SOME ALGEBRAIC PROPERTIES OF ℓp(β)

In this paper we consider a generalized Cauchy product ⋄ on ℓp(β) and then we characterized some Banach algebra structures for ℓp(β). Also some classic properties of ⋄-multiplication operator M⋄,z on ℓp(β) will be investigated.

On the Construction of Large Algebras Not Contained in the Image of the Borel Map

The Borel map $$j^{\infty }$$j∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $$j^{\infty }$$j∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $$j^{\infty }$$j∞ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $$j^{\infty }$$j∞ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic

Probabilistic evolution theory (PREVTH) forms a framework for the solution of explicit ODEs. The purpose of the paper is two-fold: (1) conversion of multinomial right-hand sides of the ODEs to purely second degree multinomial right-hand sides by space extension; (2) decrease the computational burden of probabilistic evolution theory by using the condensed Kronecker product. A first order ODE set with multinomial right-hand side functions may be converted to a first order ODE set with purely second degree multinomial right-hand side functions at the expense of an increase in the number of equations and unknowns. Obtaining purely second degree multinomial right-hand side functions is important because the solution of such equation set may be approximated by probabilistic evolution theory. A recent article by the authors states that the ODE set with the smallest number of unknowns can be found by searching. This paper gives the details of a way to search for the optimal space extension. As for the second purpose of the paper, the computational burden can be reduced by considering the properties of the Kronecker product of vectors and how the Kronecker product appears within the recursion of PREVTH: as a Cauchy product structure.

Evaluation of diagnostic thresholds dependability for tribologic signals received in the environment disturbed by vibroacoustic and functional signals

Determination of dependable diagnostic thresholds for tribologic signals received e.g. from antifriction bearings (in particular for insufficient number of measurements, only 4÷5) is a really difficult task due to complexity of working environment where such bearings are operated. Typical working environment for such objects must take account for operation time under various working conditions and accompanying (and disturbing) signals, e.g. vibroacoustic ones. The sought assessment of the relationship between diagnostic signals and environmental noise can be determined from convolution of both diagnostic and environments signals that make up the complete set of received information. The convolution of these two series of signals can be obtained from an algorithm based on the Cauchy product. Then one has to find the coherence factor and the square of amplitude gain for the set of diagnostic signals with reference to various sets of signals received from environment, which makes it possible to evaluate cohesion of the investigated series of signals, thus their suitability to determine diagnostic threshold for tribologic signals intended for the analysis.

Statistical Diagnostic Thresholds of Rolling Bearings with Correlation of the Bearings Signal and its Environment Based on Cauchy Product

During diagnostic thresholds research, sequence of numbers emerging from proper measurements, are analysed. These are sequences characterizing environment (e.g. amount of hours worked sequence x and sequence of numbers emerging from diagnostic signal measurement sequence y). Hence, two sequences exist: {x0, x1,, xm} and {y0, y1,, ym}. Relation between these sequences might be determined by Cauchy product. The Cauchy product can significantly expand opportunities of applications of signal processing and analysis. Auto and cross Cauchy products of signals as well as quotients and subtraction result of the diagnostic and environment signals, for the standard condition and the condition resulting from the current diagnosis can be determined. These allow to make diagnostics more precise, particularly in case of small number of measurements, and low accuracy of the diagnostics thresholds determined from them.