Let $A$ be a Banach ternary algebra over a scalar field $Bbb R$ or $Bbb C$ and $X$ be a ternary Banach $A$--module. Let $sigma,tau$ and $xi$ be linear mappings on $A$, a linear mapping $D:(A,[~]_A)to (X,[~]_X)$ is called a Lie ternary $(sigma,tau,xi)$--derivation, if $$D([a,b,c])=[[D(a)bc]_X]_{(sigma,tau,xi)}-[[D(c)ba]_X]_{(sigma,tau,xi)}$$ for all $a,b,cin A$, where $[abc]_{(sigma,tau,xi)}=ata...

The theory of S2(δ) family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The δ deformation of the xi function is formulated in terms of the S2(δ) distribution and shown to satisfy Riemann’s functional equation. Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are form...

It is shown that any triangular derivation on k[X1, X2, X3, X4] sending Xi to a monomial has kernel generated by at most four elements, hence is finitely generated. An explicit formula for the generators is given.

The sets Sj are the sets of points to which μj is the closest center. In each step of the algorithm the potential function is reduced. Let’s examine that. First, if the set of centers μj are fixed, the best assignment is clearly the one which assigns each data point to its closest center. Also, assume that μ is the center of a set of points S. Then, if we move μ to 1 |S| ∑ i∈S xi then we only r...

This document provides additional description to certain con1 tent of the paper and additional results of the A field-based 2 flow visualization. 3 1. Detailed Derivation of Relation between FTLE and Path4 line Orientation Vector 5 Recall that in Section 5.2 of the paper, the Lagrangian accu6 mulation is applied to collect vector-valued properties. Specif7 ically, we use it to accumulate the fl...

Web Appendix A: Derivation of P (∆i = δi, Ri = j|Xi) Clearly the pair of random variables (∆i, Ri), or equivalently (∆i,Ri+1, Ri), is completely determined by (Ti, Ci). In particular, the set {∆i,Ri+1 = 0, Ri = j} is equivalent to observing the event in (tj−1, tj], which in turn is equivalent to the set {Ti ∈ (tj−1, tj], Ci ≥ tj}; and the set {∆i,Ri+1 = 1, Ri = j} is equivalent to censoring the...

the gravity method is one of the first geophysical techniques used in oil and gas exploration. an algorithm is developed for a fast quantitative interpretation of gravity data generated by geometrically simple but also the estimated depths and other model parameters of a buried structure. following abdelrahman et al (1989). the general gravity anomaly expression produced by a sphere, an infinit...

let  be a banach algebra. let  be linear mappings on . first we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple banach algebras, in certain case. afterwards we define a new vocabulary called “-higher double derivation” and present a relation between this subject and derivations and finally give some ...

A locally nilpotent linear derivation δ of the commutative polynomial algebra K[Xd ]=K[x1,…,xd ] rank d is called Weitzenböck. It well known that subalgebra ]^δ consisting polynomials which are sent to zero by finitely generated. Let Weitzenböck act on K[Xd,Yd such δ(yi )=xi, δ(xi )=0, i=1,…,d. The explicit form generators was conjectured Nowicki in 1994. In this study, we consider conjecture W...

Derivation of PCA I: For a set of d-dimensional data vectors {x}i=1, the principal axes {e}qj=1 are those orthonormal axes onto which the retained variance under projection is maximal. It can be shown that the vectors ej are given by the q dominant eigenvectors of the sample covariance matrix S, such that Sej = λjej . The q principal components of the observed vector xi are given by the vector ...