Richard W. Hamming

Richard W. HammingRichard W. HammingRichard W. HammingRichard W. Hamming

Learning to LearnLearning to LearnThe Art of Doing Science and EngineeringThe Art of Doing Science and Engineering

Superintendent’s Guest LectureSuperintendent’s Guest Lecture

30 April 1990 30 April 1990

Editorial noteEditorial noteEditorial noteEditorial note

Dr. Hamming’s Superintendent’s Guest Dr. Hamming’s Superintendent’s Guest Lecture (SGL) presentation is an excellent Lecture (SGL) presentation is an excellent summary of the many diverse, challenging summary of the many diverse, challenging ideas tackled in his book and course.ideas tackled in his book and course.

These slides summarize DrThese slides summarize Dr Hamming’s Hamming’s background, contributions, awards and background, contributions, awards and publications as listed in his curriculum vita.publications as listed in his curriculum vita.

Dr. Hamming’s Superintendent’s Guest Dr. Hamming’s Superintendent’s Guest Lecture (SGL) presentation is an excellent Lecture (SGL) presentation is an excellent summary of the many diverse, challenging summary of the many diverse, challenging ideas tackled in his book and course.ideas tackled in his book and course.

These slides summarize DrThese slides summarize Dr Hamming’s Hamming’s background, contributions, awards and background, contributions, awards and publications as listed in his curriculum vita.publications as listed in his curriculum vita.


Born: 11 Feb 1915 in Chicago Illinois USABorn: 11 Feb 1915 in Chicago Illinois USA

Died: 7 Jan 1998 in Monterey California USADied: 7 Jan 1998 in Monterey California USA

Degrees ReceivedDegrees Received

• 1942, Ph.D., Mathematics, University of Illinois1942, Ph.D., Mathematics, University of Illinois

• 1939, M.A., Mathematics, University of Nebraska1939, M.A., Mathematics, University of Nebraska

• 1937, B.S., Mathematics, University of Chicago1937, B.S., Mathematics, University of Chicago

Born: 11 Feb 1915 in Chicago Illinois USABorn: 11 Feb 1915 in Chicago Illinois USA

Died: 7 Jan 1998 in Monterey California USADied: 7 Jan 1998 in Monterey California USA

Degrees ReceivedDegrees Received

• 1942, Ph.D., Mathematics, University of Illinois1942, Ph.D., Mathematics, University of Illinois

• 1939, M.A., Mathematics, University of Nebraska1939, M.A., Mathematics, University of Nebraska

• 1937, B.S., Mathematics, University of Chicago1937, B.S., Mathematics, University of Chicago


Work ExperienceWork Experience

• 1945-46 Los Alamos Lab, Manhattan Project1945-46 Los Alamos Lab, Manhattan Project

– Atomic Bomb Numerical Calculation ResearchAtomic Bomb Numerical Calculation Research

• 1946-1976 Bell Laboratories1946-1976 Bell Laboratories

– Mathematics and computing as applied to military Mathematics and computing as applied to military and telephone researchand telephone research

– Adjunct Professor of Statistics, Princeton, 3 yearsAdjunct Professor of Statistics, Princeton, 3 years

• 1976-1998, Naval Postgraduate School, Professor1976-1998, Naval Postgraduate School, Professor

Work ExperienceWork Experience

• 1945-46 Los Alamos Lab, Manhattan Project1945-46 Los Alamos Lab, Manhattan Project

– Atomic Bomb Numerical Calculation ResearchAtomic Bomb Numerical Calculation Research

• 1946-1976 Bell Laboratories1946-1976 Bell Laboratories

– Mathematics and computing as applied to military Mathematics and computing as applied to military and telephone researchand telephone research

– Adjunct Professor of Statistics, Princeton, 3 yearsAdjunct Professor of Statistics, Princeton, 3 years

• 1976-1998, Naval Postgraduate School, Professor1976-1998, Naval Postgraduate School, Professor


HonorsHonors

• President, Association Computing MachineryPresident, Association Computing Machinery

• Turing Prize of ACMTuring Prize of ACM

• Fellow IEEEFellow IEEE

• Piore Prize, 1979Piore Prize, 1979

• National Academy of Engineering, 1980National Academy of Engineering, 1980

• Pender Prize, 1981Pender Prize, 1981

HonorsHonors

• President, Association Computing MachineryPresident, Association Computing Machinery

• Turing Prize of ACMTuring Prize of ACM

• Fellow IEEEFellow IEEE

• Piore Prize, 1979Piore Prize, 1979

• National Academy of Engineering, 1980National Academy of Engineering, 1980

• Pender Prize, 1981Pender Prize, 1981


HonorsHonors

• IEEE R. W. Hamming Gold Medal namesakeIEEE R. W. Hamming Gold Medal namesake

– First recipient, $10,000 prize, 1996First recipient, $10,000 prize, 1996

– ““For exceptional contributions to information For exceptional contributions to information sciences and systems”sciences and systems”

• Vice President, Math Section, American Association Vice President, Math Section, American Association for the Advancement of Science (AAAS)for the Advancement of Science (AAAS)

• Editor of numerous journalsEditor of numerous journals

HonorsHonors

• IEEE R. W. Hamming Gold Medal namesakeIEEE R. W. Hamming Gold Medal namesake

– First recipient, $10,000 prize, 1996First recipient, $10,000 prize, 1996

– ““For exceptional contributions to information For exceptional contributions to information sciences and systems”sciences and systems”

• Vice President, Math Section, American Association Vice President, Math Section, American Association for the Advancement of Science (AAAS)for the Advancement of Science (AAAS)

• Editor of numerous journalsEditor of numerous journals

Quotable quotes 1Quotable quotes 1Quotable quotes 1Quotable quotes 1

The purpose of computing is insight, not The purpose of computing is insight, not numbers. numbers.

It is better to do the right problem the wrong It is better to do the right problem the wrong way, than the wrong problem the right way. way, than the wrong problem the right way.

The purpose of computing is insight, not The purpose of computing is insight, not numbers. numbers.

It is better to do the right problem the wrong It is better to do the right problem the wrong way, than the wrong problem the right way. way, than the wrong problem the right way.


Mathematics is the language of clear Mathematics is the language of clear thinking.thinking.

If the prediction that an airplane can stay up If the prediction that an airplane can stay up depends on the difference between Riemann depends on the difference between Riemann and Lebesgue integration, then I don’t want and Lebesgue integration, then I don’t want to fly in it.to fly in it.

Mathematics is the language of clear Mathematics is the language of clear thinking.thinking.

If the prediction that an airplane can stay up If the prediction that an airplane can stay up depends on the difference between Riemann depends on the difference between Riemann and Lebesgue integration, then I don’t want and Lebesgue integration, then I don’t want to fly in it.to fly in it.


Mathematics is an interesting intellectual Mathematics is an interesting intellectual sport but it should not be allowed to stand sport but it should not be allowed to stand in the way of obtaining sensible information in the way of obtaining sensible information about physical processes.about physical processes.

• Quoted in N. Rose Quoted in N. Rose Mathematical Maxims and MinimsMathematical Maxims and Minims (Raleigh NC 1988). (Raleigh NC 1988).

Mathematics is an interesting intellectual Mathematics is an interesting intellectual sport but it should not be allowed to stand sport but it should not be allowed to stand in the way of obtaining sensible information in the way of obtaining sensible information about physical processes.about physical processes.

• Quoted in N. Rose Quoted in N. Rose Mathematical Maxims and MinimsMathematical Maxims and Minims (Raleigh NC 1988). (Raleigh NC 1988).


I bugged Claude Shannon for years and I bugged Claude Shannon for years and years to write a book on Information Theory years to write a book on Information Theory so that everyone would understand it. He so that everyone would understand it. He refused, so I went ahead & wrote it.refused, so I went ahead & wrote it.

A good theoretician can account for almost A good theoretician can account for almost any result that is produced, right or wrong.any result that is produced, right or wrong.

I bugged Claude Shannon for years and I bugged Claude Shannon for years and years to write a book on Information Theory years to write a book on Information Theory so that everyone would understand it. He so that everyone would understand it. He refused, so I went ahead & wrote it.refused, so I went ahead & wrote it.

A good theoretician can account for almost A good theoretician can account for almost any result that is produced, right or wrong.any result that is produced, right or wrong.


My doctoral dissertation was 27 pages long.My doctoral dissertation was 27 pages long.

– Some Problems in the Boundary Value Theory of Some Problems in the Boundary Value Theory of Linear Differential EquationsLinear Differential Equations

– University of Illinois at Urbana-Champaign, 1942 University of Illinois at Urbana-Champaign, 1942

My doctoral dissertation was 27 pages long.My doctoral dissertation was 27 pages long.

– Some Problems in the Boundary Value Theory of Some Problems in the Boundary Value Theory of Linear Differential EquationsLinear Differential Equations

– University of Illinois at Urbana-Champaign, 1942 University of Illinois at Urbana-Champaign, 1942

[“… uhh, why so short, Dr. Hamming??”][“… uhh, why so short, Dr. Hamming??”][“… uhh, why so short, Dr. Hamming??”][“… uhh, why so short, Dr. Hamming??”]

There was a lot less to know in 1942.There was a lot less to know in 1942.– see chapter 1, exponential growth of knowledgesee chapter 1, exponential growth of knowledge

There was a lot less to know in 1942.There was a lot less to know in 1942.– see chapter 1, exponential growth of knowledgesee chapter 1, exponential growth of knowledge


If you don't work on important problems, it's If you don't work on important problems, it's not likely that you'll do important work. not likely that you'll do important work.

Usually expressed as a thought-provoking Usually expressed as a thought-provoking challenge to the audience:challenge to the audience:

• Who here wants to do important work in their career?Who here wants to do important work in their career?

• Who here is working on important problems?Who here is working on important problems?

• (for those not raising a hand) Why not?(for those not raising a hand) Why not?

If you don't work on important problems, it's If you don't work on important problems, it's not likely that you'll do important work. not likely that you'll do important work.

Usually expressed as a thought-provoking Usually expressed as a thought-provoking challenge to the audience:challenge to the audience:

• Who here wants to do important work in their career?Who here wants to do important work in their career?

• Who here is working on important problems?Who here is working on important problems?

• (for those not raising a hand) Why not?(for those not raising a hand) Why not?

Reflections on Los AlamosReflections on Los Alamos(Manhattan Project, World War II)(Manhattan Project, World War II)Reflections on Los AlamosReflections on Los Alamos(Manhattan Project, World War II)(Manhattan Project, World War II)

• Wartime call to service.Wartime call to service.

• “ “Janitor of science” and first computers.Janitor of science” and first computers.

• To perform great work, study the masters.To perform great work, study the masters.

• No two histories of the time are consistent.No two histories of the time are consistent.

• Thoughts on the accuracy of the computed Thoughts on the accuracy of the computed OO1818 radiative-absorption cross section for radiative-absorption cross section for slow neutrons, considered on the day prior slow neutrons, considered on the day prior to the detonation of the first atomic device.to the detonation of the first atomic device.

• Wartime call to service.Wartime call to service.

• “ “Janitor of science” and first computers.Janitor of science” and first computers.

• To perform great work, study the masters.To perform great work, study the masters.

• No two histories of the time are consistent.No two histories of the time are consistent.

• Thoughts on the accuracy of the computed Thoughts on the accuracy of the computed OO1818 radiative-absorption cross section for radiative-absorption cross section for slow neutrons, considered on the day prior slow neutrons, considered on the day prior to the detonation of the first atomic device.to the detonation of the first atomic device.

Namesake Concepts 1Namesake Concepts 1Namesake Concepts 1Namesake Concepts 1

Hamming error-correcting codes:Hamming error-correcting codes:

Carefully added redundancy which allows Carefully added redundancy which allows automatic detection & correction of errors.automatic detection & correction of errors.

• Hamming bits: redundancy built into hardwareHamming bits: redundancy built into hardware

Forward error correction:Forward error correction:

Sufficient redundancy that a receiver (on Sufficient redundancy that a receiver (on forward side) can correct errors without forward side) can correct errors without retransmission by original sender.retransmission by original sender.

Hamming error-correcting codes:Hamming error-correcting codes:

Carefully added redundancy which allows Carefully added redundancy which allows automatic detection & correction of errors.automatic detection & correction of errors.

• Hamming bits: redundancy built into hardwareHamming bits: redundancy built into hardware

Forward error correction:Forward error correction:

Sufficient redundancy that a receiver (on Sufficient redundancy that a receiver (on forward side) can correct errors without forward side) can correct errors without retransmission by original sender.retransmission by original sender.


Hamming distance: Hamming distance:

The number of positions which differ when The number of positions which differ when comparing corresponding bits between two comparing corresponding bits between two code words.code words.

Sometimes used as a measure of Sometimes used as a measure of conceptual distance, rather than spatial or conceptual distance, rather than spatial or numerical distance.numerical distance.

Hamming distance: Hamming distance:

The number of positions which differ when The number of positions which differ when comparing corresponding bits between two comparing corresponding bits between two code words.code words.

Sometimes used as a measure of Sometimes used as a measure of conceptual distance, rather than spatial or conceptual distance, rather than spatial or numerical distance.numerical distance.


Hamming predictor-corrector (PC) set for Hamming predictor-corrector (PC) set for ordinary differential equations:ordinary differential equations:

One of numerous numerical-analysis One of numerous numerical-analysis techniques used to make computer techniques used to make computer computations accurate by damping out computations accurate by damping out progressive roundoff errors. Overcomes a progressive roundoff errors. Overcomes a fundamental limitation of discrete arithmetic fundamental limitation of discrete arithmetic processing implemented in computer processing implemented in computer systems.systems.

Hamming predictor-corrector (PC) set for Hamming predictor-corrector (PC) set for ordinary differential equations:ordinary differential equations:

One of numerous numerical-analysis One of numerous numerical-analysis techniques used to make computer techniques used to make computer computations accurate by damping out computations accurate by damping out progressive roundoff errors. Overcomes a progressive roundoff errors. Overcomes a fundamental limitation of discrete arithmetic fundamental limitation of discrete arithmetic processing implemented in computer processing implemented in computer systems.systems.


Hamming digital filterHamming digital filter

Application of digital techniques (discrete Application of digital techniques (discrete computer processing) to perform signal computer processing) to perform signal processing, historically possible only with processing, historically possible only with analog electronic circuitry.analog electronic circuitry.

Hamming digital filterHamming digital filter

Application of digital techniques (discrete Application of digital techniques (discrete computer processing) to perform signal computer processing) to perform signal processing, historically possible only with processing, historically possible only with analog electronic circuitry.analog electronic circuitry.


Books AuthoredBooks Authored

• Numerical Methods for Scientists and EngineersNumerical Methods for Scientists and Engineers

• Computers and SocietyComputers and Society

• Introduction for Applied Numerical AnalysisIntroduction for Applied Numerical Analysis

• Calculus and the Computer RevolutionCalculus and the Computer Revolution

• Digital Filters (3Digital Filters (3rdrd Edition, January 1989) Edition, January 1989)

• The Art of Doing Science and Engineering, Learning The Art of Doing Science and Engineering, Learning to Learnto Learn

Books AuthoredBooks Authored

• Numerical Methods for Scientists and EngineersNumerical Methods for Scientists and Engineers


• Introduction for Applied Numerical AnalysisIntroduction for Applied Numerical Analysis

• Calculus and the Computer RevolutionCalculus and the Computer Revolution

• Digital Filters (3Digital Filters (3rdrd Edition, January 1989) Edition, January 1989)

• The Art of Doing Science and Engineering, Learning The Art of Doing Science and Engineering, Learning to Learnto Learn


Papers AuthoredPapers Authored

• A Class of Integration FormulasA Class of Integration Formulas

• The Computer as an Experimental ToolThe Computer as an Experimental Tool

• A Computer Scientist Looks at StatisticsA Computer Scientist Looks at Statistics

• Error Detecting and Error Correcting CodesError Detecting and Error Correcting Codes

• Impact of ComputersImpact of Computers

• Educational Implications of the Computer RevolutionEducational Implications of the Computer Revolution



• The Computer as an Experimental ToolThe Computer as an Experimental Tool

• A Computer Scientist Looks at StatisticsA Computer Scientist Looks at Statistics

• Error Detecting and Error Correcting CodesError Detecting and Error Correcting Codes

• Impact of ComputersImpact of Computers

• Educational Implications of the Computer RevolutionEducational Implications of the Computer Revolution



• Intellectual Implications of the Computer RevolutionIntellectual Implications of the Computer Revolution

• One Man’s View of Computer ScienceOne Man’s View of Computer Science

• Introduction to “Fundamental Theory of Introduction to “Fundamental Theory of Servomechanisms”Servomechanisms”

• The Mechanization of ScienceThe Mechanization of Science

• A Note on the Location of the Binary Point in a A Note on the Location of the Binary Point in a Computing MachineComputing Machine


• Intellectual Implications of the Computer RevolutionIntellectual Implications of the Computer Revolution

• One Man’s View of Computer ScienceOne Man’s View of Computer Science

• Introduction to “Fundamental Theory of Introduction to “Fundamental Theory of Servomechanisms”Servomechanisms”

• The Mechanization of ScienceThe Mechanization of Science

• A Note on the Location of the Binary Point in a A Note on the Location of the Binary Point in a Computing MachineComputing Machine



• Nuclear Magnetic Resonance in CrystalsNuclear Magnetic Resonance in Crystals

• Numerical Analysis vs. MathematicsNumerical Analysis vs. Mathematics

• Pitfalls in Numerical Analysis- IEEE Talk, March 19, Pitfalls in Numerical Analysis- IEEE Talk, March 19, 19681968

• Numerical Evaluation of Electron Image Phase Numerical Evaluation of Electron Image Phase ContrastContrast

• Stable Predictor-Corrector Methods for Ordinary Stable Predictor-Corrector Methods for Ordinary Differential EquationsDifferential Equations


• Nuclear Magnetic Resonance in CrystalsNuclear Magnetic Resonance in Crystals

• Numerical Analysis vs. MathematicsNumerical Analysis vs. Mathematics

• Pitfalls in Numerical Analysis- IEEE Talk, March 19, Pitfalls in Numerical Analysis- IEEE Talk, March 19, 19681968

• Numerical Evaluation of Electron Image Phase Numerical Evaluation of Electron Image Phase ContrastContrast

• Stable Predictor-Corrector Methods for Ordinary Stable Predictor-Corrector Methods for Ordinary Differential EquationsDifferential Equations



• The Impact of Computer technology on Management The Impact of Computer technology on Management Concepts, Planning, and Decision MakingConcepts, Planning, and Decision Making

• Checking Techniques for Digital ComputersChecking Techniques for Digital Computers

• Social Implications of the Computer RevolutionSocial Implications of the Computer Revolution

• The Effects of Computers Upon Engineering The Effects of Computers Upon Engineering EducationEducation


• The Impact of Computer technology on Management The Impact of Computer technology on Management Concepts, Planning, and Decision MakingConcepts, Planning, and Decision Making

• Checking Techniques for Digital ComputersChecking Techniques for Digital Computers

• Social Implications of the Computer RevolutionSocial Implications of the Computer Revolution

• The Effects of Computers Upon Engineering The Effects of Computers Upon Engineering EducationEducation


Papers AuthoredPapers Authored• Mathematical NotesMathematical Notes

• Controlling the Digital ComputerControlling the Digital Computer

• Computer Appreciation CoursesComputer Appreciation Courses

• Convergent Monotone SeriesConvergent Monotone Series

• Monotone SeriesMonotone Series

• An Essay on Computer Science Training ProgramsAn Essay on Computer Science Training Programs

• The Transcendental Character of COS XThe Transcendental Character of COS X

Papers AuthoredPapers Authored• Mathematical NotesMathematical Notes

• Controlling the Digital ComputerControlling the Digital Computer

• Computer Appreciation CoursesComputer Appreciation Courses

• Convergent Monotone SeriesConvergent Monotone Series

• Monotone SeriesMonotone Series

• An Essay on Computer Science Training ProgramsAn Essay on Computer Science Training Programs

• The Transcendental Character of COS XThe Transcendental Character of COS X



• On the Distribution of NumbersOn the Distribution of Numbers

• Modern Control TheoryModern Control Theory



• An Elementary Discussion of the Transcendental An Elementary Discussion of the Transcendental Nature of the Elementary Transcendental FunctionsNature of the Elementary Transcendental Functions

• Contributing to Modern Science and EngineeringContributing to Modern Science and Engineering


• On the Distribution of NumbersOn the Distribution of Numbers

• Modern Control TheoryModern Control Theory



• An Elementary Discussion of the Transcendental An Elementary Discussion of the Transcendental Nature of the Elementary Transcendental FunctionsNature of the Elementary Transcendental Functions

• Contributing to Modern Science and EngineeringContributing to Modern Science and Engineering



• Note on the Teaching of TrigonometryNote on the Teaching of Trigonometry

• Effects of Computers Upon Engineering EducationEffects of Computers Upon Engineering Education

• Electronic Digital Computer as an Intellectual ToolElectronic Digital Computer as an Intellectual Tool

• General Purpose SystemGeneral Purpose System

• Standard for Computer MathematicsStandard for Computer Mathematics


• Note on the Teaching of TrigonometryNote on the Teaching of Trigonometry

• Effects of Computers Upon Engineering EducationEffects of Computers Upon Engineering Education

• Electronic Digital Computer as an Intellectual ToolElectronic Digital Computer as an Intellectual Tool

• General Purpose SystemGeneral Purpose System

• Standard for Computer MathematicsStandard for Computer Mathematics



• Thinking Big Even with Small ComputersThinking Big Even with Small Computers

• Limitations of ComputersLimitations of Computers

• A Philosophy of Computer Science of My Prejudices A Philosophy of Computer Science of My Prejudices and Confusionsand Confusions

• How Do You Know the Simulation is Relevant?How Do You Know the Simulation is Relevant?

• A History of Computing in the United StatesA History of Computing in the United States

• Fifth Generation Computers and BeyondFifth Generation Computers and Beyond


• Thinking Big Even with Small ComputersThinking Big Even with Small Computers

• Limitations of ComputersLimitations of Computers

• A Philosophy of Computer Science of My Prejudices A Philosophy of Computer Science of My Prejudices and Confusionsand Confusions

• How Do You Know the Simulation is Relevant?How Do You Know the Simulation is Relevant?

• A History of Computing in the United StatesA History of Computing in the United States

• Fifth Generation Computers and BeyondFifth Generation Computers and Beyond



• Invariance and Bertrands ParadoxInvariance and Bertrands Paradox

• The Role of the Digital Computer in Scientific The Role of the Digital Computer in Scientific Research, Past, Present, and FutureResearch, Past, Present, and Future

• Gaussian Quadrature as a Minimization PrincipleGaussian Quadrature as a Minimization Principle

• Error Correcting CodesError Correcting Codes

• The Role of the Technical Societies in the Field of The Role of the Technical Societies in the Field of Computer MeasurementComputer Measurement


• Invariance and Bertrands ParadoxInvariance and Bertrands Paradox

• The Role of the Digital Computer in Scientific The Role of the Digital Computer in Scientific Research, Past, Present, and FutureResearch, Past, Present, and Future

• Gaussian Quadrature as a Minimization PrincipleGaussian Quadrature as a Minimization Principle

• Error Correcting CodesError Correcting Codes

• The Role of the Technical Societies in the Field of The Role of the Technical Societies in the Field of Computer MeasurementComputer Measurement



• Statistical Estimation of Error Propagation Through Statistical Estimation of Error Propagation Through Multiplication and DivisionMultiplication and Division

• A Systems Approach to Software TestingA Systems Approach to Software Testing

• Noninterpolatory Quadrature FormulasNoninterpolatory Quadrature Formulas

• The Frequency Approach to Numerical AnalysisThe Frequency Approach to Numerical Analysis

• Compumetrics: The Way AheadCompumetrics: The Way Ahead

• Computers and Computing in the ’70’sComputers and Computing in the ’70’s


• Statistical Estimation of Error Propagation Through Statistical Estimation of Error Propagation Through Multiplication and DivisionMultiplication and Division

• A Systems Approach to Software TestingA Systems Approach to Software Testing

• Noninterpolatory Quadrature FormulasNoninterpolatory Quadrature Formulas

• The Frequency Approach to Numerical AnalysisThe Frequency Approach to Numerical Analysis

• Compumetrics: The Way AheadCompumetrics: The Way Ahead

• Computers and Computing in the ’70’sComputers and Computing in the ’70’s



• Commencement Talk to Engineering School, Commencement Talk to Engineering School, University of California, IrvineUniversity of California, Irvine

• The Distribution of Numbers- ApplicationsThe Distribution of Numbers- Applications

• The Distribution of Numbers- Mathematical TheoryThe Distribution of Numbers- Mathematical Theory

• The Distribution of Numbers- Computer TheoryThe Distribution of Numbers- Computer Theory

• The Distribution of Numbers- Physical TheoryThe Distribution of Numbers- Physical Theory


• Commencement Talk to Engineering School, Commencement Talk to Engineering School, University of California, IrvineUniversity of California, Irvine

• The Distribution of Numbers- ApplicationsThe Distribution of Numbers- Applications

• The Distribution of Numbers- Mathematical TheoryThe Distribution of Numbers- Mathematical Theory

• The Distribution of Numbers- Computer TheoryThe Distribution of Numbers- Computer Theory

• The Distribution of Numbers- Physical TheoryThe Distribution of Numbers- Physical Theory



• Some Thoughts on SimulationSome Thoughts on Simulation

• Band Limited FunctionsBand Limited Functions

• Velocity Dependence on Contrast in Electron Images Velocity Dependence on Contrast in Electron Images of Periodic Structuresof Periodic Structures


• Some Thoughts on SimulationSome Thoughts on Simulation

• Band Limited FunctionsBand Limited Functions

• Velocity Dependence on Contrast in Electron Images Velocity Dependence on Contrast in Electron Images of Periodic Structuresof Periodic Structures

Favorite quoteFavorite quoteFavorite quoteFavorite quote

Luck favors the prepared mind.Luck favors the prepared mind.

• Louis Pasteur, 1822-1895Louis Pasteur, 1822-1895

Luck favors the prepared mind.Luck favors the prepared mind.

• Louis Pasteur, 1822-1895Louis Pasteur, 1822-1895

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Richard W. Hamming