Joel Lee Brenner

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Joel Lee Brenner
Born(1912-08-02)August 2, 1912
Matrix theory
Scientific career
FieldsMathematics
Thesis The Linear Homogeneous Group Modulo P  (1936)
Doctoral advisorGarrett Birkhoff

Joel Lee Brenner ((1912-08-02)August 2, 1912 – (1997-11-14)November 14, 1997) was an American

Stanford Research Institute from 1956 to 1968. He published over one hundred scholarly papers, 35 with coauthors, and wrote book reviews.[1][2][3]

Academic career

In 1930 Brenner earned a B.A. degree with major in

American Mathematical Monthly.[4]

In 1951 Brenner published his findings about matrices with

characteristic root of a quaternion matrix (an eigenvalue) and shows that they must exist. He also shows that a quaternion matrix is unitarily-equivalent to a triangular matrix
.

In 1956 he became a Senior Mathematician at

Stanford Research Institute
. Brenner, in collaboration with Donald W. Bushaw and S. Evanusa, assisted in the translation and revision of Felix Gantmacher's Applications of the Theory of Matrices (1959).[6]

Brenner translated

Aleksei Fedorovich Filippov
.

Brenner translated Problems in Higher Algebra

roots of unity, as well as some linear algebra and abstract algebra
.

In 1959 Brenner generalized propositions by

G. B. Price on minors of a diagonally dominant matrix.[8] His work is credited with stimulating a reawakening of interest in the permanent of a matrix.[9]

One of the challenges in linear algebra is to find the

University of Wisconsin—Madison, working in the Mathematics Research Center, he produced a technical report New root-location theorems for partitioned matrices.[11]

In 1968 Brenner, following

Alston Householder, published "Gersgorin theorems by Householder’s proof".[12] In 1970 he published the survey article (21 references) "Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices".[13] The article was extended with "Some determinantal identities".[14]

In 1971 Brenner extended his geometry of the spectrum of a square complex matrix deeper into abstract algebra with his paper "Regularity theorems and Gersgorin theorems for matrices over rings with valuation".

ring of polynomials
has a valuation ... a different type of regularity ..."

Collaborations

Joel Lee Brenner was a member of the American Mathematical Society from 1936.

Beasley relates that he

was a graduate student and [Brenner] was visiting the University of British Columbia in 1966-67. Shortly after arriving at UBC, Joel circulated a memo to all the graduate students, informing them that he had several open problems in various areas of mathematics and would share them with willing students. Hoping to get a problem in group theory that I might work into a thesis, I went to his office and inquired about the problems. He presented me the Van der Waerden conjecture, which he informed me would be quite difficult, and after defining the permanent for me sent me off with several problems concerning the permanent function. His encouragement and enthusiasm persevered through several "proofs" of the Van der Waerden conjecture, and soon some of the less well-known problems had been solved. He would always tell me how a proposed attack would work and leave me to fight out the details. Those exchanges led to the publication of my first paper, and I became his thirteenth coauthor. By the time Joel had left UBC in the spring of 1967, I was firmly entrenched in matrix theory.[3]: 3 

In 1981 Brenner and

H. W. Kuhn for proving the fundamental theorem of algebra. In the solution by Eric S. Rosenthal to a problem in the American Mathematical Monthly posted by Harry D. Ruderman,[16] Kuhn's work from 1974 was cited. A query was made and prompted an article by Brenner and Lyndon.[17]
The version of the fundamental theorem stated was as follows:

Let P(z) be a non-constant polynomial with complex coefficients. Then there is a positive number S > 0, depending only on P, with the following property:
for every δ > 0 there is a complex number z such that |z| ≤ S and |P(z)| < δ .

Brenner ultimately acquired 35 coauthors in his publications.

Alternating group

Given an ordered set Ω with n elements, the

even permutations on it determine the alternating group An. In 1960 Brenner proposed the following research problem in group theory:[18] For which An does there exist an element an such that every element g is similar to a commutator
of an? Brenner states that the property is true for 4 < n < 10; in symbols it may be expressed

The alternating groups are

cycle type of cyclic permutations, and when An ⊂ C C, where C is a conjugacy class of a certain type.[19][20][21]

In 1977 he posed the question, "What permutations in An can be expressed as a product of permutations of periods k and l" ?[22]

Works

In 1987

Linear Algebra and its Applications published a list of 111 articles by J.L. Brenner, and the four books he translated.[3]

Research

Book reviews

References

  1. ^ "Mathematics People" (PDF). Notices of the AMS. 45 (4). American Mathematical Society. 1998. Retrieved 18 December 2012.
  2. ^ "Brenner, J. L. (Joel Lee)". Retrieved 1 January 2013.
  3. ^
    Linear Algebra and its Applications
    90:1–13
  4. American Mathematical Monthly
    86: 359–6
  5. doi:10.2140/pjm.1951.1.329
    .
  6. ^ George Weiss (1960) Review Applications of the Theory of Matrices, Science 131: 405,6, issue #3398
  7. ^ Сборник задач по высшей алгебре
  8. doi:10.1215/S0012-7094-59-02653-5
    .
  9. ^ Henryk Minc (1978) Permanents, page 13, Encyclopedia of Mathematics and its Applications volume 6, Addison-Wesley
  10. ^ Brenner (January 1964) Theorems of Gersgorin type, citation from Defense Technical Information Center.
  11. ^ J. L. Brenner (1967) New root-location theorems for partitioned matrices, citation from Defense Technical Information Center
  12. ^ Brenner (1968) Gersgorin theorems by Householder’s proof, Bulletin of the American Mathematical Society 74:3, link from Project Euclid
  13. ^ Brenner (1970) "Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices", SIAM Journal for Applied Mathematics 19(2)
  14. ^ Brenner (1971) ) Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices, and some determinantal identities, Pacific Journal of Mathematics 39(1), link from Project Euclid
  15. ^ Brenner (1971) Regularity theorems and Gersgorin theorems for matrices over rings with valuation, Rocky Mountain Journal of Mathematics 1(3), link from Project Euclid
  16. ^ Solution to problem #6192, American Mathematical Monthly 86: 598
  17. R. C. Lyndon
    (1981) "Proof of the Fundamental Theorem of Algebra", American Mathematical Monthly 88(4):254–6
  18. ^ Brenner (1960) Research Problem in Group Theory, Bulletin of the American Mathematical Society 66(4):275
  19. ^ Brenner, R.M. Cranwell, and J. Riddell (1975) Covering theorems: V, Pacific Journal of Mathematics 58: 55–60
  20. ^ Brenner & L. Carlitz (1976) Rendiconti del Seminario Matematico della Università di Padova 55:81–90
  21. Journal of the Australian Mathematical Society
    25A: 210–14
  22. ^ Brenner & J. Riddell (1977) American Mathematical Monthly 84(1): 39–40
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