**Rufus Isaacs**' parents were Philip Simon Isaacs and Alice Cohn. Philip Isaacs came from a lowly background but became a successful businessman. Rufus was brought up in New York City where he attended the Franklin School. This school, founded by Julius Sachs in 1872, was at 18 West 89

^{th}Street. Isaacs [9]:-

He graduated from the Franklin School in 1932 and, later that year, entered the Massachusetts Institute of Technology in Cambridge, Massachusetts. He had been persuaded by his father to concentrate on 'air conditioning', a topic his father believed would lead to a well paid career in an area of great practical importance. He graduated with a Bachelor of Science degree in General Engineering in 1936. However, his interests were broad with a great love of the arts and literature. When at the Massachusetts Institute of Technology, he painted and wrote dramas which were performed by a student drama group. After graduating, Isaacs was employed by the Carrier Corporation of Syracuse, New York. This firm, a manufacturer and distributor of heating, ventilating and air conditioning systems, had been founded by Willis Carrier in 1915. Willis Carrier is said to be the inventor of modern air conditioning systems. Isaacs writes about his experiences at the Carrier Corporation in [4]:-... began writing short stories while he was in high school, where he served as co-editor-in-chief of the school magazine and yearbook.

After two years with the Carrier Corporation, where his talent and problem solving abilities were greatly appreciated, Isaacs enrolled in the Master's mathematics programme at Columbia University in New York. He was advised by Edward Kasner and, on 25 February 1939, submitted a paper entitledIn the Carrier Engineering Department, just after my engineering BS, there was a problem to be solved by a differential equation. Now, I had taken a course in differential equations; it told us how to set one up germane to a real-world problem and how to solve it; I was a master. Therefore, I wrote my equation including in it every detail of the problem. In average handwriting, its length was about two feet. And, hard as I find it to believe now, I was chagrined to find no solution rattling out as they had done for the homework problems in the textbook. Please restrain the polite snicker. The course had given no hint that not all differential equations can be solved in elementary closed form.

*A geometric interpretation of the difference quotient of polygenic functions*to the American Mathematical Society. The paper was eventually published as

*The finite differences of polygenic functions*in the

*Bulletin*of the American Mathematical Society in 1941. Philip Franklin writes in a review:-

Isaacs was awarded his Master's degree (an M.A.) in 1940 and continued to undertake research at Columbia University for his Ph.D. He was awarded his doctorate by Columbia University in 1943 for his thesisThe author shows how the values of the difference quotients of a polygenic function, formed at a given point, may be represented by the points of a surface in three space. He studies the geometry of this surface, one of whose plane sections is the Kasner derivative circle.

*A Finite Difference Function Theory*[2]:-

In 1941, even before the award of his Ph.D., Isaacs had taken up a job with Hamilton Standard Propellers, Pittsburgh, Pennsylvania. This firm was a major manufacturer of aircraft propellers and Isaacs was employed as a Senior Analytical Engineer. In 1942 he married Rose Bicov (born in Hartford, Ford, Connecticut on 6 February 1912, died in Baltimore, Maryland on 5 April 2004), the daughter of Benjamin Bicov and Sarah Beatman. Rose was [9]:-In his Ph.D. Thesis, he developed an extension of the calculus of finite differences to complex variables. Isaacs constructed a theory of functions of a complex variable which springs entirely and basically from the concept of the difference quotient(instead of the derivative)and called these functions monodiffric. Monodiffric functions are the analogue of the monogenic functions introduced by Euler. In his Ph.D. Thesis, Isaacs presented29new theorems with proofs and applications ...

Ellen was a medical doctor. Frances graduated from Tufts University, worked in occupational health and safety, married John Gilmore, and has published a book of poetry.... a charming lady. The marriage is considered by Isaacs as the prime event of his life. They have two lovely daughters - Ellen Sara(born1944)and Frances Carol.

Isaacs joined Hamilton Standard Propellers around the time the United States entered World War II. His work became war related, solving military problems involving aerodynamics, elasticity, and vibrations. Isaacs wrote [4]:-

In 1945, after the war ended, Isaacs was appointed as an assistant to Karl Menger in the Mathematics Department of Notre Dame University, South Bend, Indiana. Menger had come to the United States to escape the Nazis fully intending to return to Vienna after the war. However, this was not possible so Menger remained in the United States but left Notre Dame University in 1947 going to the Illinois Institute of Technology in Chicago. Once he knew that Menger was leaving and finding it hard to support his family on an academic salary, Isaacs decided to take a position with North American Aviation, Los Angeles. Back once again in the aviation industry, Isaacs was involved in the design of intercontinental guided missiles. He worked on subsonic and supersonic aerodynamics, missile guidance systems, and transstability flutter writing the 1948 reports 'Transstability Flutter of Supersonic Aircraft Panels' and 'A Static Stability Analysis for Thin Buckled Beams at Supersonic Speeds'. However, Isaacs was not very happy working for North American Aviation [4]:-At the outbreak of World War II, aircraft propeller blades were made of solid aluminum. Hamilton Standard proposed a hollow steel construction: a central narrow core was to be brazed along its length to an outer steel shell of the ultimate aerodynamic shape. When an eight-foot blade of such construction rotates at umpty thousand RPM, the centrifugal force load is born over the length of the braze as a shearing stress. How is it distributed? This urgent question was handed to me for a quick answer.

In 1948 he joined the Mathematics Department of the RAND Corporation and he worked there until 1954. It was at RAND that he developed the idea of differential games, the idea for which he is most famed today. Isaacs explained [1]:-At my first jobs, I was a lone mathematician, first among engineers, then among physicists.

His first RAND report on differential games wasThen under the auspices of the U.S. Air Force, RAND was concerned largely with military problems and, to us there, this syllogism seemed incontrovertible:(1)game theory is the analysis of conflict;(2)conflict analysis is the means of warfare planning;(3)therefore, game theory is the means of warfare planning. The early RAND discussions proposed and discarded schemes for a pursuing missile to seek an enemy aircraft. Collision course was good? Yes. But if the targeted craft did this(or that), the missile won't hit. Constant bearing navigation? That seems easier for the missile. But what if the evader does that(or this)? What optimal pursuit scheme for the missile? The crucial idea dawned: the question was unanswerable unless one also asked "what optimal evasive scheme for the aircraft?". The whole subject was game theory.

*Games of Pursuit*(1951). Here are examples of the types of problems he studied in that report (quoted in [2]):

Game 2: Vicariously, we observe the chase through regularly laid out city streets of a pedestrian fugitive by a police car. The car, although faster, must abide by local traffic rules which prohibit left (and also U) turns. Our model is played discretely on the vertices of an infinite square lattice. If E is at a point, he may move vertically or horizontally to any one of the four adjoining points; P moves two spaces but only in a direction the same as or to the right of that of his previous move.

He explains in [1] why he left RAND:-Through the Rand seminars, my colleagues were privy to my progress. Enthusiasm seemed ardent at first, but it dwindled. A proposed joint research project did not materialise. From my fellow mathematicians, I heard "Too hard", "Not rigorous", or "No existence theorem". ... 'Games of Pursuit'(1951)contained the tenet of transition but its presentation of continuous game techniques was crude and a bit erroneous. Understandably, it was rejected by a mathematical journal. I was beginning to grasp the innovations inherent in my incipient theory ... ; its novel flavour could simply not be conveyed in a brief article.

His work at RAND on differential games became the basis for his famous bookI felt overwhelmed by the difficulties in making my new subject fulfill its rich promise. I wanted Rand to devote substantial resources to a joint effort, or, failing that, to support a long-term investigation by myself alone. Neither was to be. Although I have never learned what went on behind administrative doors, my six-year stay at Rand ended in1954. When opportunities had long seemed nonexistent for simultaneously pursuing the subject and earning a living, I was offered a teaching fellowship abroad in1963. The teaching load was light; here was a chance to set down the ideas I had been carrying in my head for ten years.

*Differential games. A mathematical theory with applications to warfare and pursuit, control and optimization*(1965). For the publisher's description, extracts from the author's preface and Introduction as well as extracts from five reviews of the book see THIS LINK.

In 1969 Isaacs wrote the paper [6] in which he ties to correct what he perceives as a general misunderstanding of the concept of differential games:-

Let us return to our description of Isaacs' career. After leaving RAND in 1954 he spent one year as Senior Scientist at Lockheed Aircraft Company before joining Hughes Aircraft Company in 1955 as a Research Mathematician. He moved again, this time after three years, taking up the position of Mathematician at the Institute of Defense Analyses. After four years, 1958-62, he took up an appointment at the Center for Naval Analyses where he spent the five years 1962-67. In addition to these positions, Isaacs was a professor at George Washington University, Washington, DC, from 1961 to 1967. After these many jobs in industry, Isaacs became a full-time academic for the final ten years of his working life. He was appointed as a professor at Johns Hopkins University in 1967 and retired, being made professor emeritus, in December 1977. The article [9], written shortly after he retired, says:-Since the appearance of my book 'Differential Games'(1965), I have felt increasing concern over a certain prevalent misunderstanding of the real nature of differential games. My own reflections, from the perspective that trails the setting down of all the complex technical details, conversations with others and their writings, and reactions to my lectures, all bolster this view. At present, there appear to be comparatively few people who have grappled in depth with theproblems of differential games. Hence, there is a rather widespread failure to grasp the distinction from classical analysis. This distinction is due to the presence of two players with contrary purposes. When each makes a decision, he must take into account his opponent's portending action toward the opposite end, his opponent's similar wariness of the first player's actions, and so forth. This situation is basically different from that of much of classical mathematics, which, as I shall argue below, consists of one-player games. Thus, my main thesis is that differential games lie in a stratum distinctively above the mathematics to which we are accustomed; accordingly, our thinking in this field must transcend the habits inculcated by traditional training. Beyond are further strata which can be expected to harbor further novelties.

We note that the article [9] is written by Po-Lung Yu who studied for a Ph.D. at Johns Hopkins University advised by Isaacs.During his retirement, Isaacs plans to continue his research in mathematics and other creative pursuits. In celebrating his great contribution to differential games, we send our best wishes to him and his family. We also would like to thank him for spending time to write the article 'On Applied Mathematics', for this issue so that his experiences and wisdom can be shared by our readers.

Isaacs returned to one of his many mathematical loves, namely pure mathematics, when he published the paper *The distribution of primes in a special ring of integers* in 1979. This paper contains some fascinating ideas as a review by John Knopfmacher shows:-

Isaacs received several honours for his innovative work. He was SIAM Lecturer from 1961 to 1963. He was awarded the Frederick W Lanchester Prize by the Operations Research Society of America in 1965. The citation reads:-The arithmetic of polynomials in one variable over a finite field GF(q), in particular the distribution of irreducible polynomials, is quite well known and has been studied by various authors .... In this article, using the representation of non-negative integers in binary decimal notation, the author considers nonstandard definitions of addition and multiplication which convert the set N of all nonnegative integers into a ring isomorphic to GF[2, x]. In particular, as the author indicates, the theory of the distribution of irreducible polynomials could then be applied to this ring. However, he prefers to give a direct discussion of the distribution of the corresponding "primes" in N. Further, by direct reference to binary decimal notation, he defines a subset of "palindromic primes" with no well-known classical analogue, and obtains a similar result about their distribution in N.

One final honour that we note is the establishment of the Rufus Isaacs prize by the International Society of Dynamic Games. First awarded in 2004, the prize is awarded to two scholars every two years for:-The1965Lanchester Prize was awarded to Rufus Isaacs for his book, 'Differential Games', John Wiley,1965.

Finally let us quote from Wilson Rugh [8] giving his sketch of Isaacs as a person:-... outstanding contribution to the theory and applications of dynamic games.

Isaacs did not have many years of retirement to enjoy since, at the age of 66, he died from cancer in Johns Hopkins Hospital in Baltimore.For those of us fortunate enough to know him personally we will always remember this quiet, unassuming, and interesting man. He loved music, was an avid artist, and his mastery of the written word is evident in his published work. His fifteen-speed bicycle was always nearby and, to the dismay of everyone, he regularly used it to commute between home and campus in Baltimore's rush hours. But perhaps most of all it will be the twinkle in his eye that will stay. Some vignettes in recent years capture images of the man friends will remember with affection: Rufus visiting the sailing ship Cutty Sark to verify the accuracy of his hand crafted model, the radio-transmitting unloseable golf ball he purchased out of curiosity and promptly lost, and his professed desire to write a book entitled 'Engineering for Mathematicians'.

**Article by:** *J J O'Connor* and *E F Robertson*