Moshe Zakai
Encyclopedia
Moshe Zakai is Distinguished Professor at the Technion, Israel
in Electrical Engineering
, member of the Israel Academy of Sciences and Humanities
and Rothschild Prize winner.
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 where he was assigned to research and development of radar
systems. From 1956–1958 he did graduate work at the University of Illinois on an Israeli Government Fellowship, and was awarded the PhD
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
and in 1970 he was appointed the holder of the Fondiller Chair in Telecommunication. He was appointed Distinguished Professor in 1985. From 1970 till 1973 he served as the Dean
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.
Moshe Zakai is married to Shulamit (Mita) Briskman, they have 3 children and 12 grandchildren.
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
" or the "Wiener process
" 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
developed a stochastic calculus
(the Ito calculus
) for such random processes.
, that if a sequence of smooth functions which present the input to a physical system converge to something like a Brownian motion
, then the sequence of outputs of the system do not converge in the classical sense. Several papers written by Eugene Wong 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.
. This led to the same problem for nonlinear dynamical systems. The results for this case were highly complicated. Around 1967, Zakai derived a considerably simpler solution for the optimal filter. It is known as the Zakai equation
, and has been the starting point for further research work in this field.
and Eduardo Mayer-Wolf) introduced new approaches and results that gave a new impact to this topic.
, Paul Malliavin
developed in the 1970s
a "stochastic calculus of variations" known as the "Malliavin calculus
". It turned out that in this setup it is possible to define a stochastic integral which will include the Ito integral. The papers of Zakai with David Nualart, Ali Süleyman Üstünel, Zeitouni and Mayer-Wolf promoted the understanding and applicability of the Malliavin calculus.
The monograph
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 and Mayer-Wolf extended to some general cases results of information theory which were known for simpler spaces.
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...
in Electrical Engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, member of the Israel Academy of Sciences and Humanities
Israel Academy of Sciences and Humanities
The Israel Academy of Sciences and Humanities, based in Jerusalem, was set up in 1961 by the State of Israel to foster contact between scholars from the sciences and humanities in Israel, to advise the government on research projects of national importance, and to promote excellence. It comprises...
and Rothschild Prize winner.
Biography
Moshe Zakai was born in Sokółka, Poland, to his parents Rachel and Eliezer Zakheim with whom he immigrated to IsraelIsrael
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...
in 1936. He got the BSc
BSC
BSC is a three-letter abbreviation that may refer to:Science and technology* Bachelor of Science , an undergraduate degree* Base Station Controller, part of a mobile phone network; see: Base Station subsystem...
degree in Electrical Engineering from the Technion – Israel Institute of Technology in 1951. He joined the Scientific Department of the Defense Minister of Israel where he was assigned to research and development of radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
systems. From 1956–1958 he did graduate work at the University of Illinois on an Israeli Government Fellowship, and was awarded the PhD
PHD
PHD may refer to:*Ph.D., a doctorate of philosophy*Ph.D. , a 1980s British group*PHD finger, a protein sequence*PHD Mountain Software, an outdoor clothing and equipment company*PhD Docbook renderer, an XML renderer...
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
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
and in 1970 he was appointed the holder of the Fondiller Chair in Telecommunication. He was appointed Distinguished Professor in 1985. From 1970 till 1973 he served as the Dean
Dean (education)
In academic administration, a dean is a person with significant authority over a specific academic unit, or over a specific area of concern, or both...
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.
Moshe Zakai is married to Shulamit (Mita) Briskman, they have 3 children and 12 grandchildren.
Major awards
- 1973 Fellow of the Institute of Electrical and Electronic Engineers (IEEE)
- 1988 Fellow of the Institute of Mathematical StatisticsInstitute of Mathematical StatisticsThe 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 of the world...
- 1989 Foreign member of the US National Academy of EngineeringNational Academy of EngineeringThe National Academy of Engineering is a government-created non-profit institution in the United States, that was founded in 1964 under the same congressional act that led to the founding of the National Academy of Sciences...
- 1993 Member of the Israel Academy of Sciences and HumanitiesIsrael Academy of Sciences and HumanitiesThe Israel Academy of Sciences and Humanities, based in Jerusalem, was set up in 1961 by the State of Israel to foster contact between scholars from the sciences and humanities in Israel, to advise the government on research projects of national importance, and to promote excellence. It comprises...
- 1993 The IEEE Control Systems AwardIEEE Control Systems AwardThe IEEE Control Systems Award is a technical field award given to an individual by the Institute of Electrical and Electronics Engineers "for outstanding contributions to control systems engineering, science or technology"...
- 1994 The Rothschild Prize in Engineering
Background
Zakai's main research concentrated on the study of the theory of stochastic processesFiltering problem (stochastic processes)
In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some observations of that system...
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
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
" or the "Wiener process
Wiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
" 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
1940s
File:1940s decade montage.png|Above title bar: events which happened during World War II : From left to right: Troops in an LCVP landing craft approaching "Omaha" Beach on "D-Day"; Adolf Hitler visits Paris, soon after the Battle of France; The Holocaust occurred during the war as Nazi Germany...
Kiyoshi Itō
Kiyoshi Ito
was a Japanese mathematician whose work is now called Itō calculus. The basic concept of this calculus is the Itō integral, and among the most important results is Itō's lemma. The Itō calculus facilitates mathematical understanding of random events...
developed a stochastic calculus
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...
(the Ito calculus
Ito calculus
Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion . It has important applications in mathematical finance and stochastic differential equations....
) for such random processes.
The relation between classical and Ito calculi
From the results of Ito it became clear, back in the 1950s1950s
The 1950s or The Fifties was the decade that began on January 1, 1950 and ended on December 31, 1959. The decade was the sixth decade of the 20th century...
, that if a sequence of smooth functions which present the input to a physical system converge to something like a Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
, then the sequence of outputs of the system do not converge in the classical sense. Several papers written by Eugene Wong 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 problem of the optimal filtering (separating the signal from the noise) of a wide class of linear dynamical system is known as the Kalman filterKalman filter
In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...
. This led to the same problem for nonlinear dynamical systems. The results for this case were highly complicated. Around 1967, Zakai derived a considerably simpler solution for the optimal filter. It is known as the Zakai equation
Zakai equation
In filtering theory the Zakai equation is a linear recursive filtering equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear recursive equation for the normalized density of the hidden state...
, and 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. Several papers written by Zakai jointly with coauthors (Ben-Zion Bobrovsky, Ofer Zeitouni, Jacob ZivJacob Ziv
Jacob Ziv is an Israeli computer scientist who, along with Abraham Lempel, developed the LZ family of lossless data compression algorithms.-Biography:...
and Eduardo Mayer-Wolf) introduced new approaches and results that gave a new impact to this topic.
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 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 Ito calculusIto calculus
Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion . It has important applications in mathematical finance and stochastic differential equations....
, Paul Malliavin
Paul Malliavin
Paul Malliavin was a French mathematician. He was Professor Emeritus at the Pierre and Marie Curie University. He was a member of the Academy of Sciences since 1979.-Domains of research:...
developed in the 1970s
1970s
File:1970s decade montage.png|From left, clockwise: US President Richard Nixon doing the V for Victory sign after his resignation from office after the Watergate scandal in 1974; Refugees aboard a US naval boat after the Fall of Saigon, leading to the end of the Vietnam War in 1975; The 1973 oil...
a "stochastic calculus of variations" known as the "Malliavin calculus
Malliavin calculus
The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....
". It turned out that in this setup it is possible to define a stochastic integral which will include the Ito integral. The papers of Zakai with David Nualart, Ali Süleyman Üstünel, Zeitouni and Mayer-Wolf promoted the understanding and applicability of the Malliavin calculus.
The monograph
Monograph
A monograph is a work of writing upon a single subject, usually by a single author.It is often a scholarly essay or learned treatise, and may be released in the manner of a book or journal article. It is by definition a single document that forms a complete text in itself...
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 and Mayer-Wolf extended to some general cases results of information theory which were known for simpler spaces.
Volume in honor of Zakai's 65 birthday
- Eddy Mayer-Wolf, Ely Merzbach, Adam Shwartz (Eds.), Stochastic Analysis: Liber Amicorum for Moshe Zakai, Academic Press, San Diego, California, 1991.
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 [17]. For later publications search for M Zakai in arXiv.