Pollen-based reconstructions of Japanese biomes at 0, 6000 and 18,000 14C yr bp

A biomization method, which objectively assigns individual pollen assemblages to biomes ( Prentice et al., 1996 ), was tested using modern pollen data from Japan and applied to fossil pollen data to reconstruct palaeovegetation patterns 6000 and 18,000 14C yr bp Biomization started with the assignment of 135 pollen taxa to plant functional types (PFTs), and nine possible biomes were defined by specific combinations of PFTs. Biomes were correctly assigned to 54% of the 94 modern sites. Incorrect assignments occur near the altitudinal limits of individual biomes, where pollen transport from lower altitudes blurs the local pollen signals or continuous changes in species composition characterizes the range limits of biomes. As a result, the reconstructed changes in the altitudinal limits of biomes at 6000 and 18,000 14C yr bp are likely to be conservative estimates of the actual changes. The biome distribution at 6000 14C yr bp was rather similar to today, suggesting that changes in the bioclimate of Japan have been small since the mid-Holocene. At 18,000 14C yr bp the Japanese lowlands were covered by taiga and cool mixed forests. The southward expansion of these forests and the absence of broadleaved evergreen/warm mixed forests reflect a pronounced year-round cooling.

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.

Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces Hs(Ω) and tildeHs(Ω), for s in R and an open Ω in R^n. We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.