Further cryptographic properties of the multiplicative inverse function

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

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Investigation of a Stochastic Inverse Method to Estimate an External Force: Applications to a Wave-Structure Interaction

Mathematical Problems in Engineering ◽

10.1155/2012/175036 ◽

2012 ◽

Vol 2012 ◽

pp. 1-25 ◽

Cited By ~ 3

Author(s):

S. L. Han ◽

Takeshi Kinoshita

Keyword(s):

External Force ◽

Inverse Method ◽

Inverse Function ◽

Wave Structure ◽

Dynamic Responses ◽

Structure Interaction ◽

External Forces ◽

Interaction Problem ◽

Wave Structure Interaction

The determination of an external force is a very important task for the purpose of control, monitoring, and analysis of damages on structural system. This paper studies a stochastic inverse method that can be used for determining external forces acting on a nonlinear vibrating system. For the purpose of estimation, a stochastic inverse function is formulated to link an unknown external force to an observable quantity. The external force is then estimated from measurements of dynamic responses through the formulated stochastic inverse model. The applicability of the proposed method was verified with numerical examples and laboratory tests concerning the wave-structure interaction problem. The results showed that the proposed method is reliable to estimate the external force acting on a nonlinear system.

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Series with Inverse Function Terms

College Mathematics Journal ◽

10.4169/college.math.j.42.4.283 ◽

2011 ◽

Vol 42 (4) ◽

pp. 283-288

Author(s):

Sergei Ovchinnikov

Keyword(s):

Inverse Function

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Algorithms for Density, Potential Temperature, Conservative Temperature, and the Freezing Temperature of Seawater

Journal of Atmospheric and Oceanic Technology ◽

10.1175/jtech1946.1 ◽

2006 ◽

Vol 23 (12) ◽

pp. 1709-1728 ◽

Cited By ~ 97

Author(s):

David R. Jackett ◽

Trevor J. McDougall ◽

Rainer Feistel ◽

Daniel G. Wright ◽

Stephen M. Griffies

Keyword(s):

Thermal Expansion ◽

Equation Of State ◽

Thermodynamic Potential ◽

Potential Temperature ◽

Inverse Function ◽

Freezing Temperature ◽

Ocean Models ◽

The Difference ◽

New Equation ◽

Density Equation

Abstract Algorithms are presented for density, potential temperature, conservative temperature, and the freezing temperature of seawater. The algorithms for potential temperature and density (in terms of potential temperature) are updates to routines recently published by McDougall et al., while the algorithms involving conservative temperature and the freezing temperatures of seawater are new. The McDougall et al. algorithms were based on the thermodynamic potential of Feistel and Hagen; the algorithms in this study are all based on the “new extended Gibbs thermodynamic potential of seawater” of Feistel. The algorithm for the computation of density in terms of salinity, pressure, and conservative temperature produces errors in density and in the corresponding thermal expansion coefficient of the same order as errors for the density equation using potential temperature, both being twice as accurate as the International Equation of State when compared with Feistel’s new equation of state. An inverse function relating potential temperature to conservative temperature is also provided. The difference between practical salinity and absolute salinity is discussed, and it is shown that the present practice of essentially ignoring the difference between these two different salinities is unlikely to cause significant errors in ocean models.

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Quantitative Spectral Data Analysis Using Extreme Learning Machines Algorithm Incorporated with PCA

Algorithms ◽

10.3390/a14010018 ◽

2021 ◽

Vol 14 (1) ◽

pp. 18

Author(s):

Michael Li ◽

Santoso Wibowo ◽

Wei Li ◽

Lily D. Li

Keyword(s):

Data Analysis ◽

Conceptual Framework ◽

Spectral Data ◽

Learning Algorithm ◽

Structural Parameters ◽

Inverse Function ◽

Principal Component ◽

Learning Task ◽

Classification Problems ◽

Spectral Data Analysis

Extreme learning machine (ELM) is a popular randomization-based learning algorithm that provides a fast solution for many regression and classification problems. In this article, we present a method based on ELM for solving the spectral data analysis problem, which essentially is a class of inverse problems. It requires determining the structural parameters of a physical sample from the given spectroscopic curves. We proposed that the unknown target inverse function is approximated by an ELM through adding a linear neuron to correct the localized effect aroused by Gaussian basis functions. Unlike the conventional methods involving intensive numerical computations, under the new conceptual framework, the task of performing spectral data analysis becomes a learning task from data. As spectral data are typical high-dimensional data, the dimensionality reduction technique of principal component analysis (PCA) is applied to reduce the dimension of the dataset to ensure convergence. The proposed conceptual framework is illustrated using a set of simulated Rutherford backscattering spectra. The results have shown the proposed method can achieve prediction inaccuracies of less than 1%, which outperform the predictions from the multi-layer perceptron and numerical-based techniques. The presented method could be implemented as application software for real-time spectral data analysis by integrating it into a spectroscopic data collection system.

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An Application of Hadamard's Inverse Function Theorem to Algebra

American Mathematical Monthly ◽

10.2307/2318301 ◽

1977 ◽

Vol 84 (1) ◽

pp. 28 ◽

Cited By ~ 2

Author(s):

William B. Gordon

Keyword(s):

Inverse Function ◽

Inverse Function Theorem

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The regulation ofOstertagia ostertagipopulations in calves: density-dependent control of fecundity

Parasitology ◽

10.1017/s0031182000057814 ◽

1987 ◽

Vol 95 (2) ◽

pp. 373-388 ◽

Cited By ~ 38

Author(s):

G. Smith ◽

B. T. Grenfell ◽

R. M. Anderson

Keyword(s):

Worm Burden ◽

Egg Production ◽

Inverse Function ◽

Single Infection ◽

Intensity Of Infection ◽

Density Dependent ◽

Helminth Infections ◽

Model Predictions ◽

Initial Peak ◽

Infection Experiments

SUMMARYThe decline in faecal egg counts, characteristic of calves which have been experimentally infected withOstertagia ostertagi, is analysed using a mathematical model in which parasite fecundity is assumed to be an inverse function of both the duration and intensity of infection. The model incorporates a description of the frequency distribution of mature parasites between hosts (which is less over-dispersed than is usual for many other helminth infections). The model provides a good overall description of the decline in faecal egg production observed during trickle and single infection experiments. The main discrepancy between a comparison of the model predictions and the results of the most detailed available series of trickle infection experiments occurs at the initial peak of egg production. The magnitude of this difference appears to be related to the worm burden at the peak of egg production. The possible mechanisms underlying density-dependent regulation of the fecundity ofO. ostertagiaare discussed.

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