scholarly journals Mathematical Modelling by Help of Category Theory: Models and Relations between Them

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1946
Author(s):  
Dmitrii Legatiuk

The growing complexity of modern practical problems puts high demand on mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice is becoming particularly important. Methods for model comparison and model choice typically used in practical applications nowadays are computation-based, and thus time consuming and computationally costly. Therefore, it is necessary to develop other approaches to working abstractly, i.e., without computations, with mathematical models. An abstract description of mathematical models can be achieved by the help of abstract mathematics, implying formalisation of models and relations between them. In this paper, a category theory-based approach to mathematical modelling is proposed. In this way, mathematical models are formalised in the language of categories, relations between the models are formally defined and several practically relevant properties are introduced on the level of categories. Finally, an illustrative example is presented, underlying how the category-theory based approach can be used in practice. Further, all constructions presented in this paper are also discussed from a modelling point of view by making explicit the link to concrete modelling scenarios.

2019 ◽  
Vol 6 (3) ◽  
pp. 181848 ◽  
Author(s):  
I. Kavrakov ◽  
D. Legatiuk ◽  
K. Gürlebeck ◽  
G. Morgenthal

Reliable modelling in structural engineering is crucial for the serviceability and safety of structures. A huge variety of aerodynamic models for aeroelastic analyses of bridges poses natural questions on their complexity and thus, quality. Moreover, a direct comparison of aerodynamic models is typically either not possible or senseless, as the models can be based on very different physical assumptions. Therefore, to address the question of principal comparability and complexity of models, a more abstract approach, accounting for the effect of basic physical assumptions, is necessary. This paper presents an application of a recently introduced category theory-based modelling approach to a diverse set of models from bridge aerodynamics. Initially, the categorical approach is extended to allow an adequate description of aerodynamic models. Complexity of the selected aerodynamic models is evaluated, based on which model comparability is established. Finally, the utility of the approach for model comparison and characterization is demonstrated on an illustrative example from bridge aeroelasticity. The outcome of this study is intended to serve as an alternative framework for model comparison and impact future model assessment studies of mathematical models for engineering applications.


Author(s):  
David I. Spivak

Category theory is presented as a mathematical modelling framework that highlights the relationships between objects, rather than the objects in themselves. A working definition of model is given, and several examples of mathematical objects, such as vector spaces, groups, and dynamical systems, are considered as categorical models.


2021 ◽  
Vol 15 (5) ◽  
pp. e0009449
Author(s):  
Maylis Layan ◽  
Simon Dellicour ◽  
Guy Baele ◽  
Simon Cauchemez ◽  
Hervé Bourhy

Background Rabies is a fatal yet vaccine-preventable disease. In the last two decades, domestic dog populations have been shown to constitute the predominant reservoir of rabies in developing countries, causing 99% of human rabies cases. Despite substantial control efforts, dog rabies is still widely endemic and is spreading across previously rabies-free areas. Developing a detailed understanding of dog rabies dynamics and the impact of vaccination is essential to optimize existing control strategies and developing new ones. In this scoping review, we aimed at disentangling the respective contributions of mathematical models and phylodynamic approaches to advancing the understanding of rabies dynamics and control in domestic dog populations. We also addressed the methodological limitations of both approaches and the remaining issues related to studying rabies spread and how this could be applied to rabies control. Methodology/principal findings We reviewed how mathematical modelling of disease dynamics and phylodynamics have been developed and used to characterize dog rabies dynamics and control. Through a detailed search of the PubMed, Web of Science, and Scopus databases, we identified a total of n = 59 relevant studies using mathematical models (n = 30), phylodynamic inference (n = 22) and interdisciplinary approaches (n = 7). We found that despite often relying on scarce rabies epidemiological data, mathematical models investigated multiple aspects of rabies dynamics and control. These models confirmed the overwhelming efficacy of massive dog vaccination campaigns in all settings and unraveled the role of dog population structure and frequent introductions in dog rabies maintenance. Phylodynamic approaches successfully disentangled the evolutionary and environmental determinants of rabies dispersal and consistently reported support for the role of reintroduction events and human-mediated transportation over long distances in the maintenance of rabies in endemic areas. Potential biases in data collection still need to be properly accounted for in most of these analyses. Finally, interdisciplinary studies were determined to provide the most comprehensive assessments through hypothesis generation and testing. They also represent new avenues, especially concerning the reconstruction of local transmission chains or clusters through data integration. Conclusions/significance Despite advances in rabies knowledge, substantial uncertainty remains regarding the mechanisms of local spread, the role of wildlife in dog rabies maintenance, and the impact of community behavior on the efficacy of control strategies including vaccination of dogs. Future integrative approaches that use phylodynamic analyses and mechanistic models within a single framework could take full advantage of not only viral sequences but also additional epidemiological information as well as dog ecology data to refine our understanding of rabies spread and control. This would represent a significant improvement on past studies and a promising opportunity for canine rabies research in the frame of the One Health concept that aims to achieve better public health outcomes through cross-sector collaboration.


2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Hong Zeng ◽  
Reza Rohani ◽  
Wei E. Huang ◽  
Aidong Yang

Abstract Background The rising consensus that the cell can dynamically allocate its resources provides an interesting angle for discovering the governing principles of cell growth and metabolism. Extensive efforts have been made in the past decade to elucidate the relationship between resource allocation and phenotypic patterns of microorganisms. Despite these exciting developments, there is still a lack of explicit comparison between potentially competing propositions and a lack of synthesis of inter-related proposals and findings. Results In this work, we have reviewed resource allocation-derived principles, hypotheses and mathematical models to recapitulate important achievements in this area. In particular, the emergence of resource allocation phenomena is deciphered by the putative tug of war between the cellular objectives, demands and the supply capability. Competing hypotheses for explaining the most-studied phenomenon arising from resource allocation, i.e. the overflow metabolism, have been re-examined towards uncovering the potential physiological root cause. The possible link between proteome fractions and the partition of the ribosomal machinery has been analysed through mathematical derivations. Finally, open questions are highlighted and an outlook on the practical applications is provided. It is the authors’ intention that this review contributes to a clearer understanding of the role of resource allocation in resolving bacterial growth strategies, one of the central questions in microbiology. Conclusions We have shown the importance of resource allocation in understanding various aspects of cellular systems. Several important questions such as the physiological root cause of overflow metabolism and the correct interpretation of ‘protein costs’ are shown to remain open. As the understanding of the mechanisms and utility of resource application in cellular systems further develops, we anticipate that mathematical modelling tools incorporating resource allocation will facilitate the circuit-host design in synthetic biology.


Crystals ◽  
2020 ◽  
Vol 10 (6) ◽  
pp. 538
Author(s):  
Maria Cristina Burla ◽  
Benedetta Carrozzini ◽  
Giovanni Luca Cascarano ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori

In this study, the properties of observed, difference, and hybrid syntheses (hybrid indicates a combination of observed and difference syntheses) are investigated from two points of view. The first has a statistical nature and aims to estimate the amplitudes of peaks corresponding to the model atoms, belonging or not belonging to the target structure; the amplitudes of peaks related to the target atoms, missed or shared with the model; and finally, the quality of the background. The latter point deals with the practical features of Fourier syntheses, the special role of weighted syntheses, and their usefulness in practical applications. It is shown how the properties of the various syntheses may vary according to the available structural model and, in particular, how weighted hybrid syntheses may act like an observed and difference or a full hybrid synthesis. The theoretical results obtained in this paper suggest new Fourier syntheses using novel Fourier coefficients: their main features are first discussed from a mathematical point of view. Extended experimental applications show that they meet the basic mission of the Fourier syntheses, enhancing peaks corresponding to the missed target atoms, depleting peaks corresponding to the model atoms not belonging to the target, and significantly reducing the background. A comparison with the results obtained via the most popular modern Fourier syntheses is made, suggesting a role for the new syntheses in modern procedures for phase extension and refinement. The most promising new Fourier synthesis has been implemented in the current version of SIR2014.


2020 ◽  
Vol 41 (6supl2) ◽  
pp. 3391-3396
Author(s):  
Valiollah Palang ◽  
◽  
Maghsoud Besharati ◽  

our mathematical models were used to describe the ruminal disappearance of dry matter (DM) and crude protein (CP) of fish meal and cottonseed meal. Results of DM degradability particularity showed that all the models fitted well (R2 > 0.95), however, considering that values below 0 or above 100 are not biologically justified in ruminal degradability, they are not acceptable. The models I and II were accepted to ruminal DM degradability of fish meal and cottonseed meal data. Only models I and II were successfully fitted to CP degradability of fish meal (R2 > 0.96), and the I, II and III models were acceptable to ruminal CP degradability of cottonseed meal (R2 > 0.98). In terms of effective degradability (ED) of DM and CP, model II generated higher values than other models. To appreciate fully the role of mathematical modelling in the biological sciences, it is necessary to consider the nature of feeds that evaluated and to review the types of models that may be constructed.


2020 ◽  
Vol XXIII (1) ◽  
pp. 8-14
Author(s):  
Vasilescu Mircea Dorin

The purpose scope of this paper is to analyse the economic aspects of making 3D printing components for the development of machine for experimental devices or stands used in laboratory or industrial activity. The author of this paper intends to make a clarification related to the approach of the productive economic aspects through the production of components through the additive technology of SLA type. The work is structured in 5 chapters. For the generation of the constructive elements, a laser printer will be used, where the components will be generated on the bottom-up principle. The analysis will be carried out both from a constructive and functional point of view, taking into account possibilities of component design and their realization. The next chapter is allocated to analysing how to perform the printer's command program and to check the part structure. Part of this chapter is devoted to the way and role of the elements of orientation and support of the component that is being made. The fourth chapter presents the economic calculation program made with practical applications related to the parts generated in the previous chapter. The last chapter is allocated to the conclusions and comparisons from the technological-economic point of view compared with other specific technologies for generating the analysed parts.


Author(s):  
N.V. Belov ◽  
U.I. Papiashwili ◽  
B.E. Yudovich

It has been almost universally adopted that dissolution of solids proceeds with development of uniform, continuous frontiers of reaction.However this point of view is doubtful / 1 /. E.g. we have proved the active role of the block (grain) boundaries in the main phases of cement, these boundaries being the areas of hydrate phases' nucleation / 2 /. It has brought to the supposition that the dissolution frontier of cement particles in water is discrete. It seems also probable that the dissolution proceeds through the channels, which serve both for the liquid phase movement and for the drainage of the incongruant solution products. These channels can be appeared along the block boundaries.In order to demonsrate it, we have offered the method of phase-contrast impregnation of the hardened cement paste with the solution of methyl metacrylahe and benzoyl peroxide. The viscosity of this solution is equal to that of water.


2009 ◽  
pp. 4-27
Author(s):  
A. Cohen ◽  
G. Harcourt

The article written by the well-known theorists and historians of economic thought contains a detailed overview of the Cambridge capital controversy, which had raged from the mid-1950-s through the mid-1970-s. The authors track the origins of the controversy and cover arguments of both sides in chronological order. From their point of view, the discussion hasnt been resolved, and its main underlying aspects were ideological beliefs and fundamental methodological controversies on the nature of equilibrium and on the role of time in economic theory. The article is published with comments written by other leading theoreticians.


2019 ◽  
pp. 22-29
Author(s):  
Н. В. Фрадкіна

The purpose and tasks of the work are to analyze the contemporary Ukrainian mass culture in terms of its value and humanistic components, as well as the importance of cultural studies and Ukrainian studies in educational disciplines for the formation of a holistic worldview of modern youth.Analysis of research and publications. Scientists repeatedly turned to the problems of the role of spirituality in the formation of society and its culture. This problem is highlighted in the publications by O. Losev, V. Lytvyn, D. Likhachev, S. Avierintsev, M. Zakovych, I. Stepanenko and E. Kostyshyn.Experts see the main negative impact of mass culture on the quality approach, which determines mass culture through the market, because mass culture, from our point of view, is everything that is sold and used in mass demand.One of the most interesting studies on this issue was the work by the representatives of Frankfurt School M. Horkheimer and T. Adorno «Dialectics of Enlightenment» (1947), devoted to a detailed analysis of mass culture. Propaganda at all socio-cultural levels in the form is similar in both totalitarian and democratic countries. It is connected, according to the authors, with the direction of European enlightenment. The tendency to unify people is a manifestation of the influence of mass culture, from cinema to pop. Mass culture is a phenomenon whose existence is associated with commerce (accumulation in any form – this is the main feature of education), in general, the fact that it exists in this form is related to the direction of the history of civilization.Modern mass culture, with its externally attractive and easily assimilated ideas and symbols, appealing to the trends of modern fashion, becomes a standard of prestigious consumption, does not require intense reflection, allows you to relax, distract, not teach, but entertains, preaches hedonism as the main spiritual value. And as a consequence, there are socio-cultural risks: an active rejection of other people, which leads to the formation of indifference; cruelty as a character trait; increase of violent and mercenary crime; increase in the number of alcohol and drug addicts; anti-patriotism; indifference to the values of the family and as a result of social orphanhood and prostitution.Conclusions, perspectives of research. Thus, we can conclude that modern Ukrainian education is predominantly formed by the values of mass culture. Namely, according to the «Dialectic» by Horkheimer and Adorno, «semi-enlightenment becomes an objective spirit» of our modern society.It is concluded that only high-quality education can create the opposite of the onset of mass culture and the destruction of spirituality in our society. It is proved that only by realizing the importance of cultivating disciplines in the educational process and the spiritual upbringing of the nation, through educational reforms, humanitarian knowledge will gradually return to student audiences.Formation of youth occurs under the influence of social environment, culture, education and self-education. The optimal combination of these factors determines both the process of socialization itself and how successful it will be. In this context, one can see the leading role of education and upbringing. It turns out that the main task of modern education is to spread its influence on the development of spiritual culture of the individual, which eventually becomes a solid foundation for the formation of the individual. Such a subject requires both philosophical and humanitarian approaches in further integrated interdisciplinary research, since the availability of such research will provide the theoretical foundation for truly modern educational and personal development.


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