A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

We revisit some classical models for dilute polymeric fluids, and we show that thermodynamically consistent models for non-isothermal flows of these fluids can be derived in a very elementary manner. Our approach is based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid of interest, which, in turn, leads to explicit formulae for the Cauchy stress tensor and for all of the fluxes involved. Having identified these mechanisms and derived the governing equations, we document the potential use of the thermodynamic basis of the model in a rudimentary stability analysis. In particular, we focus on finite amplitude (nonlinear) stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

TIGHT ANALYSIS OF SHORTEST PATH CONVERGECAST IN WIRELESS SENSOR NETWORKS

We consider the convergecast problem in wireless sensor networks where each sensor has a reading that must reach a designated sink. Since a sensor reading can usually be encoded in a few bytes, more than one reading can readily fit into a standard transmission packet. We assume that each packet hop consumes one unit of energy. Our objective is to minimize the total energy consumed to send all readings to the sink. We show that this problem is NP-hard even when all readings are of fixed size. We then study a class SPEP of distributed algorithms that is completely defined by two properties. Firstly, the packets hop along some shortest path to the sink. Secondly, the nodes use an elementary packing algorithm to pack readings into packets. Our main technical contribution is a lower bound. We show that no algorithm for UCCP that either follows the shortest path or packs in an elementary manner is a (2 − ϵ)-approximation, for any fixed ϵ > 0. To complement this, we show that SPEP algorithms are [Formula: see text]-approximation for UCCP and 3-approximation for CCP, where k ≥ 2 is the number of readings that can fit within a packet. We conclude with some special cases and experimental observations.

Natural Language in Multimodal and Multimedia Systems

Recent years have witnessed a rapid growth in the development of multimedia applications. Improving technology and tools enable the creation of large multimedia archives and the development of completely new styles of interaction. This article provides a survey of multimedia applications in which natural language plays a significant role. Conventional multimodal systems usually do not maintain explicit representations of the user's input and handle mode integration only in elementary manner. This article shows how the generalization of techniques and representation formalisms developed for the analysis of natural language can help to overcome some of these problems. It surveys techniques for building automated multimedia presentation systems drawing upon lessons learned during the development of natural language generators. Finally, it argues that the integration of natural language technology can lead to a qualitative improvement of existing methods for document classification and analysis.

A class of integral equations

It was pointed out by Copson(1) in 1947 that the solution of the integral equation for the electrostatic problem for a circular disc could be reduced to the solution of Abel integral equations and hence a solution obtained in a fairly elementary manner. This result was obtained independently by Lebedev(2), who also obtained a similar result for the electrostatic problem for the spherical cap. The solution for the spherical cap was also obtained independently by Collins (3). In view of the relative simplicity of the approach it seems to be of interest to examine whether there exist other integral equations which can be treated in a similar fashion. In the present paper one such class of integral equations will be considered.

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

A New Form of Boyle's Law Tube

The following apparatus for verifying Boyle's Law in an elementary manner is of interest from the point of view of pure mathematics as well as that of the laboratory. The method of construction, which I have described in the July (1910) number of the School World, presents a little difficulty in the first attempt, but is easily mastered in the second.

The Bolyai-Lobatschewsky Non-Euclidean Geometry: an Elementary Interpretation of this Geometry, and some Results which follow from this Interpretation

Much has been written in recent years on the foundations of geometry, chiefly in Germany and Italy, and the relations of the various Non-Euclidean geometries to the Euclidean system are now more generally known among mathematicians. But most of these writings involve a knowledge of more advanced mathematics, while it has been found difficult to represent even the simplest Non-Euclidean geometry—that of Bolyai-Lobatschewsky—in an elementary manner.

III. On the proof of the law of errors of observations

The object of this Paper is to give the mathematical proof, in its most general form, of the law of single errors of observations, on the hypothesis that each error in practice arises from the joint operation of a large number of independent sources of error, each of which, did it exist alone, would occasion errors of extremely small amount as compared generally with those actually produced by all the sources combined. This proof is contained in a process given for a different object, namely, Poisson’s generalization of Laplace’s investigation of the law of the mean results of a large number of observations, to be found in the Connaissance des Temps’ for 1827, and also in his 'Recherches sur la Probabilité des Jugements ’ it is also re­produced in Mr. Todhunter’s able History of the Theory of Probability.’ It is not therefore pretended that any new results are arrived at in the present Paper. Considering, however, the importance and celebrity of the question, and the refined and difficult character of Poisson’s analysis, it will not probably be deemed superfluous to show how the same law may be demonstrated with equal generality, in a much more simple and elementary manner. The difficulty of the general proof seems indeed to have been so extensively felt, that several attempts have been made to simplify it. However, so far as the present writer is aware, no proof has been given, except Poisson’s, which is not open to grave objection, as based upon unjustifiable assumptions, or as unduly limiting, the generality of the in­vestigation.