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<\/figure>\n<\/div>\n\n\nAs a rule, a good mathematical model is complex, so it requires a good knowledge and work with certain numerical software.\u00a0After creating a mathematical model and obtaining a numerical solution, opportunities for simulations open up.\u00a0Simulation represents the execution of operations on the system model in order to realise its behavior.\u00a0This is performed in a simulation software characterized by a user-friendly graphical interface.\u00a0It also allows modifications of the model parameters and components, and provides a representation of the obtained results in the form of graphs, tables, diagrams and similar ways. <\/p>\n\n\n\n
The advantages of the mathematical model compared to the classical experiment are numerous.\u00a0Experiments are sometimes impossible to implement, for example in<\/p>\n\n\n\n
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- climatology - study of sea currents and winds, <\/a><\/li>\n\n\n\n
- economics - market development research,<\/li>\n\n\n\n
- medicine - making new materials for implants, planning operations.<\/a><\/li>\n<\/ul>\n\n\n
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<\/figure>\n<\/div><\/div><\/div>\n\n\n\n\nOn the other hand, in some situations experiments are undesirable, for example in<\/p>\n\n\n\n
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- working with radioactive material,<\/li>\n\n\n\n
- testing the stability of buildings,<\/li>\n\n\n\n
- studying natural disasters. <\/li>\n<\/ul>\n\n\n\n
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<\/figure>\n<\/div><\/div><\/div>\n\n\n\n\nExperiments can be expensive and time-consuming, for example for the purpose of studying<\/p>\n\n\n\n
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- the effect of radiation on genetic material,<\/li>\n\n\n\n
- crystal structures and macromolecules,<\/li>\n\n\n\n
- turbulence in an aerodynamic wind tunnel,<\/a><\/li>\n\n\n\n
- the behavior of cars in a crash.<\/a><\/li>\n<\/ul>\n\n\n
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<\/figure>\n<\/div>\n\n\nIt is clear that the previously mentioned problems can only be solved by the application of Numerical mathematics, which will offer adequate numerical methods and the error estimation.<\/p>\n\n\n\n
Some of the innumerable problems that, after successfully setting up of a mathematical model, are solved by methods of Optimization are the following: <\/p>\n\n\n\n
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- the problem of choosing combat strategies and assets in the army,<\/li>\n\n\n\n
- the problem of determining locations for the construction of schools, hospitals, warehouses, industrial plants, bus stations, airports, shopping centers and similar public buildings,<\/li>\n\n\n\n
- designing transport networks: railway and road systems, postal networks, fast delivery systems, passenger and goods transportation in air traffic,<\/li>\n\n\n\n
- the problem of product packaging.<\/li>\n<\/ul>\n\n\n\n
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<\/figure>\n<\/div><\/div><\/div>\n\n\n\n\nIn the following examples, we illustrate some problem formulations that we can solve using the methods of Variation calculus and Optimal control:<\/p>\n\n\n\n
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- optimal travel time of light,<\/li>\n\n\n\n
- optimal trajectory of a ship or plane,<\/li>\n\n\n\n
- maximization of company profits.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<\/div><\/div>\n\n\n\n
Modelling of many practical problems requires use of the particular mathematical objects known as graphs, i.e., structures represented by a set of vertices and a set of edges. <\/p>\n\n\n\n
In real applications, the vertices represent particular objects (geographical locations, parts of the organizational system, persons or groups of persons), while the edges indicate the existence of some relationship between them.\u00a0Depending on the situation, the edges can represent a geographical connection, a communication channel, the existence of family ties between people, the possibility of transferring goods between locations, and so on. <\/p>\n\n\n\n
Graphs appear in many scientific disciplines.\u00a0They are especially important in electrical engineering and computer science in the construction of computer and telecommunication networks, organization and protection of data, presentation of electrical schemes, image analysis and shape recognition, etc.\u00a0It would not be reasonable to deal with mathematical models, numerical mathematics and optimization with bypassing the very significant, and closely related, field of graph theory.<\/p>\n\n\n\n
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\u201eThe way around the world\u201c on the graph of the dodecahedron<\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n \n![\"\"](\"http:\/\/nmo.matf.bg.ac.rs\/wp-content\/uploads\/2022\/02\/graf1.jpg\")
Determining the shortest paths between every two nodes<\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n In the last period, graphs have been used more and more in solving biological and social problems (preventing the spread of viral infections, increasing the positive influence on social networks...).\u00a0Graph theory includes a creation of algorithms in order to get an insight into the existing relations between entities and to recognize hidden patterns in the modelled system.<\/p>\n\n\n\n
Finding the shortest paths, dominant sets, strongly connected components are just some of the important tasks for graph analysis.<\/p>\n\n\n\n
- the behavior of cars in a crash.<\/a><\/li>\n<\/ul>\n\n\n