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A set of fitted polynomial equations for calculating the physical variables density, entropy, heat capacity and potential temperature of a thermal saline fluid for a temperature range of 0–3741C, pressure range of 0.1–100 MPa and absolute salinity range of 0–40 g/kg is established. The freshwater components of the equations are extracted from the recently released tabulated data of freshwater properties of Wagner and Prus [2002. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. Journal of Physical and Chemical

We revisit a problem solved in 1963 by Zaanen & Luxemburg in this MONTHLY: What is the largest possible length of the graph of a monotonic function on an interval? Is there such a function that attains this length? This is an interesting and intriguing problem with a somewhat surprising answer that should be of interest to a broad spectrum of mathematicians, starting with upper-level undergraduates. Our proof is very elementary, as opposed to the proof of Zaanen & Luxemburg. We are also able to give a pleasing geometric interpretation of our proof that is not possible with the proof of Zaanen & Luxemburg. 1. INTRODUCTION. This article, accessible to advanced undergraduates, concerns a simply stated but fascinating problem in analysis. The problem concerns monotonic functions, which are functions that are either nonincreasing or nondecreasing. Arc Length Problem. Let (a,b) be a closed interval. What is the maximum arc length that can be attained by a monotonic function f :( a,b) → ( f (a), f (b))? Is there a function that attains this maximum?

This book is a comprehensive study of the Radon transform, which operates on a function by integrating it over hyperplanes. The book begins with an elementary and graphical introduction to the Radon transform, tomography and CT scanners, followed by a rigorous development of the basic properties of the Radon transform. Next the author introduces Grassmann manifolds in the study of the k-plane transform (a version of the Radon transform) which integrates over k-dimensional planes rather than hyperplanes. The remaining chapters are concerned with more advanced topics, such as the attenuated Radon transform and generalized Radon transforms defined by duality of homogeneous spaces and double fibrations. Questions of invertibility and the range of the Radon transform are dealt with and inversion formulas are developed with particular attention to functions on L2 spaces and some discussion of the case of Lp spaces.

Introduction The purpose of this chapter is to give an informal introduction to the subject of tomography. There are very few mathematical requirements for this chapter, so readers who are not specialists in the field, indeed who are not mathematicians or scientists, should find this material accessible and interesting. Specialists will find a graphic and intuitive presentation of the Radon transform and its approximate inversion. Tomography is concerned with solving problems such as the following. Suppose that we are given an object but can only see its surface. Could we determine the nature of the object without cutting it open? In 1917 an Austrian mathematician named Johann Radon showed that this could be done provided the total density of every line through the object were known. We can think of the density of an object at a specific point as the amount of material comprising the object at that point. The total density along a line is the sum of the individual densities or amounts of material. In 1895 Wilhelm Roengten discovered x-rays, a property of which is their determining of the total density of an object along their line of travel. For this reason, mathematicians call the total density an x-ray projection . It is immaterial whether the x-ray projection was obtained via x-rays or by some other method; we still call the resulting total density an x-ray projection.

There is a great deal of current interest in inverting attenuated Radon transforms which occur in single photon emission tomography. These transforms are special cases of generalized Radon transforms $R_\mu $ which are defined by integrating a function over lines with respect to given positive $C^2$ measures $\mu $.As a positive result, we show $R_\mu $ is locally invertible. However, on the negative side, we present counterexamples to show that some smoothness assumptions on the measures are crucial for invertibility. The second example shows that limited angle tomography is not possible in general, even with somewhat smoother measures.

A variably attenuated x-ray transform is shown to be invertible via an integral formula for the inversion of the exponential x-ray transform.The attenuation must be known and constant in a convex set containing the unknown emitter. However the attenuation can be otherwise arbitrary.If $\mu $ denotes the attenuation constant of the exponential x-ray transform then the integral formula computes the Fourier transform of the emitter on all of $R^n $ from the values of the Fourier transform on the set $A^\mu = \{ {\sigma + i\mu \omega \in C^n |\omega \in S^{n - 1} ,\sigma \bot \omega } \}$. Of course F. Natterer [Numer. Math., 32 (1979), pp. 431–438] showed that the values of the Fourier transform of the emitter can be obtained from the Fourier transform of the exponential x-ray transform. In essence however the basic method is analytic continuation from the set $A^\mu $.A consequence of the integral formula is a uniqueness theorem for attenuated x-ray transforms of the type considered here: if the transforms ...

Analytic families of differential complexes, which are useful for proving blowing down and direct image theorems for regular families of strongly pseudoconvex manifolds, are defined and shown to have an associated deformation theory. The results of this deformation theory include a criteria for the coherence of the cohomology sheaves of an analytic family of complexes, a theorem on the upper semi-continuity of the dimensions of the cohomology of the fibers of the family and a result on the constancy of the Euler characteristic along the fibers.