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1. Structuralism and the conformity of mathematics and nature.

Author

Stemeroff, Noah

Subjects
*CALCULUS of variations, *CONFORMITY, *APPLIED mathematics, *STRUCTURALISM, *MATHEMATICS
Abstract

Structuralists typically appeal to some variant of the widely popular 'mapping' account of mathematical representation to suggest that mathematics is applied in modern science to represent the world's physical structure. However, in this paper, I argue that this realist interpretation of the 'mapping' account presupposes that physical systems possess an 'assumed structure' that is at odds with modern physical theory. Through two detailed case studies concerning the use of the differential and variational calculus in modern dynamics, I show that the formal structure that we need to assume in order to apply the mapping account is inconsistent with the way in which mathematics is applied in modern physics. The problem is that a realist interpretation of the 'mapping' account imposes too severe of a constraint on the conformity that must exist between mathematics and nature in order for mathematics to represent the structure of a physical system. • This paper presents a critique of the 'mapping' account of mathematical representation. • The 'mapping' account requires that physical systems possess an assumed structure. • This structure is at odds with modern physical theory. • I argue that a realist interpretation of the account imposes a severe constraint on the conformity between math and nature. • The argument is defended through two case studies concerning the differential and variational calculus. [ABSTRACT FROM AUTHOR]

Published
2021
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2. Hamiltonian Systems in Dynamic Reconstruction Problems

Author

Nina N. Subbotina

Subjects
State variable, Current (mathematics), FUNCTIONALS, Dynamical systems theory, AUXILIARY PROBLEM, 02 engineering and technology, 01 natural sciences, CALCULUS OF VARIATIONS, Hamiltonian system, STATE VARIABLES, Applied mathematics, 0101 mathematics, Mathematics, DYNAMIC RECONSTRUCTION, Inverse problem, 021001 nanoscience & nanotechnology, HAMILTONIAN SYSTEMS, 010101 applied mathematics, CALCULATIONS, Nonlinear system, DYNAMICAL SYSTEMS, Control and Systems Engineering, HAMILTONIANS, OPTIMALITY CONDITIONS, Key (cryptography), Calculus of variations, 0210 nano-technology, INVERSE PROBLEMS, KEY ELEMENTS
Abstract

We consider dynamic reconstruction (DR) problems for controlled dynamical systems linear in controls and nonlinear in state variables as inaccurate current information about real motions is known. A solution of this on-line inverse problem is obtained with the help of auxiliary problems of calculus of variations (CV) for integral discrepancy functionals. Key elements of the constructions are solutions of hamiltonian systems obtained with the help of optimality conditions for the CV problems. An illustrating example is exposed. © 2018 17-01-00074, 18-01-00221 The work is supported by Russin Foundation for Basic Research (project no. 17-01-00074 and no. 18-01-00221).

Published
2018

3. A New Approximate Method for Construction of the Normal Control

Author

Evgeniy A. Krupennikov

Subjects
State variable, Approximations of π, NORMAL CONTROLS, 01 natural sciences, CALCULUS OF VARIATIONS, 0103 physical sciences, STATE VARIABLES, Applied mathematics, 0101 mathematics, CONTROL SYSTEMS, Lp space, Normal control, Mathematics, Computer simulation, VARIATIONAL PROBLEMS, CONTROL PARAMETERS, 010102 general mathematics, NORMAL CONTROL, NONLINEAR CONTROL SYSTEMS, HAMILTONIAN SYSTEMS, CALCULATIONS, Control and Systems Engineering, HAMILTONIANS, Control system, Norm (mathematics), APPROXIMATE METHODS, 010307 mathematical physics, NECESSARY OPTIMALITY CONDITION
Abstract

The problem of construction of the normal control (namely, the control with the least norm in L2 space) that generates a given trajectory of a control system is considered. A new method for constructing approximations of the normal control is suggested for a class of control systems with dynamics linear in controls and non-linear in state coordinates where the dimension of the control parameter is greater than or equal to the dimension of the state variables. This method relies on necessary optimality conditions in auxiliary variational problems. An illustrating example is exposed. The results of numerical simulation are compared with the results obtained with another approach. © 2018 Russian Foundation for Basic Research, RFBR 17-01-00074 Russian Foundation for Basic Research, RFBR Russian Foundation for Basic Research, RFBR: 17-01-00074, PRAS-18-01 This work is supported by the Russian Foundation for Basic Research (project no. 17-01-00074) and by the Program of the Presidium of the Russian Academy of Sciences 01 ’Fundamental Mathematics and its Applications’ under grant PRAS-18-01.

Published
2018