which will be a crucial element of high-end brain–computer interfaces.

Separable solutions to nonlinear anisotropic diffusion equation in elliptic coordinates.

Two- and three-dimensional exact solutions of the nonlinear diffusion equation are proved to exist in elliptic coordinates subject to an arbitrary piecewise constant azimuthal anisotropy. Degrees of freedom traditionally used to satisfy boundary conditions are instead employed to ensure continuity and conservation of mass across contiguity surfaces between subdomains of distinct diffusivities. Not all degrees of freedom are exhausted thereby, and conditions are given for the inclusion of higher harmonics. Degrees of freedom associated with one isotropic subdomain are always available to satisfy boundary conditions. The second harmonic is pivotal in the solution construction as well as the identification of partial symmetries in the domain partition. The anisotropy gives rise to an unconventional mixed type critical point that combines saddle and node-like characteristics. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

A hybrid decision-making method using robust programming and interval-valued fuzzy sets for sustainable-resilient supply chain network design considering circular economy and technology levels

This study introduces a new three-stage optimization approach with a circular economy perspective for sustainable-resilient supply chain network design for perishable products. The proposed model specifies the number of facilities and the number of products flowing within the supply chain in case of disruption. The three contributions of the mathematical model are considering product lifetime, monitoring financial resources, and selecting technology levels. It incorporates suppliers’ green image and circular economy rating in the supply chain network structure. Epistemic uncertainty is considered in the optimization model to deal with the unknown capacity, cost, and demand. The study includes three main stages. In the first stage, to determine suppliers' green image and circular economy rating, a new interval-valued fuzzy (IVF)-compromise decision-making method is presented based on possibilistic mean and standard deviation evaluations. In the second stage a new multi-objective mathematical model is proposed to reduce total costs, reduce environmental impacts, and increase social sustainability. To deal with uncertainties in the mathematical model, a new IVF-robust solution approach is introduced. In the third stage, the solution is incorporated with the AUGMECON2 method to produce separate Pareto-optimal solutions, offering a hybrid of trade-offs between cost, emissions, and social responsibility. A case study in the food, dairy, and drink industry is presented to show how the suggested approach might be applied. Finally, several sensitivity analyses and comprehensive comparisons of the proposed approach with the literature were performed.

Spontaneous Imbibition and Core Flooding Experiments of Enhanced Oil Recovery in Tight Reservoirs with Surfactants

Despite the implementation of hydraulic fracturing technologies, the oil recovery in tight oil reservoirs is still poor. In this study, cationic, anionic, and nonionic surfactants of various sorts were investigated to improve oil recovery in tight carbonate cores from the Middle Bakken Formation in the Williston Basin. Petrophysical investigations were performed on the samples prior to the imbibition and core-flooding experiments. The composition of the minerals was examined using the XRD technique. To investigate the pore-size distribution and microstructures, nitrogen adsorption and SEM techniques were applied. The next step involved brine and surfactant imbibition for six Bakken cores and two Berea sandstone cores. The core samples were completely saturated with Bakken crude oil prior to the experiments. The core plugs were then submerged into the brine and surfactant solutions. The volume of recovered oil was measured using imbibition cells as part of experiments involving brine and surfactant ingestion into oil-filled cores. According to the findings, oil recovery from brine imbibition ranges from 4.3% to 15%, whereas oil recovery from surfactant imbibition can range from 9% to 28%. According to the findings, core samples with more clay and larger pore diameters produce higher levels of oil recovery. Additionally, two tight Bakken core samples were used in core-flooding tests. Brine and a separate surfactant solution were the injected fluids. The primary oil recovery from brine flooding on core samples is between 23% and 25%, according to the results. The maximum oil recovery by second-stage surfactant flooding is approximately 33% and 35%. The anionic surfactants appear to yield a better oil recovery in tight Bakken rocks, possibly due to their higher carbonate mineral concentrations, especially clays, according to both the core-scale imbibition and flooding experiments. For studied samples with larger pore sizes, the oil recovery is higher. The knowledge of the impacts of mineral composition, pore size, and surfactant types on oil recovery in tight carbonate rocks is improved by this study.

Gregory-Laflamme and superradiance encounter black resonator strings

We construct novel black strings that are neither time-translation invariant, nor axisymmetric, nor translationally invariant in the string direction, but nevertheless have a helical Killing vector field. These solutions branch from the superradiant instability of D = 6 Myers-Perry black strings with equal angular momenta. We coin these solutions as black resonator strings and we find that they have more entropy than Myers-Perry black strings for the energies and angular momenta where both solutions coexist. We also construct Kaluza-Klein geons, which share the symmetries of black resonator strings, but are horizonless. Unlike in other superradiant systems, Kaluza-Klein geons are not the horizonless limit of black resonator strings and are instead entirely separate solutions.

Application of a three-fluid nozzle for the preparation of amorphous solid dispersions

In this work, the manufacture of amorphous solid dispersions of poorly water-soluble carbamazepine using a laboratory spray drying setup with a three-fluid nozzle and two separate liquid feed solutions of (a) the active and (b) the polymeric excipient (copovidone or basic butylated methacrylate copolymer) was investigated and compared to a traditional two-fluid nozzle setup. Results show that particles with the desired drug content were manufactured using the three-fluid nozzle setup. Amorphous solid dispersion could be obtained up to a drug load of 50% (w/w) processing solutions with miscible solvents and copovidone as polymer. Higher active concentrations, the use of immiscible solvents or switching of the capillaries led to a substantial crystallization of carbamazepine as seen in XRD and DSC analysis. The polymorphic form IV of carbamazepine was formed in the presence of copovidone at high solid concentrations of the liquid feed.

THE SEPARATION OF VARIABLES METHOD FOR SOLVING THE KLEIN-GORDON EQUATION IN CURVED SPACETIME

This study was carried out with the aim of finding a solution to the Klein-Gordon equation in a curved spacetime using the separation of variables method. The first step involved transforming the equation into a form that was suitable for separation of variables. This was achieved by using a Fourier transform to separate the time and spatial variables. Next, a separable solution was assumed in the form of █(φ(t,x,y,z)=T(t)X(x)Y(y)Z(z),)which was then substituted into the Klein-Gordon equation. The variables were separated by multiplying both sides of the equation by █(X(x)Y(y)Z(z)), resulting in four separate ordinary differential equations (ODEs). These ODEs were solved using standard methods such as separation of variables or characteristic equations. The general solution for the Klein-Gordon equation was found by combining the solutions of the four separate ODEs into a single solution of the form █(φ(t,x,y,z)=∑c_n T_n (t)X_n (x)Y_n (y)Z_n (z)), where c_n are constants and █(T_n (t),X_n (x),Y_n (y)), and █(Z_n (z))are the solutions of the separate ODEs. The final step was to determine the values of the constants █(c_n ) that satisfied the boundary conditions for the Klein-Gordon equation. This was done by using methods such as the method of eigenfunctions or the method of Greens functions. The results of this study showed that the separation of variables method is an effective way to solve the Klein-Gordon equation in a curved spacetime. These findings have important implications for our understanding of quantum field theory in curved spacetime and provide a basis for further research in this area. KEYWORDS: Klein-Gordon equation, Separation of variables method, Curved spacetime, Quantum field theory, Ordinary differential equations

Integrated capacity and production planning in the pharmaceutical supply chain: Framework and models

Production planning is widely studied, and mathematical formulations have been proposed for various problems. The capacity level is usually fixed which requires a separate solution of the capacity planning problem. In this paper, a methodology is proposed for setting up and solving integrated capacity and production planning problems for a given manufacturing network. Mixed-integer linear programming problems are assembled from a constraint and objectives database based on a system characterization and their solution determines whether capacity levels should be changed. An indirect approach to solving the problem is suggested, in which the required capacities are determined with no knowledge of specific ways on how the capacity could be changed corresponding to an early stage of capacity planning. This contrasts with the traditional direct approach, in which a set of options are given and a subset of these chosen through optimization. The methodology is applied to three case studies from primary pharmaceutical manufacturing. A literature example is used to validate the method and two larger case studies show the method's ability to solve problems of industrial scale and relevance.

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays.