Iskanje po katalogu digitalne knjižnice

Opis: The main goal of the master's thesis was the analysis of air resistance on the body in motion in a model that does not require solving the Navier-Stokes equations, but works on the basis of mechanics and statistical physics. The model was a Monte Carlo (MC) simulation of the motion of ideal gas molecules in a closed container in which a body was placed, moving along one of the axes. For the most part of calculations, the approach was used when the body was fixed in the middle of the simulation cell, and one of the components of the molecular velocity had an additional term that simulated the flow, as if the body was moving at this speed in the opposite direction. First of all, a linear dependence of the drag force on speed was found for low flow speed for a flat plate, which was predicted by linear drag law. For high molecular flow rates, the quadratic dependence predicted by the Bernoulli equation was clearly observed. The results of calculating the corresponding resistivity coefficients for the flat plate were in agreement with the analytical values for both regimes of speeds. By analogy, a simulation was made for a spherical body, which also demonstrated a strong quadratic dependence at high speeds and the drag coefficient value is approximately equal to the analytical one. In the following, we studied systematically ellipsoids with circular cross-section, where we varied the ratio between semiaxes in the direction of motion and perpendicular direction, respectively. The results for the ellipsoid showed that the drag coefficient value is maximum for a flat plate (a limiting case of an ellipsoid, when the semiaxis in the direction of motion tends to 0) and decreases with stretching of the body along the flow axis. When the Maxwell distribution of molecular speeds that was mainly used was replaced with uniform Root-Mean-Square (RMS) speed the results for drag coefficient were slightly different.
Ključne besede: Air resistance, drag force, quadratic drag law, drag coefficient, Monte Carlo (MC) simulation, Maxwell distribution.
Objavljeno v DKUM: 13.10.2021; Ogledov: 458; Prenosov: 32
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Opis: When considering friction or resistance, many physical processes are mathematically simulated by quadratic systems of ODEs or discrete quadratic dynamical systems. Probably the most important problem when such systems are applied in engineering is the stability of critical points and (non)chaotic dynamics. In this paper we consider homogeneous quadratic systems via the so-called Markus approach. We use the one-to-one correspondence between homogeneous quadratic dynamical systems and algebra which was originally introduced by Markus in (1960). We resume some connections between the dynamics of the quadratic systems and (algebraic) properties of the corresponding algebras. We consider some general connections and the influence of power-associativity in the corresponding quadratic system.
Ključne besede: quadratic systems, nonlinear systems
Objavljeno v DKUM: 14.06.2017; Ogledov: 787; Prenosov: 359
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Opis: Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the well-studied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size.
Ključne besede: matematično programiranje, problem kvadratičnega prirejanja, kopozitivno programiranje, semidefinitna poenostavitev, quadratic assignment problem, copositive programming, semidefinite relaxations, lift-and-project relaxations
Objavljeno v DKUM: 10.07.2015; Ogledov: 1146; Prenosov: 96
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Opis: A polynomial ▫$f$▫ in non-commuting variables is trace-positive if the trace of ▫$f(underline{A})$▫ is positive for all tuples ▫$underline{A}$▫ of symmetric matrices of the same size. The investigation of trace-positive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from statistical quantum mechanics and the embedding conjecture of Alain Connes concerning von Neumann algebras. First, results on the question of when a trace-positive polynomial in two non-commuting variables can be written as a sum of hermitian squares and commutators are presented. For instance, any bivariate trace-positive polynomial of degree at most four has such a representation, whereas this is false in general if the degree is at least six. This is in perfect analogy to Hilbert's results from the commutative context. Further, a partial answer to the Lieb-Seiringer formulation of the BMV conjecture is given by presenting some concrete representations of the polynomials ▫$S_{m,4}(X^2; Y^2)$▫ as a sum of hermitian squares and commutators. The second part of this work deals with the tracial moment problem. That is, how can one describe sequences of real numbers that are given by tracial moments of a probability measure on symmetric matrices of a fixed size. The truncated tracial moment problem, where one considers only finite sequences, as well as the tracial analog of the ▫$K$▫-moment problem are also investigated. Several results from the classical moment problem in Functional Analysis can be transferred to this context. For instance, a tracial analog of Haviland's theorem holds: A traciallinear functional ▫$L$▫ is given by the tracial moments of a positive Borel measure on symmetric matrices of a fixed size s if and only if ▫$L$▫ takes only positive values on all polynomials which are trace-positive on all tuples of symmetric ▫$s times s$▫-matrices. This result uses tracial versions of the results of Fialkow and Nie on positive extensions of truncated sequences. Further, tracial analogs of results of Stochel and of Bayer and Teichmann are given. Defining a tracial Hankel matrix in analogy to the Hankel matrix in the classical moment problem, the results of Curto and Fialkow concerning sequences with Hankel matrices of finite rank or Hankel matrices of finite size which admit a flat extension also hold true in the tracial context. Finally, a relaxation for trace-minimization of polynomials using sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, the tracial analogs of the results of Curto and Fialkow give a sufficient condition for the exactness of this relaxation.
Ključne besede: matematika, algebra, polinomi s pozitivno sledjo, prosta algebra, nekomutativni polinom, centralna enostavna algebra, reducirana sled, polinomska identiteta, kvadratna forma, prosta pozitivnost, vsota hermitskih kvadratov, problem momentov, mathematics, algebra, free algebra, noncommutative polynomial, central simple algebra, (reduced) trace, polynomial identity, central polynomial, quadratic form, free positivity, sum of hermitian squares, (truncated) moment problem
Objavljeno v DKUM: 10.07.2015; Ogledov: 1279; Prenosov: 120
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Ključne besede: linear matrix inequality (LMI), completely positive, semidefinite programming, Positivstellensatz, Gleichstellensatz, archimedean quadratic module, free positivity
Objavljeno v DKUM: 10.07.2015; Ogledov: 865; Prenosov: 43
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