Velvel Kahan's informal name in Yiddish, װעלװעל, means "little wolf." If he needs a more formal name, Velvel uses William.

Contents

First Meeting

I first met Velvel in George Forsythe's office at Stanford in 1962. Velvel had come to California from Toronto for a meeting of the IBM computer users group, SHARE, and visited Stanford on his way.

When Forsythe introduced us, Velvel said, "Oh, Cleve Moler, I have wanted to see you. Do you have a minute?" We went to my office and Velvel asked "Do you have a copy of the report you wrote at JPL last summer?" I retrieved the report from my meager collection. Velvel opened it to page 12, pointed to an equation, and said

        "That's wrong."

He was right.

Anonymous Referee

I submitted the manuscript for my first paper, "Iterative Refinement in Floating Point", to the Journal of the ACM in 1966. Forsythe was the editor who handled the submission. He sent it to an anonymous referee

The manuscript was 10 pages long. The referee's report was 12. Guess who refereed the paper.

Close enough

In 1968, Velvel was thinking of leaving the University of Toronto. Forsythe told me he considered offering Kahan a position at Stanford, but he was afraid Velvel would be too disruptive. "I hope he comes to Berkeley," George said. "That's close enough."

Velvel Kahan, 1968. (Photo Credit: George M. Bergman, Archives of the Mathematisches Forschungsinstitut Oberwolfach.)

Householder VII

I always met Velvel at Householder seminars. In 1977, the Householder seminar was at Asilomar, a California state park south of Monterey.

Velvel would begin talks about the importance of reliable floating-point arithmetic with a story that went something like this.

A physician leaves Sacramento late one evening. He flies his private plane home to Santa Barbara. After setting the autopilot, he falls asleep. When the plane is over the mountains north of the airport, the autopilot fails with a floating-point overflow and the plane crashes. The doctor's estate sues you because your company wrote the autopilot software.

This photo shows Velvel telling that story with an overhead projector.

Householder XVII

In 1993, the Householder seminar was in California again, this time at Lake Tahoe. Here is a rare moment when Velvel is actually listening to me.

The Office

Both Velvel and Beresford Parlett have been on the faculty at U. C. Berkeley since 1968. Velvel's student, Jerome Coonen, took this photo when he volunteered to help clear out Velvel's office.

Turing Award

The citation for Velvel's Turing Award in 1989 reads:

For his fundamental contributions to numerical analysis. One of the foremost experts on floating-point computations. Kahan has dedicated himself to "making the world safe for numerical computations!"

https://amturing.acm.org/award_winners/kahan_1023746.cfm

IEEE 754

My friend Axel Ruhe was from Sweden. He once asked me where he should visit on his first trip to the western United States. I recommended

The meetings of the IEEE floating-point committee were legendary. They would go on for hours at various spots in Silicon Valey. Suffering from jetlag, Axel went to sleep on the floor of the conference room.

Here is Velvel, remembering the committee. https://www.youtube.com/watch?v=ATCpecsyPE8.

Photo Credit: Alaina G. Levine, Heidelberg Laureate Forum.

Bibliography

Nelson Beebe has compiled this bibliography.

Oral History

The manuscript of Thomas Haigh's interview of Velvel in 2005 for SIAM's oral history project is 229 pages long. Take a look. https://history.siam.org/oralhistories/kahan.htm.

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Linocuts by Henk van der Vorst.

Beresford Parlett passed away last Saturday, February 7. He was 93 years old. I have known him since graduate school at Stanford.

Contents

Stanford

Source: George M. Bergman, Berkeley.

Beresford was born in 1932 in London. He received his bachelor's degree in mathematics from Oxford in 1955.

Beresford came to graduate school at Stanford in 1958. He had very little money and no place to live. At first, he slept in the math faculty women's lounge and showered in the gymnasium. Fortunately, after two quarters, the math made him a teaching assistant, and he was able to move out of the lounge.

I came to Stanford in 1961, three years after Beresford. We were both students of George Forsythe, and are both on the Forsythe tree.

Career

Source: EECS, Berkeley.

Beresford received his Ph.D. in math from Stanford in 1962.

He was a postdoc for two years at NYU's Courant Institute and for one year at the Stevens Institute of Technology

Beresford joined the U. C Berkeley math faculty in 1965 and remained there until his retirement.

Holy Grail

Many researchers, including Parlett, contributed to the search for the Holy Grail of matrix eigenvalue computation, an algorithm with O(n) time complexity. The quest culminated in the MRRR algorithm, the subject of the Ph.D. thesis of one of Beresford's students, Inderjit Dhillon.

Beresford's specialities include

Nelson Beebe has compiled this bibliography

Beresford's book, The Symmetric Eigenvalue Problem, was published in 1980 by Prentice-Hall and reprinted in 1987 by SIAM.

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I was amazed by two recent YouTube videos about the double pendulum.

Contents

Double Pendulum

The classic double pendulum consists of one pendulum attached to a fixed pivot and a second pendulum attached to the end of the first. The system is governed by Newton's laws, and its motion is completely determined by the initial conditions. There is no randomness. However, its behavior is sometimes difficult to predict. It may be chaotic.

Cleve's Lab

Ten years ago, I included the double pendulum in Cleve's Lab.

Drew's Campfire

Drew's Campfire is one recent video about the double pendulum. Here is a link to the video.

Here is a sample frame.

2Swap

The second video comes from 2Swap. Here is a link to the video.

Here is a frame featuring the Pretzel pendulum.

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My first blog about SVD and Partisanship in the U. S. Senate was five years ago. Today's post updates that to 2025.

Contents

SVD Measures Partisanship

Many observers have noted that the United States Senate has become increasingly partisan in recent years. Votes are being made more frequently along strict political party lines. The singular value decomposition, the SVD, of matrices derived from records of roll call votes in the senate can measure this partisanship.

The Senate

Senators serve six-year terms and every two years about one-third of the senate stands for reelection or new senators are elected. So, a new "Congress" convenes every two years and has two year-long "Sessions". The first session was in 1789. The 101-st Congress began in 1989 and last year, 2025, is the First Session of the 119-th Congress.

This bar chart shows the division between the Republican and Democratic parties over our time period. The data from the web site party division comes with the caveat, "The actual number of senators representing a particular party often changes during a Congress, due to the death or resignation of a senator, or as a consequence of a member changing parties."

The Data

The web site www.senate.gov has year-by-year records of voting in the senate. We are going to look at 37 years, from 1989 to 2025. For example, the URL

https: //www.senate.gov/legislative/LIS/roll_call_votes/vote1162/vote_116_2_00033.xml

points to the record of a roll call vote taken in the senate early in 2020. The congress number, 116, and the session, 2, each appear twice. The 33 signifies this is the 33rd vote of this session.

Each roll-call vote is recorded in an XML file. XML is the abbreviation for eXtensible Markup Language, an important standard from the WWW Consortium defining a format for encoding documents that is both human and machine readable. I was pleased to find MATLAB Central File Exchange 28518-xml2struct by Wouter Falkena. As the function name implies, this handy code "converts an XML file into a MATLAB structure for easy access to the data."

You can also use your browser to access the example URL and (a) examine the XML, and (b) see why bills 33 and 34 from year 2020 are not typical measures before the Senate.

The Matrices

Once you find your way through the layers of structure that results from the XML, you will have many details about each senate roll call vote. In particular, for each senator who voted, you will have a structure with fields full name, last name, first name, party, state, vote, and member id.

For each year, we create a matrix of +1s, -1s, and 0s. A +1 is for a vote of "Yea" or "Guilty", a -1 is a "Nay" or "Not Guilty", and a 0 is for anything else, which includes "Not Voting", "Present" (abstain), and something too complicated to explain called "Present, Giving Live Pair". Because there are 100 senators, the matrix has 100 rows. The number of columns is the number of roll call votes taken that year. The matrices range in size from 100-by-163 for 2016 to 100-by-613 for 1995.

The XML is organized using last names of senators in alphabetic order -- Alexander (R-TN), Baldwin (D-WI), Barrasso (R-WY), ..., Young(R-IN). At first, I tried to create matrices with rows in this order. But senators retire or die in midsession, so this ordering proved to be difficult to manage correctly. Ultimately, I decided to order the rows by states. Each state is allocated two rows; the name of the specific senator occupying each row is irrelevant. After the matrix for a given year is complete, I reorder the rows so that all the senators from each party are grouped together. The one or two senators who declare themselves to be independent (but who caucus with a major party) are between the two parties.

Here are the matrices, 37 of them, one for each session from 1989 to 2025. A good way to see the structure of these matrices is to view them as images. Red is "Yea" or "Guilty", gold is "Nay" or "Not Guilty", and white is not present, abstain or other.

Partisanship

The SVD of these matrices provides the basis for our analysis. The singular values decrease very rapidly with increasing index, so the matrices can be well approximated by matrices of rank two. When viewed from the record of roll call votes, the U. S. Senate is nearly two-dimensional.

Let A(y) be the matrix of 1s, -1s and 0s summarizing the roll call votes in year y. Let

sigma=svd(A(y))

Then the partisanship for year y is defined usinng the third singular value.

partianship=1-sigma(3)/sigma(1)

This quantity measures how close the voting pattern is to being two-dimensional.

Compare

For example, 1991 is one of the least partisan years in our sample, and 2025 is one of the most partisan years. You can immediately see that 2025 has much less detailed structure than 1991; it is closer to being two-dimensional

Increasing Partisanship

This animation shows the distribution of the first ten singular values for the years from 1989 to 2025. The singular values are normalized so that 1-sigma(1) = 0. Then the third singular value is highlighted.

Finally, here is this measure of partisanship for each year from 1989 to 2025, and a linear least squares fit. With this much scatter is the values, the exact fit can't be taken too seriously. But there is definitely an upward trend. Partisanship in the U. S. Senate has increased from less than 70% thirty years ago to more than 80% today.

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I admire the color scheme in the New York Times Games section and occasionally make it my colororder.

https://www.nytco.com/games

Contents

Colormaps

It is important to distinguish between "color map" and "color order". A colormap defines the colors used by surface, image, and patch objects. MATLAB provides 22 color maps. parula is the default.

    disp(colormaps)

    "parula"
    "turbo"
    "hsv"
    "hot"
    "cool"
    "spring"
    "summer"
    "autumn"
    "winter"
    "gray"
    "bone"
    "copper"
    "pink"
    "sky"
    "abyss"
    "nebula"
    "jet"
    "lines"
    "colorcube"
    "prism"
    "flag"
    "white"

Palettes

The colormap defines the palette of colors used by the plot function for multiple lines. MATLAB provides 10 palettes. gem is the default.

    disp(palettes')

    "gem"
    "gem12"
    "glow"
    "glow12"
    "sail"
    "reef"
    "meadow"
    "dye"
    "earth"
    "default"

Color Order

The colors used by MATLAB for plotting multiple lines need to be well separated. These colors are given by the colororder property of the current axes. By default, MATLAB cycles through the seven colors in the gem palette.

    gcaco=get(gca,"ColorOrder");
    show_rgb(gcaco)

    close all
    clear

New York Times Games

The New York Times Games originated with the newspaper's crossword puzzle in 1942. NYT Games was established in 2014, with the addition of the Mini Crossword. Wordle was added in 2022 and I have been playing Wordle almost daily since its launch.

NYT Games now has over 10 million daily players. According to one member of staff, "the half joke that is repeated internally is that The New York Times is now a gaming company that also happens to offer news."

NYT Color Order

I like the color scheme that the Times uses in the Games section. It provides a subtler color order than the default gem.

    nyt_palette
    

ans =
   184   228    68
   248   211     0
   121   160   238
   184   113   198
   160    82    45
   235   235   200

    nytco = nyt_palette/256
    ribbon(6)
    set(gcf,'position',[200 200 440 300])
    set(gca,'colororder',nytco)
    drawnow
   

nytco =
    0.7188    0.8906    0.2656
    0.9688    0.8242         0
    0.4727    0.6250    0.9297
    0.7188    0.4414    0.7734
    0.6250    0.3203    0.1758
    0.9180    0.9180    0.7812

    show_rgb(nytco)
    drawnow

Tangrams

In October, I used NYT colors for the kitty in my blog post about Tangrams.

NYT Qube

The NYT colors give the Rubiks Cube a new look.

Gallery

The MiniGallery with NYT colors.

fftmatrix

The Fast Finite Fourier Matrix with the NYT colors.

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Rob Schrieber is one of my very best friends.

Contents

Career

Hewlett-Packard

Rob worked at Hewlett Packard for more than 20 years, https://trimaran.org/car/rob_schreiber.

Sparse

Iain Duff is known for his work on algorithms and software for sparse matrix computation. In 1990, he visited Stanford and gave a talk in the numerical analysis seminar. I attended the talk. So did John Gilbert, who was then at Xerox Palo Alto Research Center, and Rob Schreiber, from Hewlett Packard Research. We all went to lunch at Stanford's Tresidder Union and there Gilbert, Schreiber and I decided it was about time for MATLAB to support sparse matrices.

In 1992 we introduced sparse matrices. One of our examples is west0479

SIAM

Rob has been a member of SIAM for 50 years, https://www.facebook.com/watch/?v=464499592673781.

NAS Benchmarks

The NAS Parallel Benchmarks (NPB) are a set of programs designed to evaluate the performance of parallel supercomputers. The benchmarks were developed as part of the Numerical Aerodynamic Simulation (NAS) Program at the NASA Ames Research Center. The benchmarks are derived from computational fluid dynamics applications and consist of five kernels and three pseudo-applications in the original "pencil-and-paper" specification. Rob is one of the 14 authors of the first specification, NAS Benchmarks.

The NAS Benchmarks received the Supercomputing Test of Time award at SC15 in Austin. Rob is next to Horst Simon, who holds the award.

Cerebras

Cerebras Systems is a ten-year old Silicon Valley startup. https://www.cerebras.ai/company.

The fellow in the center is holding the Cerebras "chip", an entire silicon wafer. Rob is on the right, in the green shirt.

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Nick Trethen is a world famous computational scientist and my good friend.

Contents

Serra House

I first met Nick in 1978 at Stanford's Serra House. Serra House was built in 1923 to be the retirement home for President David Starr Jordan. By the late 1970's it had been assigned to the Computer Science Department and housed the Numerical Analysis group. Professors George Forsythe and Gene Golub had their offices there at one time or another, as did various grad students, post docs, and visitors.

There was a lovely patio, a volleyball court and a real feeling of camaraderie in the Serra House group. In the academic year 1978/79 a particularly strong group of grad studies were working in Serra House. Nick Trefethen was one of them. Marsha Berger, Michael Overton and Randy LeVeque were others.

In 1978/79 Gene Golub was away from Stanford on sabbatical and I took his place for the year. I taught the graduate numerical analysis course and used my new Fortran "matrix laboratory". It was just a primitive matrix calculator. It was not impressive numerical analysis nor impressive computer science.

Nick and Randy were enrolled in that course and Marsha and Michael sat in sometimes. This was their first exposure to MATLAB. It would have to become a more mature tool before they would find it useful. A few grad students from engineering also took my course and found that primitive MATLAB useful for work in areas like control theory and signal processing, areas that I knew nothing about at the time. That eventually led to Jack Little and to MathWorks.

Nick

https://www.henkvandervorst.nl/computing.html

Linocut by Henk van der Vorst.

Nick is a renowned scholar of computational mathematics. His curriculum vitae provides a list of his achievements. He has authored eight books, published hundreds of journal articles and written dozens of essays, He is a fellow of the US National Academy of Engineering and the British Royal Society. He served as president of SIAM and as editor and associate editor for a number of journals.

Here is a sample of a few of his achievements.

Academic Career

Nick received his Stanford Ph.D.in 1982. His thesis supervisor was Joe Oliger. From 1982 to 1997 he was on the faculty of NYU, MIT and Cornell.

In 1997, he moved to Oxford University to succeed Leslie Fox as Professor of Numerical Analysis and Fellow of Balliol College. Nick's essay An American at Oxford is fascinating.

Wilkinson

Nick gave this talk about Jim Wilkinson in 1979.

https://www.youtube.com/watch?v=kOMrRn2tdCs

60th Birthday

A conference for Nick's birthday as held in 2015 in Oxford's new Andrew Wiles Building. That's Andre Weideman in the front row with Nick and me.

https://blogs.mathworks.com/cleve/2015/09/07/trip-report-trefethen-birthday-conference

Chebfun

Chebfun is a powerful system for dealing with functions and operators in the same way MATLAB deals with vectors and matrices. The web site: https://www.chebfun.org.

The logo:

The original authors, Nick, Toby Driscoll, Ricardo Pachón and Rodrigo Platte:

100 Digits

In 2002, Nick specified ten problems, each with a numerical solution. He offered a $100 prize to whoever produced the most accurate solutions, measured up to ten significant digits. Ultimately twenty teams solved all of the problems perfectly. An anonymous donor aided in producing the required prize money. The challenge and its solutions were described in detail by Folkmar Bornemann, Dirk Laurie, Stan Wagon and Jörg Waldvogel in

The SIAM 100-Digit Challenge.

One of the problems is to find the global minimum of

f = @(x,y) exp(sin(50*x)) + sin(60*exp(y)) + sin(70*sin(x)) + sin(sin(80*y)) - sin(10*(x+y)) + (x.^2+y.^2)./4;

Here is a surf plot of f(x,y).

Lightning Laplace Solver

The Lightning Laplace Solver is a MATLAB code that solves the Laplace equation on a polygon.

https://people.maths.ox.ac.uk/trefethen/lightning.html

AAA

AAA is a new algorithm for rational approximation. The name AAA stands for “adaptive Antoulas-Anderson” in honor of the authors who introduced the idea.

https://people.maths.ox.ac.uk/trefethen/AAAfinal.pdf

Body-Mass Index

In 2013 Nick wrote this one-page note proposing an alternative to the standard Body-Mass Index, which is widely used to assess obesity.

https://people.maths.ox.ac.uk/trefethen/bmi.pdf

Index Cards

Nick writes:

For many years, since 1970 in fact, I have been writing notes on social, mathematical, and other subjects, which I store on index cards. A collection of several hundred of these notes was published ... as Trefethen’s Index Cards: Forty Years of Notes about People, Words and Mathematics (World Scientific, 2011).

See: https://trefethen.net

Harvard

In 2023, Nick returned to Harvard, where he had been an undergraduate fifty years earlier. (Bill Gates was in the same class.) This time around, Nick is Professor of Applied Mathematics in Residence at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS). The announcement: Trefethen Returns to Harvard.

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Jack Dongarra is my student, colleague and friend.

Contents

Turing Award

The ACM Turing Award is an annual prize given by the Association% for Computing Machinery (ACM) for contributions of lasting and major technical importance to computer science. It is generally recognized as the highest distinction in the field of computer science and is often referred to as the "Nobel Prize of Computing".

Jack Dongarra won the Turing Award in 2021. Here is his award lecture, a very personal story. https://amturing.acm.org/vp/dongarra_3406337.cfm

Jack

https://www.henkvandervorst.nl/computing.html

Linocut by Henk van der Vorst.

I first met Jack at Argonne in 1970 when we began work on the EISPACK project.

In 1979, he came to the University of New Mexico to work on a Ph.D. degree under my supervision. His thesis is "Improving the Accuracy of Computed Eigenvalues and Eigenvectors".

PC-AT

Jack and the IBM PC-AT in 1984.

Jack has been the primary implementor or principal investigator for many libraries including LINPACK, BLAS, LAPACK, ScaLAPACK, PLASMA, MAGMA, and SLATE.

Argonne

The LINPACK Guys.

The authors of LINPACK -- Jack Dongarra, Cleve Moler, Pete Stewart and Jim Bunch -- at James Lyness's house near Argonne in the summer of 1978.

The first snapshot was in the New York Times when Jack won the Turing Award.

New Mexico

The 1978 New Mexico plate on Jack's Car in the NYT photo is gold. Jack had two 1979 turquoise plates made. He has one of them and he gave the other to me.

Accidental Benchmark

The LINPACK benchmark is an accidental offspring of the development of the LINPACK software package. We asked two dozen universities and laboratories to test the software on a variety of main frame machines that were then available in central computer centers. We also asked them to measure the time required to solve a 100-by-100 system of simultaneous linear equations.

Appendix B of the LINPACK Users' Guide has the timing results. The hand-written notes shown here are Jack's calculation of the megaflop rate, millions of floating point operations per second. With a matrix of order n , the megaflop rate for a factorization by Gaussian elimination plus two triangular solves is

megaflops = (2/3*n^3 + 2*n^2)/time/10^6

Jack and colleagues have continued to collect results. Twice a year, the TOP500 determines the world's fastest computer.

Spokesman

Jack has become a spokeman for high performance computing. He is often quoted by main-stream media.

SC Conference

Jack has been very active in the Super Computer Conferences. SC22 gave him a rousing local-kid-makes-good welcome after he won the Turing Award.

ICL

Jack established the Innovative Computing Laboratory in 1989 when he accepted a dual appointment as a Distinguished Professor at the University of Tennessee and a Distinguished Scientist at Oak Ridge National Laboratory. Since then, ICL has grown into an internationally recognized research laboratory, specializing in Numerical Linear Algebra, Distributed Computing, and Performance Evaluation and Benchmarking. The lab now employs nearly twenty researchers, students, and staff.

Tradition

John Gilbert, Jack, me and Rob Shriber traditionally meet for dinner once a year at the Super Computer conference. This photo was at SC18 in Dallas.

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Pete Stewart is one of my closest colleagues.

Contents

Pete

Pete's formal name is Gilbert W. Stewart. So why do we all call him Pete? It's a little vague. Because Pete's father was nicknamed Pete. Why? Because Pete's father's father was nickednamed Pete. That's all the explanation I ever got.

LINPACK

The authors of LINPACK are Jack Dongarra, Cleve Moler, Pete Stewart and Jim Bunch. This is one of the snapshots taken in 1978 when all of us spent the summer at Argonne. Another of the snapshots was in the New York Times when Jack won the Turing award.

Here we are 30 years later. Jack has lost the most hair, but I have the coolest shirt.

Householder

Pete's Ph. D. thesis adviser was Alston Householder. Here are Pete, Mike Heath, Alston, and Bob Funderlic in 1970.

Old Timers

Paul van Dooren, Bo Kågström, Pete Stewart, Chris Paige, Michael Saunders, Bob Plemmons and Cleve Moler in 2017 at the Householder Symposium XX, Virginia Tech.

Bibliograhy

Here is a Linocut by Henk van der Vorst and a link to a bibliography compiled by Nelson Beebe

https://www.henkvandervorst.nl/computing.html

A Bibliography compiled by Nelson Beebe.

Collected Works

Dianne O'Leary and Mischa Kilmer edited a volume of Pete's collected works.

https://link.springer.com/book/10.1007/978-0-8176-4968-5

Oral History

Thomas Haigh conducted several dozen oral histories of computational scientists for SIAM. Here is Pete's Oral History.

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The tangram is an ancient Chinese puzzle with seven pieces.

Contents

Variety

Tangram puzzles can be made with paper, plastic or wood.

Schools

School kids all over the world enjoy tangram puzzles.

Figures

Thousands of figures, like these animals, can be made.

Puzzle

My MATLAB code Tangram opens with a screen showing seven pieces, which are called tans. There are two large triangles, one medium triangle, two small triangles, one square, and one parallelogram.

The puzzle is interactive. You can move the tans.

Kitty

The kitty is a newcomer's favorite.

Swan

The parallelogram is the only tans that does not have reflection symmetry. It needs to be reversed with the flipper button when forming shapes like this swan. The swan at the top of this blog is looking in the opposite direction and does not need its parallelogram flipped.

Square

Here are the starting moves for making a square. You can finish the square yourself by downloading Tangram_mzip.

Or, if you are not in a do-it-yourself mood and just want the answer, see square_2.gif.

Web

There are hundreds of web sites about Tangrams. Here are two that I enjoyed.

https://www.pinterest.com/ivahilator/tangram.

https://kubiyagames.com/products/tangram-puzzle-two-sets-with-65-challenge-cards.

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