Budget constrained tax policy under price uncertainty: implications for supply control

Tax policies that constrain net transfers between the farm sector and the fisc are modeled under price uncertainty. Increasing the level of tax on profits causes the firm to expand output. Implications are derived for supply control and the distributions of profits and net receipts at the fisc.

Alternating sums of binomial coefficients with unit fraction arithmetic sequence coefficients

Expressions for finite sums involving the binomial coefficients with unit fraction coefficients whose denominators form an arithmetic sequence are determined.

The interval and indoor playmaking

Although early modern acting companies were adept at using different kinds of venue, performing indoors imposed a significant change in practice. Since indoor theatres required artificial lighting to augment the natural light admitted via windows, candles were employed; but the technology was such that candles could not last untended throughout an entire performance. Performing indoors thus introduced a new component into stage practice: the interval. This article explores what extant evidence (such as it is) might tell us about the introduction of act breaks, how they may have worked, and the implications for actors, audiences and dramatists. Ben Jonson's scripting of the interval in two late plays, The Staple of News and The Magnetic Lady, is examined for what it may suggest about actual practice, and the ways in which the interval may have been considered integral to composition and performance is explored through a reading of Middleton and Rowley's The Changeling. The interval offered playwrights a form of structural punctuation, drawing attention to how acts ended and began; actors could use the space to bring on props for use in the next act; spectators might use the pause between acts to reflect on what had happened and, perhaps, anticipate what was to come; and stage-sitters, the evidence indicates, often took advantage of the hiatus in the play to assert their presence in the space to which all eyes naturally were drawn.

The contributions of domain-general and numerical factors to third-grade arithmetic skills and mathematical learning disability

Explanations of the marked individual differences in elementary school mathematical achievement and mathematical learning disability (MLD or dyscalculia) have involved domain-general factors (working memory, reasoning, processing speed and oral language) and numerical factors that include single-digit processing efficiency and multi-digit skills such as number system knowledge and estimation. This study of third graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in three significant arithmetic skills: basic calculation fluency, written multi-digit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill and their contributions are both independent of each other and other factors. Different domain-general factors independently account for variation in each skill. Numeral comparison, a single digit processing skill, uniquely accounts for variation in basic calculation. Subsamples of children with MLD (at or below 10th percentile, n = 29) are compared with low achievement (LA, 11th to 25th percentiles, n = 42) and typical achievement (above 25th percentile, n = 187). Examination of these and subsets with persistent difficulties supports a multiple deficits view of number difficulties: most children with number difficulties exhibit deficits in both domain-general and numerical factors. The only factor deficit common to all persistent MLD children is in multi-digit skills. These findings indicate that many factors matter but multi-digit skills matter most in third grade mathematical achievement.

Evaluating interval forecasts of high-frequency financial data

A number of methods of evaluating the validity of interval forecasts of financial data are analysed, and illustrated using intraday FTSE100 index futures returns. Some existing interval forecast evaluation techniques, such as the Markov chain approach of Christoffersen (1998), are shown to be inappropriate in the presence of periodic heteroscedasticity. Instead, we consider a regression-based test, and a modified version of Christoffersen's Markov chain test for independence, and analyse their properties when the financial time series exhibit periodic volatility. These approaches lead to different conclusions when interval forecasts of FTSE100 index futures returns generated by various GARCH(1,1) and periodic GARCH(1,1) models are evaluated.

Numerical analyses of Runge–Kutta implicit–explicit schemes for horizontally explicit, vertically implicit solutions of atmospheric models

With the prospect of exascale computing, computational methods requiring only local data become especially attractive. Consequently, the typical domain decomposition of atmospheric models means horizontally-explicit vertically-implicit (HEVI) time-stepping schemes warrant further attention. In this analysis, Runge-Kutta implicit-explicit schemes from the literature are analysed for their stability and accuracy using a von Neumann stability analysis of two linear systems. Attention is paid to the numerical phase to indicate the behaviour of phase and group velocities. Where the analysis is tractable, analytically derived expressions are considered. For more complicated cases, amplification factors have been numerically generated and the associated amplitudes and phase diagnosed. Analysis of a system describing acoustic waves has necessitated attributing the three resultant eigenvalues to the three physical modes of the system. To do so, a series of algorithms has been devised to track the eigenvalues across the frequency space. The result enables analysis of whether the schemes exactly preserve the non-divergent mode; and whether there is evidence of spurious reversal in the direction of group velocities or asymmetry in the damping for the pair of acoustic modes. Frequency ranges that span next-generation high-resolution weather models to coarse-resolution climate models are considered; and a comparison is made of errors accumulated from multiple stability-constrained shorter time-steps from the HEVI scheme with a single integration from a fully implicit scheme over the same time interval. Two schemes, “Trap2(2,3,2)” and “UJ3(1,3,2)”, both already used in atmospheric models, are identified as offering consistently good stability and representation of phase across all the analyses. Furthermore, according to a simple measure of computational cost, “Trap2(2,3,2)” is the least expensive.

Trans-floating-point arithmetic removes nine quadrillion redundancies from 64-bit IEEE 754 floating-point arithmetic

IEEE 754 floating-point arithmetic is widely used in modern, general-purpose computers. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. Modifying the IEEE arithmetic so that it uses transreal arithmetic has a number of advantages. It removes one redundant binade from IEEE floating-point objects, doubling the numerical precision of the arithmetic. It removes eight redundant, relational,floating-point operations and removes the redundant total order operation. It replaces the non-reflexive, floating-point, equality operator with a reflexive equality operator and it indicates that some of the exceptions may be removed as redundant { subject to issues of backward compatibility and transient future compatibility as programmers migrate to the transreal paradigm.

Transreal arithmetic as a consistent basis for paraconsistent logics

Paraconsistent logics are non-classical logics which allow non-trivial and consistent reasoning about inconsistent axioms. They have been pro- posed as a formal basis for handling inconsistent data, as commonly arise in human enterprises, and as methods for fuzzy reasoning, with applica- tions in Artificial Intelligence and the control of complex systems. Formalisations of paraconsistent logics usually require heroic mathe- matical efforts to provide a consistent axiomatisation of an inconsistent system. Here we use transreal arithmetic, which is known to be consis- tent, to arithmetise a paraconsistent logic. This is theoretically simple and should lead to efficient computer implementations. We introduce the metalogical principle of monotonicity which is a very simple way of making logics paraconsistent. Our logic has dialetheaic truth values which are both False and True. It allows contradictory propositions, allows variable contradictions, but blocks literal contradictions. Thus literal reasoning, in this logic, forms an on-the- y, syntactic partition of the propositions into internally consistent sets. We show how the set of all paraconsistent, possible worlds can be represented in a transreal space. During the development of our logic we discuss how other paraconsistent logics could be arithmetised in transreal arithmetic.

Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.

Broad range of 2050 warming from an observationally constrained large climate model ensemble

Incomplete understanding of three aspects of the climate system—equilibrium climate sensitivity, rate of ocean heat uptake and historical aerosol forcing—and the physical processes underlying them lead to uncertainties in our assessment of the global-mean temperature evolution in the twenty-first century1,2. Explorations of these uncertainties have so far relied on scaling approaches3,4, large ensembles of simplified climate models1,2, or small ensembles of complex coupled atmosphere–ocean general circulation models5,6 which under-represent uncertainties in key climate system properties derived from independent sources7–9. Here we present results from a multi-thousand-member perturbed-physics ensemble of transient coupled atmosphere–ocean general circulation model simulations. We find that model versions that reproduce observed surface temperature changes over the past 50 years show global-mean temperature increases of 1.4–3 K by 2050, relative to 1961–1990, under a mid-range forcing scenario. This range of warming is broadly consistent with the expert assessment provided by the Intergovernmental Panel on Climate Change Fourth Assessment Report10, but extends towards larger warming than observed in ensemblesof-opportunity5 typically used for climate impact assessments. From our simulations, we conclude that warming by the middle of the twenty-first century that is stronger than earlier estimates is consistent with recent observed temperature changes and a mid-range ‘no mitigation’ scenario for greenhouse-gas emissions.

Coordinated Cluster and ground-based instrument observations of transient changes in the magnetopause boundary layer during an interval of predominantly northward IMF: relation to reconnection pulses and FTE signatures

We study a series of transient entries into the low-latitude boundary layer (LLBL) of all four Cluster spacecraft during an outbound pass through the mid-afternoon magnetopause ([X(GSM), Y(GSM), Z(GSM)] approximate to [2, 7, 9] R(E)). The events take place during an interval of northward IMF, as seen in the data from the ACE satellite and lagged by a propagation delay of 75 min that is well-defined by two separate studies: (1) the magnetospheric variations prior to the northward turning (Lockwood et al., 2001, this issue) and (2) the field clock angle seen by Cluster after it had emerged into the magnetosheath (Opgenoorth et al., 2001, this issue). With an additional lag of 16.5 min, the transient LLBL events cor-relate well with swings of the IMF clock angle (in GSM) to near 90degrees. Most of this additional lag is explained by ground-based observations, which reveal signatures of transient reconnection in the pre-noon sector that then take 10-15 min to propagate eastward to 15 MLT, where they are observed by Cluster. The eastward phase speed of these signatures agrees very well with the motion deduced by the cross-correlation of the signatures seen on the four Cluster spacecraft. The evidence that these events are reconnection pulses includes: transient erosion of the noon 630 nm (cusp/cleft) aurora to lower latitudes; transient and travelling enhancements of the flow into the polar cap, imaged by the AMIE technique; and poleward-moving events moving into the polar cap, seen by the EISCAT Svalbard Radar (ESR). A pass of the DMSP-F15 satellite reveals that the open field lines near noon have been opened for some time: the more recently opened field lines were found closer to dusk where the flow transient and the poleward-moving event intersected the satellite pass. The events at Cluster have ion and electron characteristics predicted and observed by Lockwood and Hapgood (1998) for a Flux Transfer Event (FTE), with allowance for magnetospheric ion reflection at Alfvenic disturbances in the magnetopause reconnection layer. Like FTEs, the events are about 1 R(E) in their direction of motion and show a rise in the magnetic field strength, but unlike FTEs, in general, they show no pressure excess in their core and hence, no characteristic bipolar signature in the boundary-normal component. However, most of the events were observed when the magnetic field was southward, i.e. on the edge of the interior magnetic cusp, or when the field was parallel to the magnetic equatorial plane. Only when the satellite begins to emerge from the exterior boundary (when the field was northward), do the events start to show a pressure excess in their core and the consequent bipolar signature. We identify the events as the first observations of FTEs at middle altitudes.

Towards a typology for constrained climate model forecasts

In recent years several methodologies have been developed to combine and interpret ensembles of climate models with the aim of quantifying uncertainties in climate projections. Constrained climate model forecasts have been generated by combining various choices of metrics used to weight individual ensemble members, with diverse approaches to sampling the ensemble. The forecasts obtained are often significantly different, even when based on the same model output. Therefore, a climate model forecast classification system can serve two roles: to provide a way for forecast producers to self-classify their forecasts; and to provide information on the methodological assumptions underlying the forecast generation and its uncertainty when forecasts are used for impacts studies. In this review we propose a possible classification system based on choices of metrics and sampling strategies. We illustrate the impact of some of the possible choices in the uncertainty quantification of large scale projections of temperature and precipitation changes, and briefly discuss possible connections between climate forecast uncertainty quantification and decision making approaches in the climate change context.