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As we grieve for our friend and colleague Hans Schneider, we come together here to celebrate his life and contributions. On this page we encourage you to share ways in which Hans made an impact on you or mathematics, anecdotes, funny stories, etc. Please send your contributions to ilasic@uregina.ca.

Tsuyoshi Ando

Olga Taussky-Todd, one of the founders of ILAS, wrote in her autobiographical essay,

"The truth, nothing but truth, but not all the truth"
 
"Then there was a symposium arranged here at Caltech by Professor Varga. In this symposium, a number of papers connected with my work were given, but in particular one speaker, H. Schneider, reported on my influence via three particular areas on which I had worked. He talked about just three of them because they particularly concerned him and he is not likely to know several others, but I was very grateful and elated about it all."
 
On reading the essay I felt that the italicized part was impolite because I knew that the research area of Hans was broad. Afterward when Hans visited my institute (at Sapporo, Japan) I asked him about this point. He said nothing but "My wife also pointed out that part."
 
One morning during the same visit, Hans and I were in the (central) station of Sapporo. It was around 8 o'clock in the morning and the station was full of commuters. Looking around, Hans cried, "Oh, only Japaneses !"

The autobiography of Olga Taussky-Tood is included in the book: D.J. Albers and G.L. Alexanderson: "Mathematical People", Birkäuser, 1985 (see p.335)

Harm Bart

In 1982, Peter Lancaster organized a Workshop on Matrix Theory in Calgary. It was a remarkable meeting, not only for the hikes in the mountains, but especially for its slow pace. There were only a couple of lectures a day with as a wonderful counterpart many informal discussions. As something unique in its sort, it has been a fond memory ever since. There was another thing that has contributed to this: at this meeting I met many extraordinary Linear Algebra people for the first time. One of them was Hans, and he made a lasting impression on me.
 
From 1987 on, the relation between us intensified. It was the year in which the so called International Matrix Group was formed, and I became the 'contact person' for The Netherlands, my home country. In 1989, the International Matrix Group as such disappeared but it got a successor in ILAS, The International Linear Algebra Society. Under the inspiring leadership of Hans, it became the organization which it is now and which has been so successful in bringing matrix people from all over the globe together. In the context of working for ILAS, Hans and I had many contacts, contacts that were further strengthened via editorial work for LAA.
 
Good memories are now what is left. I will miss Hans at those occasions where we used to see each other. And this is not only true for me personally, but also for my wife Greetje who is not at all a mathematician. Indeed, the sadness that comes with his passing away does not only come from missing Hans as the strong mathematician he was but -- even more importantly -- from having lost him as a person.

Carl de Boor

When I interviewed in the spring of 1972 at University of Wisconsin-Madison for a job involving the Math Research Center, its director, J Barkley Rosser, invited Hans & Miriam Schneider to a dinner with me (presumably because they both spoke German even though, by that time, I hardly spoke it at all) and my wife.
 
The two became cherished friends, through many Sunday brunches which always involved, with apologies to the ladies present, some math. Whatever LAA questions I had, I could count on Hans to point the way to the answer. His knowledge of the literature was truly remarkable.
 
Yet, six weeks ago, when I tried to thank Hans for all the help he had given me over the years, he looked at me strangely and only said 'What help?'.
 
So, for the record, I say it again: Thank you, Hans!

Rafael Bru

In winter of 1991 Hans visited Miki Neumann and at that moment I was a visiting professor at University of Connecticut. Hans shared my office during those days. I remember that every day Hans expended a lot of time writing, answering and sending emails from his PC. I took the opportunity to discuss with him on different topics of Linear Algebra. In particular, he was always very interesting in talking about matrices.

Richard Brualdi

Hans Schneider had a profound impact on my career in many ways but in particular, in 1978 when he invited me to join him as co-editor-in-chief of the journal Linear Algebra and its Applications. I learned a great deal from him which has served me very well over many years. He and his wife Miriam were dear friends of my wife Mona and me. At our wedding, Hans was my "best man" and Miriam was Mona's "best lady", each holding our wedding rings.

 

Peter Butkovic

I met Hans in Madison in 2001. It was the beginning of 13 years of our fruitful collaboration. Over all those years he was for me an invaluable source of professional advice which had deep impact on my academic life. I feel very privileged that I had the opportunity to work with Hans. I will always remember Hans as a wonderful friend, full of energy, bright ideas but also kindness. We shared many views on academic matters and life in general. I will very much miss Hans and often remember him for his wisdom and friendship.

David Carlson

I first met Hans Schneider in 1959, taking a course from him on Perron-Frobenius theory of nonnegative matrices. It was a great introduction to Hans as a teacher and to research in linear algebra. That led to his becoming my major professor. What a gentle, encouraging and inspiring mentor he was!
 
Throughout my career he was a guiding light for me and for many others, in his research, his Editorship of LAA, and his organization of ILAS. When he started his career, linear algebra and matrix theory were looked down upon as elementary and dead-end subjects. (Who was it that said that Emmy Noether �freed algebra from the dull and uninteresting subjects of matrices and determinants�?) Today, though his work as much as anyone�s, they are active research areas in their own right, and are basic building blocks of applications of mathematics in the sciences and engineering.
 
As a human being, he was modest and unassuming, yet had the kindness to encourage young mathematicians, the self-confidence to reach out to people he hardly knew, and the vision to imagine needed projects and to involve others in them.
 
And, finally, I will miss his cheery �Hello Dave� when we started phone conversations.

Froil�n Dopico

"I only chatted twice with Hans Schneider but...."
 
I remember both conversations very well, since Hans did in both occasions original, constructive, and positive comments, which have had a deep impact on my view on the field of Linear Algebra and on talks on Linear Algebra. Both conversations are related to my work as Associate Editor of LAA. In October 19, 2010, I received an email from Richard Brualdi (on behalf also of Volker Mehrmann and Hans Schneider) inviting me to join the Editorial Board of LAA. No need to say that I felt very honored and very happy to become a member of this Editorial Board, where I could read names of present and past Editors that are mathematical heroes for me. My first "social" activity as Editor of LAA was a dinner offered by Elsevier to the Editors of LAA attending to the ILAS Conference held in August 2011 in Braunschweig (Germany). There, Hans came to chat with me for a few minutes and essentially to tell me that he was very happy that I joined the Editorial Board, that he was sure that I would do a very good job, and that it was very important for LAA to incorporate continuously new Editors doing research in many different areas of Linear Algebra. He was extremely warm and I think that his words reflect very well his extremely constructive spirit and why he was one of the most important driving forces in our community during more than 50 years.
 
My second conversation was during a meeting organized in Madison (Wisconsin, USA) by Richard Brualdi on the occasion of Hans Schneider's retirement as an Editor-in-Chief of LAA in October, 2012. This was a meeting for current & former Editors of LAA. Well, I was very excited about attending the meeting and about presenting a talk in that meeting, because to have an audience completely formed by Editors of LAA was in my mind a great opportunity to spread out some of my recent results to a very important set of researchers. The title of my talk was "Inverses and condition numbers of Fiedler companion matrices" (Miroslav Fiedler was also in the audience) and I put a lot of effort to prepare that talk, really a lot! However, it was one of the worst talks I have ever presented. I wanted to make a big impression to all the great researchers in Linear Algebra in the audience and I included such huge amount of results that was impossible to explain anything in the 30 minutes of the talk. Some friends told me exactly that after the talk (I remember Daniel Szyld, Andreas Frommer, Michele Benzi, Charlie Johnson and others...my thanks to all of them!). I went so fast that it was impossible to follow the explanations even from the third slide on (you still can find the talk in my web page). However, Hans came to chat with me after my talk during some of the coffee-breaks and told me, more or less, the following: Great talk, Froil�n! Now, I know many things on Fiedler matrices and it is clear for me that they are very interesting, but with a bit less of information I would get the same feeling and perhaps I would even know more!

Ludwig Elsner

My connection with Hans started in 1974, when Hans spent a sabbatical in Munich. He phoned me and asked if he can visit me at the university of Erlangen. Few weeks later he showed up here in a very old car. My wife Margrit was really worried that he can return safely with this car to Munich. Fortunately he did. Since then we had a very fruitful cooperation mathematically and a private friendship. So I worked among others as Associate Editor of LAA. I visited him several times in Madison and met him often at conferences, e.g. in Oberwolfach, Calgary, Maynooth, Barcelona, Berlin and Bielefeld. In the last months my wife and I corresponded with him privately by e-mail. This was for all of us a special time. Thank you, Hans.

Gernot Michael Engel

From the time Hans became my thesis advisor to his last days we shared many adventures. I learned much from him. He was a student of history and used his insights to promote policies that thwarted tribalism and provided fairness.
 
He told me that he would have preferred studying languages to mathematics. He understood how to simplify a proof by making the right definitions. When working on a joint paper he would go over a proof again and again until he was sure he understood its essence and then he would make just the right definition to expose what was essential. I miss his enthusiasm and companionship. I wish I could still be working with him.

Shaun Fallat

I first met Hans at the 1995 ILAS meeting in Atlanta -- like many folks I met there, Hans was very approachable and even asked what topic I was studying for my Master's degree. I was so surprised; I do not think I responded. I never forgot his friendly nature and huge heart for young people studying linear algebra. Later in 2002 just before I was to present an invited talk in Auburn, he could tell I was nervous and took the time to calm me down -- after my talk he approached me and said "Top notch talk Shaun". I will never forget that! In 2003, I was offered an Associate Editor position with LAA, and I have suspected for some time that Hans was instrumental in getting me that position. While Hans was editor-in-chief of LAA, I worked under him, and am so grateful for his mentor-ship -- he responded to every single query I asked of him, even while he was in Hawaii during the winter. It seemed that he always had time for me, and I know that he treated everyone else in the same manner...Even when his health was failing; he took the time to respond to my emails, and congratulated me when I presented the inaugural Hans Schneider ILAS lecture last June in Kingston, Ont. It is an honour that I will treasure for the rest of my life.
 
Hans: a true professional mathematician, a valued mentor, a kind gentleman, and a dear friend...There is a special place in my heart for Hans...
 
Linear algebra has lost a wonderful soul, but we are much better off as a society because of the effort and passion of Hans Schneider... May he rest in peace.

Karl Gustafson

As I recall, I only met Hans Schneider once, and long ago. It would have been in the early 1970s, possibly at either the MRC Nonlinear Functional Analysis Symposium in Madison in 1971 or at the Research Conference on Numerical Ranges of Operators on Normed Linear Spaces in Aberdeen Scotland in July 1971. However, I often remember faces for a long time, and I remember his from then, a healthy radiant friendly face on a happy man of not-short height and with out-going disposition. At least, that is my memory.
 
Some years later, my very good friend John Maybee (who was an expert on sign-positive matrices among other things) encouraged me to join ILAS. I remember his saying � Hans Schneider is a good guy��.
 
Then in 1994 Hans wrote to me to contribute to The Collected Works of Helmut Wielandt Vol. 2 (eds. B. Huppert and H. Schneider), De Gruyters, 1996. As I protested that I had never seen Wielandt�s 1967 University of Wisconsin Lecture notes upon which he wanted me to write a commentary, he made a photocopy and sent it to me.
 
The 1968 Aequationes Math. 2 paper by Bonsall, Cain, Schneider showed that the Banach Space numerical range V(T) for a bounded operator T is always connected but need not be convex. That was the first paper I saw by Hans and I always regarded it as a fundamental result.

Nicolas Higham

I have posted some thoughts about Hans on my blog at: https://nickhigham.wordpress.com/2014/11/04/hans-schneider-1927-2014/.

Alan Hoffman

I do not remember when Hans and I first met, but I do remember we were both ardent admirers of Olga Taussky. Over the years, Hans and I became good friends united by (among other pacts) a common commitment: there would be no nonsense in letters of recommendation, or in refereeing of papers. At conferences, we would take private walks, and I loved to hear his combination of Scottish, English and Viennese German. We would also discuss mathematics, but that was done more frequently in letters than in conversation. In his last weeks, his letters contained commentary on his declining health, which he described bravely. The linear algebra community has lost its leader, and I have lost a friend.

Leslie Hogben

It has hard to write about Hans and his impact because what is accurate sounds like an exaggeration. I cannot imagine linear algebra without Hans' profound mathematical contributions and leadership, the latter including the International Linear Algebra Society and Linear Algebra and its Applications. Hans built a culture in linear algebra that is very welcoming, and as a result linear algebra has become a magnet for people changing research direction, as I did, and encourages and supports young researchers.

Stephen Kirkland

I finished my Ph.D. in 1989, the same year that ILAS was founded. Throughout my entire professional life, I have benefited from the work of ILAS, a professional society dedicated to advancing and promoting linear algebra, and to developing a sense of community and camaraderie among the matrix enthusiasts of the world. I know that a good many people have worked hard to contribute to ILAS's success, but I'm of the view that the lion's share of the credit belongs to Hans. His vision for the Society and his passion for linear algebra continue to shape our discipline. I for one think that the discipline is the better for it.
 
During the time I served as ILAS President, I sometimes consulted informally with Hans on ILAS matters, wanting to be sure that the Society stayed true to its original intentions. Hans always struck the right note with his advice -- he would give me his views on the matter, then give me the space to make my own way with the decision. Many thanks, Hans. I am proud to have known you.

Peter Lancaster

I am not sure when I first met Hans - probabaly in the 1970's. We always got along very well and felt some common roots - starting with undergraduate studies in the UK in the post-war late 1940's, and we both had some connections with A.C. Aitken, Alexander Ostrowski, Olga Taussky, and Jack Todd in the 1950's and '60's. I always admired Hans' courage and dedication in taking a leadership role with "Linear Algebra and its Applications" in 1973. We met quite frequently on the conference circuit, including a memorable one that I organised in Calgary in the 1980's. Another happy connection came through Hans' and Miriam's daughter Barbara - who is a musician with the Calgary Philharmonic Orchestra. I count myself fortunate indeed to have won the Hans Schneider prize of 2004, and am very grateful to Hans for creating that award.
 
He has had a great career and a huge creative influence on our discipline.

Bas Lemmens

It was at the ILAS meeting in Amsterdam in 2006 when I first met Hans. I was just staring my research career and, of course, knew some of his outstanding work in Perron-Frobenius theory. After my talk on certain nonlinear Perron-Frobenius theorems we had a very pleasant and interesting discussion. It was one of those rare occasions when you meet someone who has really understood what you were talking about and has a genuine interest in the work. Since then we kept in contact. I learnt a lot from Hans and was always impressed by his knowledge, open-mindedness, and generosity, which is a rare combination of virtues among mathematicians. I will certainly miss meeting Hans at conferences, his warm smile, and his big-heartedness.

Judi McDonald

I was fortunate enough to work on a PhD thesis under the supervision of Hans Schneider. A visit to his office always meant walking through the front doors of Bascom Hall and climbing the grand staircase. The grand staircase was made of hardwood, well able to stand the test of time - just like Hans and his life work. It was polished to a beautiful shine, showing the intricacies in the grain of the wood, reminding me how much Hans appreciated and shared the beauty and intricacy of mathematics and life. The design was open and inviting much like Hans was open and inviting to anyone who wanted to explore linear algebra. The staircase has several corners, promising things yet to be discovered as long as we continue to climb. Hans was always eager to see what was around the next corner. The grand staircase will always remind me of this grand man. Now that the staircase isn't tall enough to visit Hans, I will miss him dearly.

Volker Mehrmann

In 1983 right after my PhD, I met Hans for the first time at an Oberwolfach meeting. He invited me to Madison and I went for a year in 1984. This was a decisive step in my career. Hans supported me in many respects and I am sure without his advice and help I would have never become a professor. Since 1983 he has been a personal friend and mentor and we had many scientific and personal collaborations. He particularly educated me on how to assess the worst times in German history in particular during our writing about Helmut Wieland's mathematical and personal life.
 
I will always remember Hans' kindness and our endless discussions.
 
Thank you Hans for being my friend and supporter.

Dianne O'Leary

Hans Schneider was truly one of the giants in the field of linear algebra. His mathematical legacy is not only his own substantial body of research but also his efforts in supporting and growing the research community though his work with ILAS and LAA. The interactions he nurtured through this tremendous service to the community continue to be fruitful. He will certainly be missed by all of us.

Siegfried Rump

I became interested in matrix theory when working on a generalization of Perron-Frobenius theory to real and complex matrices. Then, it was inevitable to meet Hans.
 
His passion for Linear Algebra, for matrix theory and in particular for non-negative matrices was inspiring for me and most fruitful.
 
Once I invited him to Hamburg, and the title of his talk was: "Why I love Perron-Frobenius theory". We share this love.
 
Every time meeting him was always extremely pleasant. I will remember Hans as an excellent mathematician and a wise man.

Peter Semrl

I started my career working on operator theory problems. Over the years I became more and more interested in the finite-dimensional case moving in the direction of matrix analysis. But when did I become a linear algebraist?
 
I took part in some ILAS conferences, not very regularly at the beginning. Finding them very fruitful I became a regular participant at some point. I have also joined ILAS. But looking backwards, the moment when I really started to feel as a member of the linear algebra community was when Hans, to whom I had not been formally introduced, unexpectedly approached me and invited me to join him and two colleagues going out for a beer after the last afternoon talk at one of the conferences.
 
Later on, I met Hans many times discussing mathematics, ILAS, LAA, and also non-mathematical topics. Our communication intensified when I joined him, Richard, and Volker as Editor-in-Chief of LAA. On several occasions he was explaining to me his views on our research field, our journal, and our society. I always enjoyed these conversations and was astonished by his full devotion to linear algebra. I know that everything I learned from Hans will be of great help in my future work for ILAS and LAA.

Sergei Sergeev

I met Hans in 2006 in Birmingham. Back then I was in the very beginning of my career, and the possibility of working with a real expert in matrix theory, learning from him, trying to work on his valuable ideas meant a lot to me. I benefitted a lot from his enthusiasm and love for matrix theory and, especially, Perron-Frobenius.
 
It was easy to make friends with this great man, due to his sincere, open, modest, and very friendly personality. I am indebted to my numerous visits to Hans� house in Madison, a beautiful house full of wise books about mathematics and philosophy, and that spirit of music, and art, and mathematics. We would meet in the morning and spend time discussing our research plans and ideas in his cabinet, very informally and with no rush. Eat a modest lunch prepared by Miriam. In the evening, there were fireflies (or snowstorms). We did some nice papers together, and life was good.
 
Hans� death is a great loss to me, and I will remember him as one of my most important teachers and one of my dearest friends.

Bryan Shader

In 1977, Hans wrote a lovely paper "Olga Taussky-Todd's Influence on Matrix Theory and Matrix Theorists" (available at http://www.math.wisc.edu/hans/paper_archive/other_papers/hs057.pdf). In this tribute to Olga, Hans said
in assessing the work of a mathematician, one must ask the classical questions: "what theorems did he prove?", "what theories did he initiate?", but one must also ask: "what is his influence on others?".
Clearly, the answer to the last is tremendous. Evidence includes his 17 PhD students, 54 descendants, over 170 publications, the papers he oversaw as Editor-in-chief for LAA, and the thriving linear algebra research community.
 
On the personal side, as a graduate student at the University of Wisconsin-Madison, I had the great pleasure and experience of taking a Matrix Theory class from Hans. Hans did not lecture in the typical way; he would use boards on three sides of the room to write on and would pace back and forth throughout the lecture. Initially, I was bothered that Hans didn't always write all the details of proofs. But then I quickly realized that what Hans was saying, but not necessarily writing down, was full of gems. Hans always presented a clear way of thinking about things and was able to identify the crux of an argument. I fondly remember his excitement about some results, e.g. the Perron-Frobenius theorem.
 
Once, Hans filled in as the instructor for a Calculus class that I was a teaching assistant for. The class of about 350 students was held in a large lecture hall. As Hans was pacing back and forth on the stage, he accidently went to far off-stage and was not visible to the students; yet his voice was still audible. When Hans re-appeared on the stage, the students burst out into laughter. About 2 minutes later this repeated as Hans exited off the other side of the stage, and then reappeared to laughter. By then, Hans had figured out what was happening, and began using it to his advantage. He would walk off-stage, and choose to discuss a particularly critical point just before he would pop back onto stage. Each time he popped back onto stage, he would give a different face or gesture.
 
Hans was a member of my PhD committee. He greatly encouraged me, as he continued to do over the last 25 years. Hans impressed me with his devotion to the linear algebra community. He saw the power of encouraging students, yet holding the community to high standards for publications. The Linear Algebra community has reached its currently robust state with Hans' vision, passion and commitment.
 
Hans greatly influences my mathematical interests; I enjoy reading and learning from his papers and have had the opportunity on several occasions to utilize a theorem or two of Hans'.
 
Hans always encouragingly mentored me in my research, refereeing, and editorial roles. Most recently, he mentioned how pleased he was to see that the continued growth in stature of the Electronic Journal of Linear Algebra.
 
I have always admired Hans' devotion to his family, and enjoyed hearing Hans talk about his family.
 
I will miss him, but I know that Hans will continue to influence me and the linear algebra community.

Daniel Szyld

I got to know Hans over the years, first through my interest in nonnegative matrices, and later through my involvement in ILAS, including the launching of ELA, while Hans was ILAS president. Every exchange we had, in person or by email, was invariably a pleasure. He was always a gentle man, a strong intellectual, and a lover of opera. I will forever remember him very fondly.

Pauline van den Driessche

With Hans' passing, the world has lost a leader in linear algebra and matrix theory, and a wonderful person with immense generosity and vision. It must be almost 30 years ago that Hans came to the University of Victoria and gave an invited lecture at a meeting that Dale Olesky and I organized. But I mostly remember one of the last times that I met Hans, it was at an AMS Sectional Meeting in April 2013 held at Iowa State University, Ames. He and Miriam had driven from Madison. Hans gave a talk in a special session at which he also contributed questions, gave suggestions and showed insight into presentations. His encouragement helped many people, his leadership promoted the discipline of linear algebra, and his knowledge and good humor enlivened us all. Thank you, Hans.

Richard S. Varga

I have known Hans Schneider for over 50 years, and considered him to be a great friend of mine. We both worked on mathematical items in linear algebra, such as Gerschgorin circles, but there was no outright rivalry between our two groups. Hans, using some of his own money, created the Hans Schneider Prize in Linear Algebra, which has become a highly honored prize. I am proud to say that I received this prize in 2005.
 
I remember when, early on, Hans was invited to give a lecture at the Case Western Reserve University, and we had the opportunity of spending several hours together. I wanted to practice my speaking knowledge of German on him, but he declined, reminding me that he was forced to leave Vienna, early in his life. Hans had 14 Ph.D. students and 54 descendants, from the Mathematics Genealogy Project,and many of his students became leaders in mathematical research. In addition, Hans worked with countless others as co-authors. But, I believe that his work, with the journal Linear Algebra and Its Applications, was one of his greatest gifts to mathematics. He was Editor-in-Chief of this journal from 1972 to 2012, some forty years, and he will be greatly remembered for this.

Robert Wilson

Hans was my advisor, and I owe him so much (and my wife Elsie and I owe so much also to Miriam).
 
When I was trying to finish my PhD, Hans was department chair. I would come in for my regular visit, prepared to tell him about the stuff I had worked hard on since last time. One or both of two things would happen: (a) As chair, he would get a call from a dean or the equivalent, that had to be taken right then. I learned a fair amount about how departments and colleges worked by sitting there,,, (b) If I could have Hans' full attention, I would start to describe what I had laboriously worked out. He would invariably ask me to draw him a picture. If he saw the right picture, he would immediately know almost all the details that were in my proofs. I still believe in that: The picture ought to be convincing, although as a student I was certainly expected to nail down every detail.
 
An aspect several have mentioned somewhat but I want to emphasize is how versatile, how mathematically broad, Hans could be. The work he led me through was divided between a part on the boundary between finite geometry and combinatorics and another piece in universal algebra. We had also worked on some problems in algebraic semigroups. Many over the years have assumed that because I was Hans' student I must be "in" linear algebra. I do love it, and am proud to be a member (although I need to renew!), and in fact my father worked in that area, but I plead relative ignorance!

Henry Wolkowicz

I met Hans at the Householder meeting at Oxford in 1981. I was just starting as an academic and was introduced by my friend/colleague/teacher George Styan. From that time on Hans always was extremely friendly, always ready to talk, and ready to go out and 'eat' with George and the gang. He made a special effort at helping out young/new academics and getting them involved in the group.
 
One point/story: Hans was extremely interested in my use of email, a novelty at that time in the early 80's. He saw that this was the future of communicating among colleagues for research and for journal work. I was at the University of Alberta, UofA, and managed to obtain 100 accounts so people could phone into UofA and use email. Hans organized the distribution of these accounts to various 'Linear Algebra' people. It was comical in a way that Hans and the other early email users in linear algebra communicated through UofA. I only remember a few of the other email users. I hope it proved helpful.

Past Presidents of ILAS

ILAS Past Presidents
Hans Schneider, Richard Brualdi, Daniel Hershkowitz, and Stephen Kirkland