A recurring theorem about pairs of quadratic forms and extensions: a survey

AbstractThis is a historical and mathematical survey of work on necessary and sufficient conditions for a pair of quadratic forms to admit a positive definite linear combination and various extensions thereof

Fleets of World War II

In this handy-sized, reasonably priced book, Richard Worth and his publisher have provided an excellent instrument of discovery for readers whose range of interest in World War II includes the many fleets that fought, and even the few that only looked on nervously

How Navies Fight, and Why

In the two and a quarter centuries since the birth of the United States, its navy has experienced, or watched closely, a fair range of the possibilities of war at and from the sea. It has been a small navy fighting a large one (the Continental navy versus Britain\u27s in the American Revolution), has seen two large ones fight each other (the French against the British in that same war), has been a small navy fighting against first legalized and then state piracy (the Quasi-War and the Barbary War), for a second time has been a small navy fighting a big one (the War of 1812), has, as a small navy, fought against a foe without a navy (the Mexican War), then as a small navy suddenly grown large fought against an enemy with but fragments of a navy (the American Civil War), became one of two medium-sized navies fighting against one another (the Spanish-American War), has been one of several large navies allied against several others (the two world wars), twice again was a large navy opposing a foe without one (the Korean and Vietnam wars), watched one very small navy fight against two others, during which it acted as a large auxiliary to one of the principal combatants (the Levantine war of1973), from a distance observed a medium-sized navy against a medium-sized air force (the South Atlantic war), watched with interest two countries with minuscule navies fighting against each other ashore and against each other\u27s economic partners or nonbelligerent allies afloat (the Iran-Iraq War), and twice took part as a large navy against a minuscule one (the U.S. quasi-war against Iran and then the full United Nations war against Iraq).

The Eight Epochs of Math as Regards Past and Future Matrix Computations

This survey paper gives a personal assessment of epoch-making advances in matrix computations, from antiquity and with an eye toward tomorrow. It traces the development of number systems and elementary algebra and the uses of Gaussian elimination methods from around 2000 BC on to current real-time neural network computations to solve time-varying matrix equations. The paper includes relevant advances from China from the third century AD on and from India and Persia in the ninth and later centuries. Then it discusses the conceptual genesis of vectors and matrices in Central Europe and in Japan in the fourteenth through seventeenth centuries AD, followed by the 150 year cul-de-sac of polynomial root finder research for matrix eigenvalues, as well as the superbly useful matrix iterative methods and Francis’ matrix eigenvalue algorithm from the last century. Finally, we explain the recent use of initial value problem solvers and high-order 1-step ahead discretization formulas to master time-varying linear and nonlinear matrix equations via Zhang neural networks. This paper ends with a short outlook upon new hardware schemes with multilevel processors that go beyond the 0–1 base 2 framework which all of our past and current electronic computers have been using