Mathematical Structure Research Articles

Building a simple multivariable filtration model to predict irreversible fouling when directly filtering municipal wastewater

This paper describes a simple generic model designed to predict membrane fouling in municipal wastewater (MWW) treatment. The work was conducted using data from a direct membrane filtration demo-system (middle/long-term filtration periods of about 35 – 124 days) to calibrate the model. Two influents were treated by the demo-system: raw pre-treated MWW and primary settler supernatant from a full-scale MWW treatment plant. A resistance-in-series mathematical model structure was proposed considering fouling due to two different mechanisms: persistent cake layer formation (from suspended material) and pore blocking (from soluble and colloidal compounds). The proposed model represented transmembrane pressure dynamics at different operating solids concentrations (around 1, 2.6, 6 and 11 gL−1) using 7 model parameters, achieving 7–28 mbar differences between the experimental data and model predictions in all cases (calculated as the root mean square error). The model was also able to match the results from two different influents (raw MWW and primary settler supernatant) by modifying 3 of the 7 parameters while low uncertainties were obtained in long-term filtrations, demonstrating its robustness. This model thus provides a good potential to generate reasonable membrane fouling predictions while its simple and open structure makes it easy to implement with complementary materials. Further research will be carried out to enhance the model’s precision and validate its potential for optimizing filtration and fouling control processes.

Study on soliton behaviors of the complex nonlinear Fokas–Lenells equation utilizing different methods

In this paper, we study the Fokas–Lenells equation, which is a derivation form of the nonlinear Schrödinger equation and can be used to describe nonlinearity in the propagation of optical pulses. To seek solutions for this nonlinearity, the Logistic method has been proposed in order to drive acceptable and understandable results regarding physical meaning. This method is defined based on the summation of an ordinary differential equation. Moreover, for further study, the Bernoulli [Formula: see text]-expansion method is used by considering hypothetical solutions as a function of the Bernoulli equation. Subsequently, solutions of the Fokas–Lenells equation are successfully acquired, displaying efficient results in hyperbolic, trigonometric, and exponential equations. The importance of these results appears in the definition of the wave function under the consideration of the appropriate coefficients. Hence, computer simulation is used for better understanding of the generated results. The dynamic behaviors of these solutions are demonstrated in 3D graphs, contour maps, and line plots, under applying various parameters, solutions of the waves show different soliton behaviors, including bright-dark, bright, and breather solitons. The results indicate that the exerted methods are novel, reliable, and effective approaches which can be employed in a wide range of nonlinear differential equations. These methods and their begotten results are far from the complexities of mathematical structures and therefore go beyond previous efforts in the literature.

Thermodynamic analysis of micropolar-casson fluid flow with PST and PHF heating condition over a curved stretching surface

In this research, we explore the flow dynamics of micropolar Casson fluids over curved surfaces, while incorporating prescribed heat and surface fluxes, as well as heat generation effects. By applying similarity transformation to the governing model, we derive individual differential equations which are then reduced to general differential equations. This transformation yields a mathematical structure conducive to problem analysis. Through the use of numerical methods, we solve ordinary differential equations to investigate the fluid dynamics and heat transfer phenomena in this specific flow configuration. The main aim of this study is to offer a comprehensive understanding of the intricate behavior displayed by micropolar Casson fluids on curved surfaces under the influence of prescribed heat and surface fluxes, along with heat generation. By employing numerical methods, this study aims to contribute to the advancement of theoretical understanding by addressing the complex interactions between fluid dynamics and thermal properties.

LFTDA: A lightweight and fault-tolerant data aggregation scheme with privacy-enhanced property in fog-assisted smart grid

In recent years, the data aggregation scheme was widely adopted in smart grids (SG) to protect data privacy while ensuring data availability. Meanwhile, asymmetric homomorphic encryption is a popular technique that guarantees data aggregation and accurate computing results. However, it often brings heavy computation overheads and costs, which increases burdens to SG. In order to lighten SG’s burden, in this paper, a lightweight and fault-tolerant data aggregation scheme (LFTDA) with privacy-enhanced property is proposed. Unlike existing schemes, our proposed scheme uses a lightweight symmetric encryption algorithm and a skillful mathematical structure to achieve data aggregation. In addition, in the proposed scheme, smart meters are allowed to dynamically join and exit from the group while guaranteeing the correctness, security, and efficiency of aggregation. More importantly, our LFTDA is fault-tolerant, which means that the data collection from other devices will not be affected even if some of the smart meters or fog nodes do not work. From the view of security, LFTDA can defend against eavesdropping attack and collusion attack. Finally, we evaluate its performance along with other related schemes in terms of aggregation, decryption, and communication costs, which shows that LFTDA is very competitive and hence is suitable for many practical SG applications.

A Graph-Based Approach for Relating Integer Programs

This paper presents a framework for classifying and comparing instances of integer linear programs (ILPs) based on their mathematical structure. It has long been observed that the structure of ILPs can play an important role in determining the effectiveness of certain solution techniques; those that work well for one class of ILPs are often found to be effective in solving similarly structured problems. In this work, the structure of a given ILP instance is captured via a graph-based representation, where decision variables and constraints are described by nodes, and edges denote the presence of decision variables in certain constraints. Using machine learning techniques for graph-structured data, we introduce two approaches for leveraging the graph representations for relating ILPs. In the first approach, a graph convolutional network (GCN) is used to classify ILP graphs as having come from one of a known number of problem classes. The second approach makes use of latent features learned by the GCN to compare ILP graphs to one another directly. As part of the latter approach, we introduce a formal measure of graph-based structural similarity. A series of empirical studies indicate strong performance for both the classification and comparison procedures. Additional properties of ILP graphs, namely, losslessness and permutation invariance, are also explored via computational experiments. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0255 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0255 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

ON THE CONCEPTS OF CONVEXITY AND WEAK O-MINIMALITY FOR PARTIALLY ORDERED STRUCTURES

In this paper, we consider a generalization of the concept of weak o-minimality to partially ordered sets. However, the concept of weak o-minimality is based on the concept of a convex set, the direct transfer of which to partial orders, as it will be shown in the work, is not, in our opinion, the most successful, since then in the class of weakly o-minimal partially ordered structures, it is possible to define any mathematical structure. Moreover, as it will be shown, this can be done using such a simple operation as the intersection of intervals. The article is devoted to the search for various generalizations of the concept of “convex set” to partial orders. Since convex sets on a line also have other properties, such as the ability to represent them as a union or intersection of intervals, convex sets are connected, all these properties can be used as the basis for the definition of a “convex set” for partially ordered structures. Thus, the representation of a convex set as a union of nested intervals (half-intervals, segments) gives us the concept of an “internally convex set,” and the intersection of intervals gives us the concept of an “externally convex set”. In the article, we will build examples that show the non-equivalence of the introduced concepts.

SAT-Based Techniques for Lexicographically Smallest Finite Models

This paper proposes SAT-based techniques to calculate a specific normal form of a given finite mathematical structure (model). The normal form is obtained by permuting the domain elements so that the representation of the structure is lexicographically smallest possible. Such a normal form is of interest to mathematicians as it enables easy cataloging of algebraic structures. In particular, two structures are isomorphic precisely when their normal forms are the same. This form is also natural to inspect as mathematicians have been using it routinely for many decades. We develop a novel approach where a SAT solver is used in a black-box fashion to compute the smallest representative. The approach constructs the representative gradually and searches the space of possible isomorphisms, requiring a small number of variables. However, the approach may lead to a large number of SAT calls and therefore we devise propagation techniques to reduce this number. The paper focuses on finite structures with a single binary operation (encompassing groups, semigroups, etc.). However, the approach is generalizable to arbitrary finite structures. We provide an implementation of the proposed algorithm and evaluate it on a variety of algebraic structures.

Unveiling new exact solutions of the unstable nonlinear Schrödinger equation using the improved modified Sardar sub-equation method

The improved modified Sardar sub-equation method is used in this study to look into the complex domain of the unstable nonlinear Schrödinger equation, which is a basic idea in quantum physics. Our main goal is to obtain new and exact solutions for the unstable nonlinear Schrödinger equation, which includes various mathematical structures such as rational, exponential, trigonometric, and hyperbolic trigonometric forms. These solutions provide insight into the complex mechanics that control wave events and their nonlinear behavior. By employing MATLAB, we display visual depictions of these solutions in different dimensions, improving the direct examination of the occurrences. This study enhances the existing knowledge of mathematical methods for solving nonlinear differential equations, which can be applied to other nonlinear wave equations in other scientific fields. The findings from this work not only enhance our comprehension of the unstable nonlinear Schrödinger equation but also open up new possibilities for studying nonlinear phenomena in several physical systems, including optics, condensed matter physics, and quantum information science.

Mathematical structures of simple and compound interest: An analysis of secondary teachers’ relational thinking

Mathematical structures of simple and compound interest: An analysis of secondary teachers’ relational thinking

Roll waves in a predictive model for open-channel flows in the smooth turbulent case

A depth-averaged model for turbulent open-channel flows with breaking roll waves on a sloping smooth bottom is derived under an assumption of independence between the wall turbulence and the roller turbulence. The model includes four variables – the water depth, the average velocity, and two variables called enstrophy, the shear enstrophy and the roller enstrophy – which take into account the deviations of the velocity with respect to its depth-averaged value due to shear effect and roller turbulence, respectively. The four equations of the model are the mass, momentum, energy and shear enstrophy balance equations, with the mathematical structure of the Euler equations of compressible fluids, with an additional transport equation and with source terms. The system is hyperbolic. The roller enstrophy is created by shocks. A non-zero value of the roller enstrophy indicates a breaking wave and a turbulent roller. The model is solved by a fast and well-known numerical scheme, with an explicit finite-volume method in one step. The model is used to simulate periodic and natural roll waves with a good agreement with existing experimental results. There is no parameter to calibrate in the model, which gives it a predictive character.

Qualitative Behavior for a Discretized Conformable Fractional-Order Lotka-Volterra Model With Harvesting Effects

The predator-prey model is a widely mathematical structure that explains the dynamics between two interacting populations: predators and prey. The predator-prey interaction represents a fundamental dynamic in nature, influencing the stability and balance of ecosystems worldwide. The purpose of this article is to provide insight into the complex interactions and feedback mechanisms between predators and prey in ecological systems via mathematical tools such as stability and bifurcation. We investigate a fractional-order Lotka-Volterra model with a harvesting effect using stability and bifurcation theory. The equilibrium points and local stability of the purposed model are presented in this article. The bifurcation analysis, which is a potent approach used to analyse the qualitative behavior of the predator-prey system as the parameter values are varied, is also explored. In particular, a Neimark-Sacker bifurcation and a period-doubling bifurcation are theoretically and numerically examined. Furthermore, we illustrate some 2D figures to show the phase portriat and bifurcations of this model at various points.

A structure theorem for generalized-noncontextual ontological models

It is useful to have a criterion for when the predictions of an operational theory should be considered classically explainable. Here we take the criterion to be that the theory admits of a generalized-noncontextual ontological model. Existing works on generalized noncontextuality have focused on experimental scenarios having a simple structure: typically, prepare-measure scenarios. Here, we formally extend the framework of ontological models as well as the principle of generalized noncontextuality to arbitrary compositional scenarios. We leverage a process-theoretic framework to prove that, under some reasonable assumptions, every generalized-noncontextual ontological model of a tomographically local operational theory has a surprisingly rigid and simple mathematical structure — in short, it corresponds to a frame representation which is not overcomplete. One consequence of this theorem is that the largest number of ontic states possible in any such model is given by the dimension of the associated generalized probabilistic theory. This constraint is useful for generating noncontextuality no-go theorems as well as techniques for experimentally certifying contextuality. Along the way, we extend known results concerning the equivalence of different notions of classicality from prepare-measure scenarios to arbitrary compositional scenarios. Specifically, we prove a correspondence between the following three notions of classical explainability of an operational theory: (i) existence of a noncontextual ontological model for it, (ii) existence of a positive quasiprobability representation for the generalized probabilistic theory it defines, and (iii) existence of an ontological model for the generalized probabilistic theory it defines.

Comparative analysis of two intergranular strain-based hypoplastic models through elemental and centrifuge testing

While hypoplastic models have demonstrated accurate predictions of sand behavior under monotonic loading, their accuracy diminishes when applied to cyclic loading conditions. To address this limitation, the intergranular strain approach is used as an extension to the model. The current investigation focuses on the analysis of two variants: the original Intergranular Strain (IS) approach proposed by Niemunis and Herle (1997) and the Intergranular Strain Anisotropy (ISA) by Fuentes et al. (2019). Although both models have the same objective, they present distinct mathematical structures and therefore different repercussions on the simulations. In this study, sand Hypoplasticity is enhanced with IS and ISA, and employed to simulate a series of experimental tests conducted on Fontainebleau sand. These tests encompass isotropic compression, drained monotonic triaxial, and undrained cyclic triaxial tests, while considering different initial densities and test characteristics. Furthermore, the calibrated models were applied to simulate a series of centrifuge tests, involving a pile embedded in the same sand, which is subjected to various episodes of monotonic and cyclic lateral loading. A comparison and discussion of the similarities and differences in elemental and finite element predictions, arising from the two intergranular strain formulations is presented.

A value for cooperative games on simplicial complexes with a filtration

The classical Shapley value for cooperative games determines a payoff vector considering that the formation of the grand coalition is made by incorporating players one by one. Later, this method was generalized for games with restricted cooperation by several known mathematical structures: partitions, graphs, convex geometries, antimatroids, matroids or simplicial complexes. In this paper we consider games over simplicial complex with an extra information about the relationships of the agents, a filtration of the complex. Filtrations are very known simplicial structures in the social and neuronal networks. We propose a Shapley value for these situations and an axiomatization.

Game-Based Learning experiences in primary mathematics education

Using game-based learning (GBL), especially digital game-based learning (DGBL), as a teaching and learning environment can be a pedagogical resource and a good strategy in the classroom to support mathematical learning. Effective manipulatives and games play a crucial role in promoting mathematical understanding. They support students in building, reinforcing and connecting varied representations of mathematical concepts. High-quality games are particularly valuable for learners as they provide them with control and adaptability. These games have properties that are adapted to cognitive and mathematical structures, facilitating the development of connections between different pieces and forms of knowledge. Digital games can help to achieve the same effects. In this paper, we conduct a quasi-experiment using games developed for this purpose. Our aim is to investigate whether non-digital games vs. digital games yield different results. Our results indicate that while students enjoyed themselves and found the task-solving enjoyable during both types of game-based learning, the use of non-digital games vs. digital games can sometimes lead to different outcomes.

Theoretical-methodological approaches and textbook design: analysis of arithmetic word problems in Spanish textbooks

Primary school textbooks can enhance the acquisition of arithmetic word problem solving skills by offering diverse problems based on their semantic-mathematical structure with targeted reasoning aids, including schematics highlighting their mathematical structure. While certain countries, such as the USA and Singapore, have made progress in improving the problems and aids found in their textbooks through the use of specific theoretical-methodological approaches, textbooks from other countries, such as Spain, have included a very limited variety of problems, with hardly any aids to reasoning. Recently, however, two of the most widely used Spanish publishers have released textbooks that adhere to these theoretical-methodological approaches. To assess whether these textbooks progressed past their predecessors in relevant aspects related to the resolution of arithmetic word problems, we conducted an analysis of the quantity of problems and their variety in terms of semantic-mathematical structure and level of difficulty, as well as the inclusion of schematic representations of their mathematical structure. The study demonstrated improvements among textbooks when publishers adopted a theoretical framework, suggesting that a reference framework could enhance textbook design. This is particularly relevant in countries such as Spain, where there are no applicable standards or official curricula for designing textbooks related to solving arithmetic word problems.

Optimized Distance-Based Algorithm for Cloud Space—Tourist Route Recommendation for Heritage Town, Pondicherry, India

Architecture, often regarded as a three-dimensional art form, relies on a multitude of abstractions during the design process. In this context, architectural elements are effectively represented through vector-based data processing techniques, offering a canvas for creative exploration. To further our understanding of spatial configurations and generate multiple design insights, mathematical structures known as graphs come into play. Graph theory, a fundamental component of this process, it is a mathematical field that studies relationships and connections between objects using vertices and edges, finding applications in computer science, network analysis, and more finds application in architectural design. This study delves into the application of graph theory in the analytical exploration of urban networks, with a focus on tourism parameters in the picturesque town of Pondicherry, Tamil Nadu, India. Traditionally, urban network analysis seeks to determine the shortest distance between origin and destination, a metric rooted in accessibility. However, in leisure destinations, a novel approach involving an optimized distance-based algorithm is introduced. This innovative methodology aims to facilitate efficient access to all tourist destinations within the spatial extent while minimizing discrete physical distances. To illustrate this concept, we take the iconic French Quarters in the White Town of Pondicherry as a case study. By applying the optimized distance-based algorithm, we endeavor to uncover the most efficient route for tourists to navigate and leisurely explore the area. The outcome of this analysis reveals a path graph structure, which, when compared to a circular graph structure, demonstrates superior efficiency in guiding tourists through the space. The implications of this tourist route extend beyond the realm of tourism itself. It has the potential to enhance the organization of tourist activities, allowing stakeholders to manage and regulate the flow of visitors. Additionally, it presents an opportunity to promote and develop the region’s often-underestimated tourist destinations, which can, in turn, have a positive impact on the local economy. In summary, the integration of graph theory into architectural and urban design methodologies enriches the approach to tourism and offers a holistic perspective on spatial optimization.

Automatic procedure for modelling, calibration, and optimization of a three-component chromatographic separation

The current landscape of biopharmaceutical production necessitates an ever-growing set of tools to meet the demands for shorter development times and lower production costs. One path towards meeting these demands is the implementation of digital tools in the development stages. Mathematical modelling of process chromatography, one of the key unit operations in the biopharmaceutical downstream process, is one such tool. However, obtaining parameter values for such models is a time-consuming task that grows in complexity with the number of compounds in the mixture being purified.In this study, we tackle this issue by developing an automated model calibration procedure for purification of a multi-component mixture by linear gradient ion exchange chromatography. The procedure was implemented using the Orbit software (Lund University, Department of Chemical Engineering), which both generates a mathematical model structure and performs the experiments necessary to obtain data for model calibration. The procedure was extended to suggest operating points for the purification of one of the components in the mixture by means of multi-objective optimization using three different objectives. The procedure was tested on a three-component protein mixture and was able to generate a calibrated model capable of reproducing the experimental chromatograms to a satisfactory degree, using a total of six assays. An additional seventh experiment was performed to validate the model response under one of the suggested optimum conditions, respecting a 95 % purity requirement. All of the above was automated and set in motion by the push of a button. With these results, we have taken a step towards fully automating model calibration and thus accelerating digitalization in the development stages of new biopharmaceuticals.

What is a mathematical structure of conscious experience?

Several promising approaches have been developed to represent conscious experience in terms of mathematical spaces and structures. What is missing, however, is an explicit definition of what a ‘mathematical structure of conscious experience’ is. Here, we propose such a definition. This definition provides a link between the abstract formal entities of mathematics and the concreta of conscious experience; it complements recent approaches that study quality spaces, qualia spaces, or phenomenal spaces; and it provides a general method to identify and investigate structures of conscious experience. We hope that ultimately this work provides a basis for developing a common formal language to study consciousness.

A multiscale modeling framework for Scenario Modeling: Characterizing the heterogeneity of the COVID-19 epidemic in the US

The Scenario Modeling Hub (SMH) initiative provides projections of potential epidemic scenarios in the United States (US) by using a multi-model approach. Our contribution to the SMH is generated by a multiscale model that combines the global epidemic metapopulation modeling approach (GLEAM) with a local epidemic and mobility model of the US (LEAM-US), first introduced here. The LEAM-US model consists of 3142 subpopulations each representing a single county across the 50 US states and the District of Columbia, enabling us to project state and national trajectories of COVID-19 cases, hospitalizations, and deaths under different epidemic scenarios. The model is age-structured, and multi-strain. It integrates data on vaccine administration, human mobility, and non-pharmaceutical interventions. The model contributed to all 17 rounds of the SMH, and allows for the mechanistic characterization of the spatio-temporal heterogeneities observed during the COVID-19 pandemic. Here we describe the mathematical and computational structure of our model, and present the results concerning the emergence of the SARS-CoV-2 Alpha variant (lineage designation B.1.1.7) as a case study. Our findings show considerable spatial and temporal heterogeneity in the introduction and diffusion of the Alpha variant, both at the level of individual states and combined statistical areas, as it competes against the ancestral lineage. We discuss the key factors driving the time required for the Alpha variant to rise to dominance within a population, and quantify the impact that the emergence of the Alpha variant had on the effective reproduction number at the state level. Overall, we show that our multiscale modeling approach is able to capture the complexity and heterogeneity of the COVID-19 pandemic response in the US.