A Convergence Relation Between Discrete and Continuous Regular Quaternionic Functions

A growing complexity of modern engineering problems emphasises the need to methods working with discrete mathe- matical structures, such as methods of discrete potential theory and discrete function theory, are gaining more and more popularity nowadays. However, solution procedure of boundary value problems by these methods requires at first a construction of the dis- cretisation of a given domain. Thus, in this paper, we present a general discretisation algorithm for discrete potential and function theories. The algorithm allows a consistent discretisation of a domain for interior and exterior boundary value problems based only on a definition of a given continuous domain; and the complete discretisation procedure is based on the application of well-known set-theoretic operations originating from the characterisation of mesh points by help of star-Laplacian. Thus, the algorithm results in a rectangular lattice established over an arbitrary two-dimensional domain and its exterior.

One confusing question over a long period of time is how transfer the discrete function transfers into continuous function. Recently the issue has been resolved but some details of the transformation process will be introduced in the paper. The correlation coefficients of 100,000 values are established from the two groups of data with the range between -1 and 1, creating a histogram from these correlation coefficient values known as “the probability mass function.” The coefficient values are brought into the discrete distribution function, so that transfers into the discrete cumulative function, next converted into a continuous cumulative function, next which is differentiated to get the density function, so that it is easy to being research analysis. A model will be established during the process of the conversion what the medium is “the least squares algorithm.” Finally, when the integral of the area within the range of the density function equals to 1, this implies that the transformation complete succeeds from the discrete function to the continuous function.

Discrete function theory is a natural extension of the continuous theory to functions defined on lattices. The idea of the discrete function theory is to work directly with discretised domains (lattices) and to transfer all important properties from the continuous case to the discrete level. In the field of boundary value problems it is more beneficial to work with rectangular lattices, i.e. allowing two different stepsizes. Thus, the aim of this paper is to present the extension of the discrete function theory to rectangular lattices. Particularly, we present the discrete analogue of the Borel–Pompeiu formula. Analogously to the continuous theory it is the core of the developed discrete function theory and the related operator calculus.

The quaternionic functions method is an analytical tool used in elasticity theory. For isotropic elasticity, there are known a few variants of three-dimensional analogues of the Kolosov-Muskhelishvili formulae. In this case, a general solution of the Lamé equation for the spatial theory of elasticity is expressed in terms of two regular quaternionic or monogenic Clifford functions. For the anisotropic elasticity, a close approach exists when equations of equilibrium are factorized by means of matrix algebra. In this report, we will discuss the quaternionic factorization method in the transversely isotropic theory of elasticity. The model of an elastic media with such symmetry is described by five elastic constants, and for example, can be used in the mechanics of rocks in permafrost conditions.

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy‐Kovalevskaya extension theorem for discrete monogenic functions defined on the grid Zhm of m‐tuples of integer multiples of a variable mesh width h. Convergence to the continuous case is investigated. As illustrative examples we explicitly construct the Cauchy‐Kovalevskaya extensions of the discrete delta function and of a discretized exponential.

This paper investigates the effects of bulk acoustic wave (BAW) radiation on the admittance of interdigital transducers (IDTs) with significant internal reflections of shear horizontal type (SH-type) leaky surface acoustic waves (SAWs) on 36/spl deg/YX-LiTaO/sub 3/ (36-LT). Theoretical analysis is made by using the discrete Green function theory, and synchronous one-port resonators are analyzed. It is shown that the BAW radiation significantly affects the IDT characteristics even for resonators; under certain circumstances, BAWs launched from an IDT are converted into SH-type SAWs by the strong internal reflections, and they interact with the BAWs radiated by the IDT. Then the net amount of the radiated BAW power is highly dependent upon the number of IDT finger pairs. For the precise simulation of devices based on the SH-type SAWs with strong internal reflections, the BAW radiation should carefully be taken into consideration.

Blotekjaer's theory, extended by Hashimoto and coworkers, for analyzing surface transverse waves has been modified in the part concerning the effect of finite electrode thickness. First order Datta-Hunsinger's surface boundary conditions have been introduced in the effective permittivity of the substrate, to account for the mechanical loading. The dispersion STW velocity curves have been obtained over the entire frequency stopband, including the interval where intensive excitation of bulk modes occurs. The effective permittivity also allows obtaining the electromechanical coupling of the grating mode. Its numerical analysis reveals a strong dependence on the metalization ratio and mass-loading, in close agreement with results from Hasimoto's discrete Green function theory.

The discrete Lotka power function describes the number of sources (e.g., authors) with n = 1, 2, 3, … items (e.g., publications). As in econometrics, informetrics theory requires functions of a continuous variable j, replacing the discrete variable n. Now j represents item densities instead of number of items. The continuous Lotka power function describes the density of sources with item density j. The discrete Lotka function one obtains from data, obtained empirically; the continuous Lotka function is the one needed when one wants to apply Lotkaian informetrics, i.e., to determine properties that can be derived from the (continuous) model. It is, hence, important to know the relations between the two models. We show that the exponents of the discrete Lotka function (if not too high, i.e., within limits encountered in practice) and of the continuous Lotka function are approximately the same. This is important to know in applying theoretical results (from the continuous model), derived from practical data.

In this paper, we develop a theory of free holomorphic functions on noncommutative Reinhardt domains , generated by positive regular free holomorphic functions f in n noncommuting variables and by positive integers m ≥ 1, where is the algebra of all bounded linear operators on a Hilbert space . Noncommutative Berezin transforms are used to study Hardy algebras H ∞ (D f, rad m ) and domain algebras A(D f, rad m ) associated with and compositions of free holomorphic functions. We obtain noncommutative Cartan type results for formal power series, in several noncommuting indeterminates, which leave invariant the nilpotent parts of the corresponding domains. As a consequence, we characterize the set of all free biholomorphic functions with F(0) = 0. We show that the free biholomorphic classification of the domains is the same as the classification, up to unital completely isometric isomorphisms having completely contractive hereditary extension, of the corresponding noncommutative domain algebras A(D f, rad m ). In particular, we prove that Ψ : A(D f, rad 1 ) → A(D g, rad 1 ) is a unital completely isometric isomorphism if and only if there is a free biholomorphic map φ ∈ Bih(D g 1 , D f 1 ) such that This implies that the noncommutative domains and are free biholomorphic equivalent if and only if the domain algebras A(D f, rad 1 ) and A(D g, rad 1 ) are completely isometrically isomorphic. Using the interaction between the theory of functions in several complex variables and our noncommutative theory, we provide several results concerning the free biholomorphic classification of the noncommutative domains and the classification, up to completely isometric isomorphisms, of the associated noncommutative domain (resp. Hardy) algebras. In particular, we characterize the unit ball of among the noncommutative domains , up to free biholomorphisms. We also obtain characterizations for the unitarily implemented isomorphisms of noncommutative Hardy (resp. domain) algebras in terms of free biholomorphic functions between the corresponding noncommutative domains.

Flows on a network play an important role in the theory of discrete harmonic functions. In the study of discrete bi-harmonic functions, we encounter a concept of bi-flows. In this paper, we are concerned with minimization problems for bi-flows which are analogous to those for flows.

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes' theorem, Cauchy's theorem and a Cauchy integral formula are established.

This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators, backward and forward partial difference, one needs to modify the main tools for a development of a discrete function theory, such as the replacement of a real Clifford algebra by a complexified Clifford algebra or of the classic Weyl relations by so-called S-Weyl relations. The main results, like Cauchy integral formula, Fischer decomposition, CK-extension, and Taylor series, will be derived. To give a better idea of the differences between the discrete and continuous case, this chapter contains the problem of discrete Hardy spaces as well as some discrete objects which do not have an equivalent object in continuous Clifford analysis, such as the CK-extension of a discrete Delta function.

Comparison between error correcting output codes and fuzzy support vector machines

In this paper,the linear space $\mathcal F$ of a special type of fractal interpolation functions (FIFs) on an interval I is considered. Each FIF in $\mathcal F$ is established from a continuous function on I. We show that, for a finite set of linearly independent continuous functions on I, we get linearly independent FIFs. Then we study a finite-dimensional reproducing kernel Hilbert space (RKHS) $\mathcal F_{\mathcal B}\subset\mathcal F$, and the reproducing kernel $\mathbf k$ for $\mathcal F_{\mathcal B}$ is defined by a basis of $\mathcal F_{\mathcal B}$. For a given data set $\mathcal D=\{(t_k, y_k) : k=0,1,\ldots,N\}$, we apply our results to curve fitting problems of minimizing the regularized empirical error based on functions of the form $f_{\mathcal V}+f_{\mathcal B}$, where $f_{\mathcal V}\in C_{\mathcal V}$ and $f_{\mathcal B}\in \mathcal F_{\mathcal B}$. Here $C_{\mathcal V}$ is another finite-dimensional RKHS of some classes of regular continuous functions with the reproducing kernel $\mathbf k^*$. We show that the solution function can be written in the form $f_{\mathcal V}+f_{\mathcal B}=\sum_{m=0}^N\gamma_m\mathbf k^*_{t_m} +\sum_{j=0}^N \alpha_j\mathbf k_{t_j}$, where ${\mathbf k}_{t_m}^\ast(\cdot)={\mathbf k}^\ast(\cdot,t_m)$ and $\mathbf k_{t_j}(\cdot)=\mathbf k(\cdot,t_j)$, and the coefficients γm and αj can be solved by a system of linear equations.

A genetic algorithm with discrete cost functions is used to design and optimise a 42 V-14 V/8 A buck converter. The use of discrete and continuous cost functions are confronted. Discrete cost functions produces better designs. Electrical time domain simulation is used for verification. Experimental measurement, electrical and thermal, validates the design.