On the Existence and Frequency Distribution of the Shell Primes

This research presents the results of a study on the existence and frequency distribution of the shell primes defined herein as prime numbers that result from the calculation of the "half-shell" of an p-dimensional entity of the form $n^p-(n-1)^p$ where power $p$ is prime and base $n$ is the realm of the positive integers. Following the introduction of the shell primes, we will look at the results of a non-sieving application of the Euler zeta function to the prime shell function as well as to any integer-valued polynomial function in general which has the ability to produce prime numbers when power $p$ is prime. One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no known way to sieve composite numbers out of the product term in this famous equation. Such would be case when an infinite series of numbers to be analyzed are calculated by a polynomial expression that yields successively increasing positive integer values and which has as its input domain the positive integers themselves. In such cases there may not be an intuitive way to eliminate the composite terms from the product term in the Euler zeta function equation by either scaling a previous prime number calculation or by employing predictable values of the domain of the function which would render outputs of the polynomial prime. So the best one may be able to hope for in these cases is to calculate some value to be added or subtracted from unity in the numerator above the product term in the Euler Zeta function to make both sides of that equation equal with the expectation that that value could be used to predict the number of prime numbers that exist as outputs of the polynomial function for some limit less than or equal to x of the input domain.Comment: Version 5: Spelling corrections only ("Riemann" on page 5

A Non-Sieving Application of the Euler Zeta Function

One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no known way to sieve composite numbers out of the product term in this famous equation. Such would be case when an infinite series of numbers to be analyzed are calculated by a polynomial expression that yields successively increasing positive integer values and which has as its input domain the positive integers themselves. In such cases there may not be an intuitive way to eliminate the composite terms from the product term on the right-hand side of the Euler zeta function equation by either scaling a previous prime number calculation or by employing predictable values of the domain of the function which would make future outputs of the polynomial prime. So the best one may be able to hope for in this case is to determine some value to be added or subtracted from unity in the numerator above the product term to make both sides of the equation equal in the hope that that value can be used to predict the number of prime numbers that exist as outputs of the polynomial function for some limit less than or equal to x of the input domain.Comment: V3 added a modified version of the big Zeta equation that adapts to integer-valued functions for which the first term generated is not unity. This version also generalizes Theorem 1 to include all integer-valued polynomials, even those which do not generate prime numbers, thus opening up the potential for the M-series to be used to predict other properties of integer-valued polynomial functions. arXiv admin note: substantial text overlap with arXiv:1510.0102

Recruitment and retention of estates and facilities staff in the NHS

Purpose – Agenda for Change is set to be the biggest reform of pay since the National Health Service (NHS) began in 1948. As well as introducing a standardised pay structure; it also aims to improve recruitment, retention and staff morale. Staff groups identified as having recruitment and retention problems include estates/works officers, qualified maintenance crafts persons and qualified maintenance technicians. The object of this research was to investigate recruitment and retention problems for estates and facilities staff currently experienced by Trusts. Design/methodology/approach – Focus groups were used as the primary method of data collection in an attempt to tap into the existing expertise of staff working at strategic and operational supervisory positions in a wide range of Trusts. Findings – Although our findings suggest that the main recruitment and retention issues fall into four main themes: social, financial, environmental and political; recruitment and retention of estates and facilities management staff is a complex problem involving a wide range of issues and these can vary from location to location. Furthermore this should also be seen as a series of issues that varies across employment groups including: domestic/housekeeping, trades, managers/officers and facilities directors, which need to be distinguished. Practical implications – There is a continuing need to raise the profile of estates and facilities management staff in the NHS to those levels enjoyed by Human Resource (HR) and Financial Management. Furthermore perceptions surrounding both recruitment and retention issues and the nature of work within estates and facilities management staff in the NHS can lead to a negative and self-perpetuating “cycle of failure” where there is an assumption of loss of control. However, there are some initiatives being undertaken that suggest it is possible to concentrate on internal matters such as more appropriate and flexible recruitment processes, improved support services for staff and greater flexibility within the job and that these can generate “cycles of success”.</p

One- and two-level filter-bank convolvers

In a recent paper, it was shown in detail that in the case of orthonormal and biorthogonal filter banks we can convolve two signals by directly convolving the subband signals and combining the results. In this paper, we further generalize the result. We also derive the statistical coding gain for the generalized subband convolver. As an application, we derive a novel low sensitivity structure for FIR filters from the convolution theorem. We define and derive a deterministic coding gain of the subband convolver over direct convolution for a fixed wordlength implementation. This gain serves as a figure of merit for the low sensitivity structure. Several numerical examples are included to demonstrate the usefulness of these ideas. By using the generalized polyphase representation, we show that the subband convolvers, linear periodically time varying systems, and digital block filtering can be viewed in a unified manner. Furthermore, the scheme called IFIR filtering is shown to be a special case of the convolver

Factorability of lossless time-varying filters and filter banks

We study the factorability of linear time-varying (LTV) lossless filters and filter banks. We give a complete characterization of all, degree-one lossless LTV systems and show that all degree-one lossless systems can be decomposed into a time-dependent unitary matrix followed by a lossless dyadic-based LTV system. The lossless dyadic-based system has several properties that make it useful in the factorization of lossless LTV systems. The traditional lapped orthogonal transform (LOT) is also generalized to the LTV case. We identify two classes of TVLOTs, namely, the invertible inverse lossless (IIL) and noninvertible inverse lossless (NIL) TVLOTs. The minimum number of delays required to implement a TVLOT is shown to be a nondecreasing function of time, and it is a constant if and only if the TVLOT is IIL. We also show that all IIL TVLOTs can be factorized uniquely into the proposed degree-one lossless building block. The factorization is minimal in terms of the delay elements. For NIL TVLOTs, there are factorable and unfactorable examples. Both necessary and sufficient conditions for the factorability of lossless LTV systems are given. We also introduce the concept of strong eternal reachability (SER) and strong eternal observability (SEO) of LTV systems. The SER and SEO of an implementation of LTV systems imply the minimality of the structure. Using these concepts, we are able to show that the cascade structure for a factorable IIL LTV system is minimal. That implies that if a IIL LTV system is factorable in terms of the lossless dyadic-based building blocks, the factorization is minimal in terms of delays as well as the number of building blocks. We also prove the BIBO stability of the LTV normalized IIR lattice