Perspex machine XI: topology of the transreal numbers

The transreal numbers are a total number system in which even, arithmetical operation is well defined even-where. This has many benefits over the real numbers as a basis for computation and, possibly, for physical theories. We define the topology of the transreal numbers and show that it gives a more coherent interpretation of two's complement arithmetic than the conventional integer model. Trans-two's-complement arithmetic handles the infinities and 0/0 more coherently, and with very much less circuitry, than floating-point arithmetic. This reduction in circuitry is especially beneficial in parallel computers, such as the Perspex machine, and the increase in functionality makes Digital Signal Processing chips better suited to general computation.

Construction of the transcomplex numbers from the complex numbers

A geometrical construction of the transcomplex numbers was given elsewhere. Here we simplify the transcomplex plane and construct the set of transcomplex numbers from the set of complex numbers. Thus transcomplex numbers and their arithmetic arise as consequences of their construction, not by an axiomatic development. This simplifes transcom- plex arithmetic, compared to the previous treatment, but retains totality so that every arithmetical operation can be applied to any transcomplex number(s) such that the result is a transcomplex number. Our proof establishes the consistency of transcomplex and transreal arithmetic and establishes the expected containment relationships amongst transcomplex, complex, transreal and real numbers. We discuss some of the advantages the transarithmetics have over their partial counterparts.

On the system of rational difference equations x(n) = A+(1)over-bar y(n-p), y(n) = A+(gamma(n-1))over-bar x(n-r)y(n-5)

In this paper, we study the behavior of the positive solutions of the system of two difference equations [GRAPHICS] where p >= 1, r >= 1, s >= 1, A >= 0, and x(1-r), x(2-r),..., x(0), y(1-max) {p.s},..., y(0) are positive real numbers. (c) 2005 Elsevier Inc. All rights reserved.

Evolutionary and revolutionary effects of transcomputation

Mathematics in Defence 2011 Abstract. We review transreal arithmetic and present transcomplex arithmetic. These arithmetics have no exceptions. This leads to incremental improvements in computer hardware and software. For example, the range of real numbers, encoded by floating-point bits, is doubled when all of the Not-a-Number(NaN) states, in IEEE 754 arithmetic, are replaced with real numbers. The task of programming such systems is simplified and made safer by discarding the unordered relational operator,leaving only the operators less-than, equal-to, and greater than. The advantages of using a transarithmetic in a computation, or transcomputation as we prefer to call it, may be had by making small changes to compilers and processor designs. However, radical change is possible by exploiting the reliability of transcomputations to make pipelined dataflow machines with a large number of cores. Our initial designs are for a machine with order one million cores. Such a machine can complete the execution of multiple in-line programs each clock tick

Transdifferential and transintegral calculus

The set of transreal numbers is a superset of the real numbers. It totalises real arithmetic by defining division by zero in terms of three def- inite, non-finite numbers: positive infinity, negative infinity and nullity. Elsewhere, in this proceedings, we extended continuity and limits from the real domain to the transreal domain, here we extended the real derivative to the transreal derivative. This continues to demonstrate that transreal analysis contains real analysis and operates at singularities where real analysis fails. Hence computer programs that rely on computing deriva- tives { such as those used in scientific, engineering and financial applica- tions { are extended to operate at singularities where they currently fail. This promises to make software, that computes derivatives, both more competent and more reliable. We also extended the integration of absolutely convergent functions from the real domain to the transreal domain.

On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Let λ1,…,λn be real numbers in (0,1) and p1,…,pn be points in Rd. Consider the collection of maps fj:Rd→Rd given by fj(x)=λjx+(1−λj)pj. It is a well known result that there exists a unique nonempty compact set Λ⊂Rd satisfying Λ=∪nj=1fj(Λ). Each x∈Λ has at least one coding, that is a sequence (ϵi)∞i=1 ∈{1,…,n}N that satisfies limN→∞fϵ1…fϵN(0)=x. We study the size and complexity of the set of codings of a generic x∈Λ when Λ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every x∈Λ has a continuum of codings. We also show that almost every x∈Λ has a universal coding. Our work makes no assumptions on the existence of holes in Λ and improves upon existing results when it is assumed Λ contains no holes.

Quantification of ruminal Clostridium proteoclasticum by real-time PCR using a molecular beacon approach

Aims: All members of the ruminal Butyrivibrio group convert linoleic acid (cis-9,cis-12-18 : 2) via conjugated 18 : 2 metabolites (mainly cis-9,trans-11-18 : 2, conjugated linoleic acid) to vaccenic acid (trans-11-18 : 1), but only members of a small branch, which includes Clostridium proteoclasticum, of this heterogeneous group further reduce vaccenic acid to stearic acid (18 : 0, SA). The aims of this study were to develop a real-time polymerase chain reaction (PCR) assay that would detect and quantify these key SA producers and to use this method to detect diet-associated changes in their populations in ruminal digesta of lactating cows. Materials and Results: The use of primers targeting the 16S rRNA gene of Cl. proteoclasticum was not sufficiently specific when only binding dyes were used for detection in real-time PCR. Their sequences were too similar to some nonproducing strains. A molecular beacon probe was designed specifically to detect and quantify the 16S rRNA genes of the Cl. proteoclasticum subgroup. The probe was characterized by its melting curve and validated using five SA-producing and ten nonproducing Butyrivibrio-like strains and 13 other common ruminal bacteria. Analysis of ruminal digesta collected from dairy cows fed different proportions of starch and fibre indicated a Cl. proteoclasticum population of 2-9% of the eubacterial community. The influence of diet on numbers of these bacteria was less than variations between individual cows. Conclusion: A molecular beacon approach in qPCR enables the detection of Cl. proteoclasticum in ruminal digesta. Their numbers are highly variable between individual animals. Signifance and Impact of the Study: SA producers are fundamental to the flow of polyunsaturated fatty acid and vaccenic acid from the rumen. The method described here enabled preliminary information to be obtained about the size of this population. Further application of the method to digesta samples from cows fed diets of more variable composition should enable us to understand how to control these bacteria in order to enhance the nutritional characteristics of ruminant-derived foods, including milk and beef.

Mirror lightpipes: daylighting performance in real buildings

Mirror lightpipes are useful for providing healthy and energy-efficient daylight into buildings where windows and skylights are unsuitable, insufficient or generate too much heat gain. The lightpipes have been installed in dozens of buildings in the UK. Field monitoring has been carried out to assess their performance in four different buildings: the headquaters of a major insurance company, a health clinic, a residential building and a college dining hall In those cases where lighipipes with moderate aspect ratios were installed, good illuminance of up to 450 lux has been obtained with internal/external illuminance ratios around 1%. When long and narrow lightpipes with many bends are used, however, the ratio reduced to around 0.1%. These results showed that lightpipes can be effective daylighting devices provided that excessive aspect ratios and numbers of bends are avoided. Lightpipes with larger diameters should be used whenever possible. The lightpipes often improved signiScantly the visual quality af the interior environment, and high user satisfaction was found even in buildings where a relatively low level of daylight was admitted through the lightpipes

Real estate portfolio size and risk reduction

The reduction of portfolio risk is important to all investors but is particularly important to real estate investors as most property portfolios are generally small. As a consequence, portfolios are vulnerable to a significant risk of under-performing the market, or a target rate of return and so investors may be exposing themselves to greater risk than necessary. Given the potentially higher risk of underperformance from owning only a few properties, we follow the approach of Vassal (2001) and examine the benefits of holding more properties in a real estate portfolio. Using Monte Carlo simulation and the returns from 1,728 properties in the IPD database, held over the 10-year period from 1995 to 2004, the results show that increases in portfolio size offers the possibility of a more stable and less volatile return pattern over time, i.e. down-side risk is diminished with increasing portfolio size. Nonetheless, increasing portfolio size has the disadvantage of restricting the probability of out-performing the benchmark index by a significant amount. In other words, although increasing portfolio size reduces the down-side risk in a portfolio, it also decreases its up-side potential. Be that as it may, the results provide further evidence that portfolios with large numbers of properties are always preferable to portfolios of a smaller size.

The persistence of real estate fund performance

This paper examines the short and long-term persistence of tax-exempt real estate funds in the UK through the use of winner-loser contingency table methodology. The persistence tests are applied to a database of varying numbers of funds from a low of 16 to a high of 27 using quarterly returns over the 12 years from 1990 Q1 to 2001 Q4. The overall conclusion is that the real estate funds in the UK show little evidence of persistence in the short-term (quarterly and semi-annual data) or for data over a considerable length of time (bi-annual to six yearly intervals). In contrast, the results are better for annual data with evidence of significant performance persistence. Thus at this stage, it seems that an annual evaluation period, provides the best discrimination of the winner and loser phenomenon in the real estate market. This result is different from equity and bond studies, where it seems that the repeat winner phenomenon is stronger over shorter periods of evaluation. These results require careful interpretation, however, as the results show that when only small samples are used significant adjustments must be made to correct for small sample bias and second the conclusions are sensitive to the length of the evaluation period and specific test used. Nonetheless, it seems that persistence in performance of real estate funds in the UK does exist, at least for the annual data, and it appears to be a guide to beating the pack in the long run. Furthermore, although the evidence of persistence in performance for the overall sample of funds is limited, we have found evidence that two funds were consistent winners over this period, whereas no one fund could be said to be a consistent loser.

Predicting the annual performance of UK real estate: timeliness versus coverage

The IPD Annual Index is the largest and most comprehensive Real Estate market index available in the UK Such coverage however inevitably leads to delays in publication. In contrast there are a number of quarterly and monthly indices which are published within days of the year end but which lack the coverage in terms of size and numbers of properties. This paper analyses these smaller but more timely indices to see whether such indices can be used to predict the performance of the IPD Annual Index. Using a number of measures of forecasting accuracy it is shown that the smaller indices provide unbiased and efficient predictions of the IPD Annual Index. Such indices also significantly outperform a naive no-change model. Although no one index performs significantly better than the others. The more timely indices however do not perfectly track the IPD Annual Index. As a result any short run predictions of performance will be subject to a degree of error. Nevertheless the more timely indices, although lacking authoritative coverage, provide a valuable service to investors giving good estimates of Real Estates performance well before the publication of the IPD Annual Index.

The real-time optimisation of DNO owned storage devices on the LV network for peak reduction

Energy storage is a potential alternative to conventional network reinforcementof the low voltage (LV) distribution network to ensure the grid’s infrastructure remainswithin its operating constraints. This paper presents a study on the control of such storagedevices, owned by distribution network operators. A deterministic model predictive control (MPC) controller and a stochastic receding horizon controller (SRHC) are presented, wherethe objective is to achieve the greatest peak reduction in demand, for a given storagedevice specification, taking into account the high level of uncertainty in the prediction of LV demand. The algorithms presented in this paper are compared to a standard set-pointcontroller and bench marked against a control algorithm with a perfect forecast. A specificcase study, using storage on the LV network, is presented, and the results of each algorithmare compared. A comprehensive analysis is then carried out simulating a large number of LV networks of varying numbers of households. The results show that the performance of each algorithm is dependent on the number of aggregated households. However, on a typical aggregation, the novel SRHC algorithm presented in this paper is shown to outperform each of the comparable storage control techniques.

A constrained recursive least squares algorithm for adaptive combination of multiple models

In this paper, we develop a novel constrained recursive least squares algorithm for adaptively combining a set of given multiple models. With data available in an online fashion, the linear combination coefficients of submodels are adapted via the proposed algorithm.We propose to minimize the mean square error with a forgetting factor, and apply the sum to one constraint to the combination parameters. Moreover an l1-norm constraint to the combination parameters is also applied with the aim to achieve sparsity of multiple models so that only a subset of models may be selected into the final model. Then a weighted l2-norm is applied as an approximation to the l1-norm term. As such at each time step, a closed solution of the model combination parameters is available. The contribution of this paper is to derive the proposed constrained recursive least squares algorithm that is computational efficient by exploiting matrix theory. The effectiveness of the approach has been demonstrated using both simulated and real time series examples.