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.

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.

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.

Transreal proof of the existence of universal possible worlds

Transreal arithmetic is total, in the sense that the fundamental operations of addition, subtraction, multiplication and division can be applied to any transreal numbers with the result being a transreal number [1]. In particular division by zero is allowed. It is proved, in [3], that transreal arithmetic is consistent and contains real arithmetic. The entire set of transreal numbers is a total semantics that models all of the semantic values, that is truth values, commonly used in logics, such as the classical, dialetheaic, fuzzy and gap values [2]. By virtue of the totality of transreal arithmetic, these logics can be implemented using total, arithmetical functions, specifically operators, whose domain and counterdomain is the entire set of transreal numbers

The origins of small wars: from special operations to ideological insurgencies

This edited volume explores the origins of the term small wars and traces it to special operations. In the 17th century, such "guerrilla/petite guerre" special operations grew out of training and winter operations of the regular forces as practiced in the 16th century. In the 18th century, they fused with a tradition going back to Antiquity, of employing special ethnic groups (such as the Hungarian Hussars) for special operations. Side by side with these special operations, however, there was the even older genealogy of uprisings and insurgencies, which since the Spanish Guerrilla of 1808-1812 has been associated with this term. All three traditions have influenced each other.

Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.