Mathematics, historical

Historical Slang Literary Terms Mathematics Twentieth Century History... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The accuracy of absolute risk results depends on (1) whether all the significant contributors to risk have been analyzed, (2) the realism of the mathematical models used to predict failure characteristics and accident phenomena, and (3) the statistical uncertainty associated with the various input data. The achievable accuracy of absolute risk results is very dependent on the type of hazard being analyzed. In studies where the dominant risk contributors can be calibrated with ample historical data (e.g., the risk of an engine failure causing an airplane crash), the uncertainty can be reduced to a few percent. However, many authors of published studies and other expert practitioners have recognized that uncertainties can be greater than 1 to 2 orders of magnitude in studies whose major contributors are rare, catastrophic events. 

A. V. Sokolov. Optical Properties ofMetaL. Elsevier, New York, 1967, Chapters 10 and 11. A very detailed, mathematical description of solutions to the wave equations, with a nice historical perspective. 

Other important historical landmarks include the founding, in 1984, of the Santa Fe Institute, which is one of the leading interdisciplinary centers for complex systems theory research the first conference devoted solely to research in cellular automata (which is a prototypical mathematical model of complex systems), organized by Farmer, Toffoli and Wolfram at MIT in 1984 [farmer84] and the first artificial life conference, organized by Chri.s Langton at Los Alamos National Laboratory, in 1987 [lang89]. 

Given that, historically, new developments in mathematics and physics have always been closely aligned, we can anticipate a strong need to develop a (possibly radically) new mathematics in order to effectively deal with these issues. 

Historically, the visible emission lines shown in Figure 15-3 were the first atomic hydrogen lines discovered. They were found in the spectrum of the sun by W. H. Wollaston in 1802. In 1862, A. J. Angstrom announced that there must be hydrogen in the solar atmosphere. These lines were detected first because of the lesser experimental difficulties in the visible spectral region. They are called the "Balmer series because J. J. Balmer was able to formulate a simple mathematical relation among the frequencies (in It S). The ultraviolet series shown in Figure 15-3 was... 

There are, however, other options for treating data from both first- and second-order kinetics. Collectively, they are known as time lag methods. These methods are primarily of historical interest, although the mathematical rearrangements provide insights into the nature of the functions involved. 

The failure to identify the necessary authigenic silicate phases in sufficient quantities in marine sediments has led oceanographers to consider different approaches. The current models for seawater composition emphasize the dominant role played by the balance between the various inputs and outputs from the ocean. Mass balance calculations have become more important than solubility relationships in explaining oceanic chemistry. The difference between the equilibrium and mass balance points of view is not just a matter of mathematical and chemical formalism. In the equilibrium case, one would expect a very constant composition of the ocean and its sediments over geological time. In the other case, historical variations in the rates of input and removal should be reflected by changes in ocean composition and may be preserved in the sedimentary record. Models that emphasize the role of kinetic and material balance considerations are called kinetic models of seawater. This reasoning was pulled together by Broecker (1971) in a paper called "A kinetic model for the chemical composition of sea water."... 

Surprisingly little attention has been given hitherto to the definition of the laboratory. A space has to be specially adapted to deserve that title. It would be easy to assume that the two leading experimental sciences, physics and chemistry, have historically depended in a similar way on access to a laboratory. But while chemistry, through its alchemical ancestry with batteries of stills, had many fully fledged laboratories by the seventeenth century, physics was discovering the value of mathematics. Even experimental physics was content to make use of almost any indoor space, if not outdoors, ignoring the possible value of a laboratory. The development of the physics laboratory had to wait until the nineteenth century... 

Jaro, M. (2003), Metal threads in historical textiles, NATO Science Series, II Mathematics, Physics and Chemistry, Vol. 117 (Molecular and Structural Archaeology Cosmetic and Therapeutic Chemicals), pp. 163-178. 

Quantitative reviews are mathematical estimations that rely upon historical evidence or estimates of failures to predict the occurrence of an event. These reviews are sometimes referred to as a Quantitative Risk Assessment (QRA). 

Having briefly noted the historical highlights of the WSL effect, we now examine the basic mathematical argument for its existence, namely, the idea that the energy spectrum of an infinite crystal is discretized by an applied held. For the present discussions, we assume that the applied field is linear, with its strength given by its gradient 7. 

This chapter will overview the historical development of the Fukui function concept from its origins in the work of Parr and Yang [1] to the present day. The recent review by Levy and one of the present authors provides a more mathematical perspective on the Fukui function [5]. 

By common agreement among many historians of science, "chemistry" and "physics" became fairly well demarcated communities or disciplines around 1830, some hundred years before the founding of the Journal of Chemical Physics2 This was about the time that Auguste Comte was embarking on his Cours de philosophie positive, in which he laid out a hierarchy of the positive sciences as he observed them in contemporary Paris. In this hierarchy, the mastery of mathematics and physics was historically and foundationally prior to chemistry. 

In influencing the history and philosophy of science of later decades, Comte s positivist classification created the conviction that the constitution of mathematics and physics was historically prior to chemistry and conceptually more fundamental than chemistry. 4 But the positivist history we often have accepted from Comte is flawed. "Chemistry" as a discipline preceded "physics." In the next two chapters, I will deal with the claim that physics is conceptually a more fundamental science than chemistry and will analyze the characteristic aims and methods of nineteenth-century chemistry, particularly as reinforced through the hegemony of organic chemistry. 

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