1,721,834 research outputs found
Generalized Logistic Equations in Covid-Related Epidemic Models
No full textThe logistic equation on population growth was proposed by Verhulst (Corresp Math Phys 10:113-126, 1838) [22], with the aim to provide a possible correction to the unrealistic exponential growth forecast by T. Malthus, (J Johnson, London, 1872) [13]. Population modeling became of particular interest in the 20 th century to biologists urged by limited means of sustenance and increasing human populations. Verhulst’s scheme was rediscovered by A. Lotka, (Elements of Mathematical Biology. Dover, New York, 1956) [12], as a simple model of a self-regulating population. Subsequently, the use of logistic dynamics spreads across a huge number of different frameworks, especially in diffusion phenomena. The logistic differential equation is a fundamental element in quantitative study of population dynamics, its use also extends to the field of epidemiology: both to describe the evolution of the infected population in deterministic models, and working in conditions of uncertainty it is the deterministic component of stochastic differential equations. This work brings a contribution to the foundational basic research on the logistic equation and its generalizations which hopefully have repercussions for epidemiologic applications
Trinomial equation: the Hypergeometric way.
Full text linkThis paper is devoted to the analytical treatment of trinomial equations of the form y^n + y = x, where y is the unknown and x∈C is a free parameter. It is well-known that, for degree n≥5, algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [1,2] for example) requires the use of power series, following the seminal work of Lagrange [3]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance
Introductory Mathematical Analysis for Quantitative Finance.
No full textThe purpose of this book is to be a tool for students, with little mathematical background, who aim to study Mathematical Finance. The only prerequisites assumed are one–dimensional differential calculus, infinite series, Riemann integral and elementary linear algebra.
In a sense, it is a sort of intensive course, or crash–course, which allows students, with minimal knowledge in Mathematical Analysis, to reach the level of mathematical expertise necessary in modern Quantitative Finance. These lecture notes concern pure mathematics, but the arguments presented are oriented to Financial applications. The n–dimensional Euclidean space is briefly introduced, in order to deal with multivariable differential calculus. Sequences and series of functions are introduced, in view of theorems concerning the passage to the limit in Measure theory, and their role in the general theory of ordinary differential equations, which is also presented. Due to its importance in Quantitative Finance, the Radon–Nykodim theorem is stated, without proof, since the Von Neumann argument requires notions of Functional Analysis, which would require a dedicated course. Finally, in order to solve the Black–Scholes partial differential equation, basics in ordinary differential equations and in the Fourier transform are provided.
We kept our exposition as short as possible, as the lectures are intended to be a preliminary contact with the mathematical concepts used in Quantitative Finance and provided, often, in a one–semester course. This book, therefore, is not intended for a specialized audience, although the material presented here can be used by both experts and non-experts, to have a clear idea of the mathematical tools used in Finance
Two approaches for evaluating the period function of some Hamiltonian systems
No full textThe theory of nonlinear oscillators is an extremely active field of contemporary research, given the vastness of the phenomena modelled through nonlinear differential equations with periodic solutions. In particular, the period function associated with Hamiltonian systems has been the subject of an outstanding piece of research.
Our contribution, here, concerns oscillatory systems ruled by odd–degree polynomial restoring forces. We present two approaches for the symbolic approximation of the energy–period function: one based on asymptotic expansions of the period function, and the other through the approximation obtained using fifth–order Chebyshev polynomials of the restoring force and the consequent closed–form solution of the approximate equation. The obtained approximation
quality is verified through different comparison methods illustrated in this chapter
FIRST ORDER DIFFERENTIAL EQUATIONS: EARLY HISTORY FOLLOWING JACOPO RICCATI AND GABRIELE MANFREDI
No full textn 1722 the Italian mathematician Jacopo Ric- cati (1676–1754) held a course of lectures on differential equations, which remained unpublished until 1761. They finally emerged, under the title Della separazione delle in- determinate nelle equazioni differenziali del primo grado e della riduzione delle equazioni differenziali del secondo e di altri gradi ulteriori, in the first of four tomes of his Opera Omnia. This treatise, following the Leibniz (1646–1716) tradition of seeking solutions finitely expressed in terms of known functions, focuses on the separation of variables, to integrate a broad class of ordinary differential equations, of first or higher order. Its first part, entitled Dei metodi inventati da varj celebri Autori per separare le indeterminate nelle equazioni differenziali del primo grado, makes a survey of all relevant integrating methods, known at that time. These are thanks to Gabriele Manfredi (1681–1761), Jakob Bernoulli (1654–1705), his younger brother Johann (1667– 1748), Vincenzo Riccati (1707–1775), son of Jacopo, and Jacopo himself: our paper considers those of them, which in our opinion are the smartest and most ingenious
The Wallis Products for Fermat Curves
Full text linkAfter revisiting the properties of generalized trigonometric functions, i.e., the trigonometric function linked to the planar (Fermat) curve , using the tool of Keplerian trigonometry, introduced in (Gambini et al.: Monatsh. Math. 195, 55–72, 2021), we present the extension to this class of functions of the Wallis product, discovering connections with the representations of ordinary trigonometric functions by means of infinite products
Unsteady roto-translational viscous flow: Analytical solution to Navier-Stokes equations in cylindrical geometry
No full textWe study the unsteady viscous flow of an incompressible, isothermal (Newtonian) fluid whose motion is induced by the sudden swirling of a cylindrical wall and is also starting with an axial velocity component. Basic physical assumptions are that the pressure axial gradient keeps its hydrostatic value and the radial velocity is zero. In such a way the Navier-Stokes PDEs become uncoupled and can be solved separately. Accordingly, we provide analytic solutions to the unsteady speed components, i.e., the axial vz(r, t) and the circumferential vθ(r, t), by means of expansions of Fourier-Bessel type under time damping. We also find: the surfaces of dynamical equilibrium, the wall shear stress during time and the Stokes streamlines
On a particular spiral similarity
Full text linkOne of the most well–known geometric transformations, constituting an important problem solving tool, is the so-called spiral similarity of a geometric figure. It allows the transformation, by means of a spiral movement around a point, of a figure F into another similar figure F', that is generated by a rotation and a simultaneous expansion of F.
Together with the rotation, the inverse transformation of spiral similarity produces, instead, a contraction of the original figure.
Aim of this work isp recisely the study of properties of this inverse transformation and its links with polygonal spirals. Here, this transformation is achieved through an entirely new construction, that can be aided and eased by means of computer algebra and graphics tools
A Structural Approach to Gudermannian Functions
Full text linkThe Gudermannian function relates the circular angle to the hyperbolic one when their cosines are reciprocal.
Whereas both such angles are halved areas of circular and hyperbolic sectors, it is natural to develop similar considerations within the study of a class of curves images of maps with constant areal speed.
After a brief exposition of some use of the Gudermannian in applied sciences, we proceed to illustrate the class of curves, called \emph{Keplerian curves}, which can be parametrised by a map \mmm = (\cos_{\mmm}, \sin_{\mmm}) whose areal speed is 1.
In the next Sections, after a detailed study of p-circular and hyperbolic Fermat curves \crv F_p and \crv F^*_p, we define the \emph{p-Gudermannian} as the primitive of the derivative of the p-hyperbolic sine divided by the square of the p-hyperbolic cosine: all the analogues of the classical identities are proven.
Having realized that such curves correspond to each other using a homology, we extend our study to a wide class of Keplerian curves and their homologues; once again, defined the Gudermannian
in an identical manner, all the analogues of classical identities subsist. Below, three examples are detailed.
The last paragraph further extends this consideration, eliminating the hypothesis that the curves
are parametrised by maps with areal speed 1.
The Appendix illustrates integrating techniques for systems defining the Fermat curves, and the determination of the inverse of their tangent function
Modeling Odd Nonlinear Oscillators with Fifth–Order Truncated Chebyshev Series
Full text linkThe aim of this work is to model the nonlinear dynamics of conservative oscillators, with restoring force originating from even-order potentials. In particular, we extend our previous findings on inverting the time-integral equation that arises in the solution of such dynamical systems, a task that is almost always intractable in exact form. This is faced and solved by approximating the restoring force with its Chebyshev series truncated to order five; such a quintication approach yields a quinticate oscillator, whose associated time-integral can be inverted in closed form. Our solution procedure is based on the quinticate oscillator coefficients, upon which a second-order polynomial is constructed, which appears in the time-integrand of the quinticate problem, and whose roots determine the expression of the closed-form solution, as well as that of its period. The presented algorithm is implemented in the Mathematica software and validated on some conservative nonlinear oscillators taken from the relevant literature
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