I.M. Ryzhik

Bio: I.M. Ryzhik is an academic researcher. The author has contributed to research in topics: Table (landform) & Elementary function. The author has an hindex of 12, co-authored 19 publications receiving 51170 citations.

11 – algebraic inequalities

Abstract: Publisher Summary This chapter presents algebraic inequalities. The algebraic inequalities involve complex numbers, real numbers, and inequalities for sets of complex numbers. The inequalities in real numbers involve different identities such as Lagrange's identity, Cauchy–Schwarz–Buniakowsky inequalities, Minkowski's inequality, and Holder's inequality. The inequalities in complex numbers pronounce different inequalities, namely, complex Cauchy–Schwarz–Buniakowsky inequality, complex Minkowski inequality, and complex Holder inequality. If α, 0 are any two real numbers, the complex number z = a + iβ with real part α and imaginary part β has for its modulus |z| a nonnegative number |z| = α2 + β.

Ordinary differential equations

Abstract: This chapter discusses the ordinary differential equations. Some growth estimates for solutions of second-order equations suppose that G(x) > 0 are continuous in (– ∞, ∞). The real function φ (t) is said to be an approximate solution, to within the error θ, of the differential equation except at points of discontinuity of the derivative. The Cauchy problem for the system is the problem of existence and uniqueness of the solution to this system satisfying the initial vector condition. A fundamental system of solutions of a homogeneous second-order linear differential equation in the canonical form is a system of two linearly independent solutions. Equations whose solutions possess an infinite number of zeros in the interval (0, ∞) are said to have oscillatory solutions.

Vector field theory

Abstract: This chapter provides an overview of the vector field theory. It presents vectors, vector operators, and integral theorems. The chapter also focuses on two different products involving pairs of vectors, namely, the scalar product and the vector product. The scalar product is written as a · b, and the vector product is written as either a × b or a Λ b. The chapter also focuses on orthogonal curvilinear coordinates, vector integral theorems, and integral rate of change theorems. It further presents the equation forms of grad, div, and curl operators.

Gaussian Processes for Machine Learning

Abstract: A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

Conditional heteroskedasticity in asset returns: a new approach

Abstract: This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.

Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator

Abstract: This paper focuses on the class of speech enhancement systems which capitalize on the major importance of the short-time spectral amplitude (STSA) of the speech signal in its perception. A system which utilizes a minimum mean-square error (MMSE) STSA estimator is proposed and then compared with other widely used systems which are based on Wiener filtering and the "spectral subtraction" algorithm. In this paper we derive the MMSE STSA estimator, based on modeling speech and noise spectral components as statistically independent Gaussian random variables. We analyze the performance of the proposed STSA estimator and compare it with a STSA estimator derived from the Wiener estimator. We also examine the MMSE STSA estimator under uncertainty of signal presence in the noisy observations. In constructing the enhanced signal, the MMSE STSA estimator is combined with the complex exponential of the noisy phase. It is shown here that the latter is the MMSE estimator of the complex exponential of the original phase, which does not affect the STSA estimation. The proposed approach results in a significant reduction of the noise, and provides enhanced speech with colorless residual noise. The complexity of the proposed algorithm is approximately that of other systems in the discussed class.

Welfare Evaluations in Contingent Valuation Experiments with Discrete Responses

Abstract: Since the work of Bishop and Heberlein, a number of contingent valuation experiments have appeared involving discrete responses which are analyzed by logit or similar techniques. This paper addresses the issues of how the logit models should be formulated to be consistent with the hypothesis of utility maximization and how measures of compensating and equivalent surplus should be derived from the fitted models. Two distinct types of welfare measures are introduced and then estimated from Bishop and Heberlein's data.