B. Mathematical Sciences
Table E.III1 summarizes international research capabilities for the major subareas of mathematical science.
Table E.III1. Mathematical Sciences
MATHEMATICAL SCIENCES 
UNITED KINGDOM 
FRANCE 
GERMANY 
OTHER COUNTRIES 
JAPAN 
PACIFIC RIM 
FSU 
Applied Analysis and Physical Mathematics 
Fluid dynamics 
Bolzman's equations, Dynamic systems,
Computer vision 
Hungary Real variables 

China 
Russia Numerical methods, Mechanics 

Computational mathematics 
Linear algebra 
Finite elements, Nonsmooth optimization 
Finite elements, Interactive methods  Israel
Computational physics Canada 
China

Russia 

Stochastic Analysis Applied Probability and Statistics 
Levy processes  Canada 
China

Russia 

Systems and Control 
Control theory 
Canada

China 
Russia 

Discrete Mathematics 
Computer algebra 
Hungary Canada Czech Republic Computational geometry 
China 
Russia 
Basic research in applied analysis and physical mathematics directly contributes to the modeling, analysis, and control of complex phenomena and systems active within the Army. Applied mathematicians define practical boundaries, set the framework of analysis, and act as collaborators for scientists and engineers on many development projects. Many nations show significant capability in a number of areas identified as having potential impact on future Army technologies. This is consistent with the fact that many advanced applied mathematics research efforts involve only a small number of researchers and have minimal hardware requirements. Thus even nations without an extremely powerful industrial or research base can have a few specific points of excellence in mathematics.
There are many examples of specific areas of computational mathematics which hold promise for military applications. Computational fluid dynamics studies in the United Kingdom, Canada, and Israel can contribute significantly to missile, rotor and explosive design. Advanced work in finite element analysis in France and Germany can be applied to the problems of the design and function of complex mechanical structures. Also of interest are international research efforts in optimization, linear algebra (France), fuzzy logic (Japan), and computational geometry (Czechoslovakia) which are applicable to the development of new computer network hardware and software platforms. Control theory work has also been used for the development of computer systems, as well as applications in robotics.
Germany, France, and the United Kingdom are all considered to be on a par with the United States in a number of these areas of mathematics research. In general, Canada and Japan are also considered to be working at or near this high level. Both China and India exhibit strong potential research efforts, which are constantly improving and will conceivably soon be worldleading. The countries of the former Soviet Union show a declining capability, largely due to a lack of resources. For example, though many important numerical methods for modeling physical phenomena were developed in the Soviet Union in the 50’s and 60’s, current research is no longer considered worldleading. Additionally, Ukraine is noted for a traditional weakness in more basic research, and tends to be stronger in development areas. Many other small countries also have very strong mathematical talent  Holland, Denmark, Hungary, Israel, Poland, Romania, Greece, Sweden, and Norway, and all could be considered for potential cooperative efforts in specific areas.
The nature of basic mathematics research implies that it often has no stated or direct tie to any specific application, military or otherwise. It is often the case that seemingly unrelated research will have effects on the development of critical technologies, for example, the influence of advances in control theory on the development of nonskid brakes.