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6 results for "Asymptote and Nonlinearity"
Nonlinearity in Life and Biomedical Sciences By Ralph W Lai
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New nonlinear concepts for data analysis in life and biomedical sciences are introduced. Biological science is loaded with data with various curves: C-curves, S-curves, and symmetric and asymmetric... More > bell curves, etc,; these curves are nonlinear curves. Different types of cuves are outcome of different nonlinear physical phenomena that can have different degree of nonlinearity and different ways of meauring nonlinearity. All these nonlinear phenomena can be assembled into groups and described by simple nonlinear equations. A nonlinear phenomenon needs to be described by multiple graphs in association with nonlinear equations. The book discusses the fundamentals of nonlinearity followed by describing the methodoogy for analyzing the nonlinear data. The book is based on two mathematical axioms and two universal standards for linear and nonlinear measurements. A series of Proportionality laws along wth the GVP (graph based, true value compared, and proportionality oriented) math system are discussed.< Less
Nonlinearity and Proportionality in Science, Medicine, and Engineering By Ralph Lai
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This book introduces a new αβ math for describing physical laws in science, medicine, and engineering. The αβ math is an extension to the traditional XY math to address the... More > nonlinearity of the continuous numbers. The book starts with introducing the fundamentals of nonlinearity followed by describing the methodology for analyzing the nonlinear phenomena.In the αβ math, the fundamentals of nonlinearity are based on two mathematical axioms and two universal standards for linear and nonlinear measurements. A series of proportionality laws, in conjunction with a Graph-based, true-Value compared, and Proportionality-oriented (GVP) math system, are introduced for assisting in data analyses. The new αβ math is built on the classification of continuous numbers into linear and nonlinear numbers. The linear numbers are defined as the continuous numbers that have no association with any asymptotes; and the nonlinear numbers are the continuous numbers that have association with one or two asymptotes.< Less
Azimuthal Modulational Instability of Vortex Solutions to the Two Dimensional Nonlinear Schrödinger Equation By Ronald Meyer Caplan
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We study the azimuthal modulational instability (MI) of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying... More > the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a nonlinear equation numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS performed on a polar coordinate finite-difference scheme.< Less
Azimuthal Modulational Instability of Vortex Solutions to the Two Dimensional Nonlinear Schrödinger Equation By Ronald Meyer Caplan
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We study the azimuthal modulational instability (MI) of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying... More > the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a nonlinear equation numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS performed on a polar coordinate finite-difference scheme.< Less
College Algebra & Trigonometry By A. A. Frempong
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This book covers both algebra and trigonometry. The topics include the following: Polynomial, Nonlinear, and Radical Equations; Sets, Relations, Functions; Absolute Value Equations and Inequalities;... More > Linear Programming; Graphs of Functions; Asymptotes; Logarithms; Exponential and Logarithmic Equations; Graphs of Exponential and Logarithmic Functions; Matrix and Matrix Methods; Determinants; Complex Numbers and Operations; Polar Form of Complex Numbers; Roots of Complex Numbers; Graphing Polar Coordinates and Equations; Conic sections;; Remainder and Factor Theorems; Rational Roots; Partial Fractions; Sequences and Series; Binomial Theorem; Permutations and Combinations; Mathematical Induction; Right Triangle Trigonometry; Trigonometry of Real Numbers; Graphs of Trigonometric Functions; Graphs of Inverse Trigonometric Functions; Trigonometric identities and Equations.< Less
College Algebra By A. A. Frempong
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This book covers college algebra. The topics include the following: Polynomial, Nonlinear, and Radical Equations; Sets, Relations, Functions; Absolute Value Equations and Inequalities; Linear... More > Programming; Graphs of Functions; Asymptotes; Logarithms; Exponential and Logarithmic Equations; Graphs of Exponential and Logarithmic Functions; Matrix and Matrix Methods; Determinants; Complex Numbers and Operations; Polar Form of Complex Numbers; Roots of Complex Numbers; Graphing Polar Coordinates and Equations; Conic sections;; Remainder and Factor Theorems; Rational Roots; Partial Fractions; Sequences and Series; Binomial Theorem; Permutations and Combinations; and Mathematical Induction;< Less