Moshe Zakai (; December 22, 1926 – November 27, 2015) was a Distinguished Professor at the Technion, Israel in

electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

, member of the

Israel Academy of Sciences and Humanities Israel Academy of Sciences and Humanities, based in Jerusalem, was established in 1961 by the State of Israel to foster contact between Israeli scholars in the sciences and humanities and create a think tank for advising the government on res ...

and

Rothschild Prize Yad Hanadiv (The Rothschild Foundation) is a Rothschild family philanthropic foundation in Israel. Goals and objectives Yad Hanadiv defines its mission as: Dedicated to creating resources for advancing Israel as a healthy, vibrant, democratic so ...

winner.

Biography

Moshe Zakai was born in

Sokółka Sokółka (; , ) is a town in northeastern Poland, seat of the Sokółka County in Podlaskie Voivodeship. It is a busy rail junction located on the international Warsaw–Białystok–Grodno line, with additional connections which go to Suwałki a ...

, Poland, to his parents Rachel and Eliezer Zakheim with whom he immigrated to Israel in 1936. He got the BSc degree in electrical engineering from the Technion – Israel Institute of Technology in 1951. He joined the scientific department of the

Defense Minister of Israel The Ministry of Defense (, acronym: ) of the government of Israel, is the governmental department responsible for defending the State of Israel from internal and external military threats. Its political head is the defense minister of Israel, ...

, where he was assigned to research and development of

radar Radar is a system that uses radio waves to determine the distance ('' ranging''), direction ( azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track ...

systems. From 1956 to 1958, he did graduate work at the

University of Illinois The University of Illinois Urbana-Champaign (UIUC, U of I, Illinois, or University of Illinois) is a public university, public land-grant university, land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United ...

on an Israeli Government Fellowship, and was awarded the

PhD A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

in electrical engineering. He then returned to the scientific department as head of the communication research group. In 1965, he joined the faculty of the Technion as an associate professor. In 1969, he was promoted to the rank of

professor Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...

and in 1970, he was appointed the holder of the Fondiller Chair in Telecommunication. He was appointed distinguished professor in 1985. From 1970 until 1973, he served as the

dean Dean may refer to: People * Dean (given name) * Dean (surname), a surname of Anglo-Saxon English origin * Dean (South Korean singer), a stage name for singer Kwon Hyuk * Dean Delannoit, a Belgian singer most known by the mononym Dean * Dean Sw ...

of the faculty of Electrical Engineering, and from 1976 to 1978 he served as vice president of academic affairs. He retired in 1998 as distinguished

professor emeritus ''Emeritus/Emerita'' () is an honorary title granted to someone who retirement, retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". ...

. Moshe Zakai was married to Shulamit (Mita) Briskman, they have 3 children and 12 grandchildren.

Major awards

* 1973 Fellow of the Institute of Electrical and Electronics Engineers (

IEEE The Institute of Electrical and Electronics Engineers (IEEE) is an American 501(c)(3) organization, 501(c)(3) public charity professional organization for electrical engineering, electronics engineering, and other related disciplines. The IEEE ...

) * 1988 Fellow of the

Institute of Mathematical Statistics The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the development, dissemination, and application of statistics and probability. The Institute currently has about 4,000 members in all parts ...

* 1989 Foreign member of the US

National Academy of Engineering The National Academy of Engineering (NAE) is an American Nonprofit organization, nonprofit, NGO, non-governmental organization. It is part of the National Academies of Sciences, Engineering, and Medicine (NASEM), along with the National Academ ...

* 1993 Member of the

* 1993 The

IEEE Control Systems Award The IEEE Control Systems Award is a technical field award given to an individual by the Institute of Electrical and Electronics Engineers (IEEE) "for outstanding contributions to control systems engineering, science or technology". It is an IEEE- ...

* 1994 The

in Engineering

Research

Background

Zakai's main research concentrated on the study of the theory of stochastic processes and its application to information and control problems; namely, problems of noise in communication radar and control systems. The basic class of random processes which represent the noise in such systems are known as "

white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...

" or the "

Wiener process In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one o ...

" where the white noise is "something like a derivative" of the Wiener process. Since these processes vary quickly with time, the classical differential and integral calculus is not applicable to such processes. In the 1940s

Kiyoshi Itō Kiyoshi, (きよし or キヨシ), is a Japanese given name, also spelled Kyoshi. Possible meanings *''Kyōshi'', a form of Japanese poetry *Kyōshi, a Japanese honorific Written forms *清, "cleanse" *淳, "pure" *潔, "undefiled" *清志, " ...

developed a

stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...

(the

Ito calculus Ito, Itō or Itoh may refer to: Places * Ito Island, an island of Milne Bay Province, Papua New Guinea * Ito Airport, an airport in the Democratic Republic of the Congo * Ito District, Wakayama, a district located in Wakayama Prefecture, Japan ...

) for such random processes.

The relation between classical and Ito calculi

From the results of Ito it became clear, back in the 1950s, that if a sequence of smooth functions which present the input to a physical system converge to something like a

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

, then the sequence of outputs of the system do not converge in the classical sense. Several papers written by

Eugene Wong Eugene Wong (born December 24, 1934, in Nanjing, China) is a Chinese-American computer scientist and mathematician. Wong's career has spanned academia, university administration, government and the private sector. Together with Michael Stonebr ...

and Zakai clarified the relation between the two approaches. This opened up the way to the application of the Ito calculus to problems in physics and engineering. These results are often referred to as Wong-Zakai corrections or theorems.

Nonlinear filtering

The solution to the problem of the optimal

filtering Filtration is a physical process that separates solid matter and fluid from a mixture. Filter, filtering, filters or filtration may also refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Fil ...

of a wide class of linear dynamical system is known as the

Kalman filter In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unk ...

. This led to the same problem for nonlinear dynamical systems. The results for this case were highly complicated and were initially studied by Stratonovich in 1959 - 1960 and later by

Kushner The surname Kushner is an English-based transliteration of the Yiddish name קושנער, a variant of קושניר (Kushnir), an occupational name stemming from קירזשנער (kirzshner), a furrier. This is related to the German word Kürschne ...

in 1964, leading to the Kushner-Stratonovich equation, a non-linear

stochastic partial differential equation Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They hav ...

(SPDE) for the conditional probability density representing the optimal filter. Around 1967, Zakai derived a considerably simpler SPDE for an unnormalized version of the optimal filter density. It is known as the

Zakai equation In filtering theory the Zakai equation is a linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear stochastic partial differential equation for the nor ...

, and it has the great advantage of being a linear SPDE. The Zakai equation has been the starting point for further research work in this field.

Comparing practical solutions with the optimal solution

In many cases the optimal design of communication or radar operating under noise is too complicated to be practical, while practical solutions are known. In such cases it is extremely important to know how close the practical solution is to the theoretically optimal one.

Extension of the Ito calculus to the two-parameter processes

White noise and Brownian motion (the Wiener process) are functions of a single parameter, namely time. For problems such as rough surfaces it is necessary to extend the Ito calculus to two parameter Brownian sheets. Several papers which he wrote jointly with Wong extend the

Ito integral Ito, Itō or Itoh may refer to: Places * Ito Island, an island of Milne Bay Province, Papua New Guinea * Ito Airport, an airport in the Democratic Republic of the Congo * Ito District, Wakayama, a district located in Wakayama Prefecture, Japan ...

to a "two-parameter" time. They also showed that every functional of the Brownian sheet can be represented as an extended integral.

The Malliavin calculus and its application

In addition to the

Paul Malliavin Paul Malliavin (; September 10, 1925 – June 3, 2010) was a French mathematician who made important contributions to harmonic analysis and stochastic analysis. He is known for the Malliavin calculus, an infinite dimensional calculus for func ...

developed in the 1970s a "stochastic calculus of variations", now known as the

Malliavin calculus In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...

. It turned out that in this setup it is possible to define a

stochastic integral Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...

which will include the Ito integral. The papers of Zakai with

David Nualart David Nualart (born 21 March 1951) is a Spanish mathematician working in the field of probability theory, in particular on aspects of stochastic processes and stochastic analysis. He is retired as Black-Babcock Distinguished Professor of Mathematic ...

Ali Süleyman Üstünel Ali ibn Abi Talib (; ) was the fourth Rashidun caliph who ruled from until his assassination in 661, as well as the first Shia Imam. He was the cousin and son-in-law of the Islamic prophet Muhammad. Born to Abu Talib ibn Abd al-Muttalib an ...

and Zeitouni promoted the understanding and applicability of the Malliavin calculus. The

monograph A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...

of Üstünel and Zakai deals with the application of the Malliavin calculus to derive relations between the Wiener process and other processes which are in some sense "similar" to the probability law of the Wiener process. In the last decade he extended to transformations which are in some sense a "rotation" of the Wiener process and with Ustunel extended to some general cases results of information theory which were known for simpler spaces.

Further information

* On his life and research, see pages xi–xiv of the volume in honor of Zakai's 65 birthday. * For the list of publications until 1990, see pages xv–xx. For publications between 1990 and 2000, see 7 For later publications search fo
M Zakai in arXiv

References

{{DEFAULTSORT:Zakai, Moshe Israeli scientists Israeli inventors Jewish scientists Academic staff of Technion – Israel Institute of Technology Members of the Israel Academy of Sciences and Humanities Israeli Jews Polish emigrants to Israel 1926 births 2015 deaths Mathematical analysts Foreign associates of the National Academy of Engineering