Pasha Zusmanovich

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Problem 47, Proposed by L.E. Dickson, Fellow in Mathematics, University of Chicago:
Prove that (-1)(-1) = +1.
I. Assuming the distributive law to hold, (-1)((+1) + (-1)), or 0, = (-1)(+1) + (-1)(-1). Assuming the commutative law, (-1)(+1) = (+1)(-1 = -1. (-1) + (-1)(-1) = 0, or (-1)(-1) = +1. [L.E. Dickson]
II. (-1)(-1) means that -1 is to be taken subtractively one time. 0 - (-1) = +1. (-1)(-1) = +1. [G.B.M. Zerr]
III. -1 × a = -a, -1 × (a-1) = -(a-1) = -a + 1. -1 × ((a-1)-a) = -a + 1 - (-a) = -a + 1 + a = 1. [M. Philbrick]
VII. According to Wood's Elementary Algebra, 17th edition, we have (-5)(-3) = +15. Here -3 is to be subtracted 5 times; that is, -15 is to be subtracted. Now, subtracting -15 is the same as adding +15. Therefore, we have to add +15. Similarly, (-1)(-1) = +1. [W.I. Taylor, F.P. Matz]
VIII. The case (-a)(-b) = +ab is purely conventional and consequently an assumption, which, however, does not deprive the result of its great importance to algebraic operations. [J.F.W. Scheffer]

Below is a list of possible research topics for students, formulated in a somewhat entertaining form suitable for, say, bachelor thesis. To get an idea about more advanced stuff, on a master or PhD level, please have a look at my writings. To start working on these topics, one should be comfortable with the basic mathematical notions studied during the first university years: matrices, polynomials, vector spaces, functions, etc.; the rest will be learned along the way. Some of the topics may involve computers, in which cases one should be comfortable in making hands dirty with computer code (in one or more common programming languages and/or computer algebra systems like GAP). The topics are given merely for orientation -- all kind of alterations, variations and additions are possible. If you do not see anything suitable to you, or not sure what you want to do, please talk to me and we will try to invent a suitable topic.

For students outside Ostrava and Czech Republic: subject availability of funding, the department announces call for PhD positions each year at spring. If interested, please contact me me with informal inquiries.

Lie algebras of small dimension

Lie algebras (named after Sophus Lie) are mathematical gadgets which appear in many areas of mathematics and physics. (Some flashy pictures may be seen in my habilitation lecture pdf TeX ). They may be defined in numerous ways. For example, the usual matrices you study during the first university year can be turned into a Lie algebra by considering the bracket \([A,B] = AB - BA\). Another way to define Lie algebras is via multiplication table (akin the multiplication table you learn in the school). For example, a gadget which is called the 2-dimensional nonabelian Lie algebra is given by the following multiplication table: \[ \begin{array}{c|cc} & x & y \\ \hline x & 0 & x \\ y & -x & 0 \end{array} \] The Lie algebras we will deal with are slightly more complicated (say, of dimension \(\le\) 15). There are many important and interesting questions about them (for example, to determine whether two algebras given by different multiplication tables are, essentially, the same). Many of these questions can be investigated with the help of computer.

Solving linear and nonlinear equations in algebras on computer

The algebras we are talking about can be Lie algebras (as above), or more conventional matrix algebras, or algebras of polynomials, etc. Many questions in mathematics and physics involve functions on algebras satisfying various equations, like, for example, derivation of an algebra which is a linear map \(D\) satisfying the usual Leibniz rule for the derivative of the product of functions: \[ D(xy) = D(x)y + xD(y) . \] The task will consist of improving existing computer programs for solving such equations, to write new ones, and experimenting with these programs.

When inverse power series have positive coefficients?

Consider the polynomial \(f(x) = x - x^8 + x^{15}\). The beginning terms of its inverse power series, i.e. a power series \(g(x)\) satisfying \(f(g(x)) = g(f(x)) = x\), are: \[ g(x) = x + x^8 + 7 x^{15} + 69 x^{22} + 790 x^{29} + 9842 x^{36} + \dots \] Is it true that all nonzero coefficients of \(g(x)\) are positive? Yes, it is, but the answer to this olympiad-looking question is highly nontrivial, obtained by ingenious combination of Lagrange inversion formula, a technique for summation of hypergeometric series, and computer calculations. There are many other innocently-looking polynomials, for which the answer to the same or similar questions is not known. These questions are related to the forefront of the current research in the area of mathematics called operad theory. (Hint: the coefficients, if positive, should have a combinatorial interpretation as dimensions of spaces of some "noncommutative polynomials").

Iterative correlation matrices

The correlation coefficient between two sequences measures how "dependent" these sequences are. If we have a number of sequences, representing, say, results of some measurements, their pairwise correlation coefficients can be arranged into correlation matrix. Now one can compute correlation between rows of the correlation matrix itself, getting "correlation matrix of a correlation matrix", and repeat this process. Such procedure is used in applied statistics. On practice, this iterative procedure "almost always" converges to a pattern consisting of \(-1\) and \(1\), but a rigorous proof of this is lacking, even in the case of \(3 \times 3\) matrices. Some initial matrices exhibit a very strange ("chaotic") behavior. There is a big freedom for computer experimentation.

History of Czech mathematics

Study life and work of some of XIX-early XX century Czech mathematicians. For example: Another topic might be to study history of first Czech mathematical journals. For all this work, you should have a taste for history, and be able to read old mathematical texts in German and French (and occasionally in Czech).

Other history

Homophonic group of Czech language

Consider the free group generated by 26 letters of the English alphabet, and take the quotient by all relations of the form A = B where A and B are two English words spelled differently, but having the same pronunciation. In the article Homophonic quotients of free groups by a group of authors including Don Zagier, it is proved that this group is trivial. The same is true for French. The task is to compute the same group for Czech language. It would be especially interesting if this group will turn out to be nontrivial (which is probable in view of the bigger number of letters in the Czech alphabet, due to diacritics).

Lewis Carroll's soriteses through polynomial systems

Consider the following:
  1. A simpleton, who is not always shouting, is sure to be a crab;
  2. None but spiders are good-humoured;
  3. No unsuccessful frog is despised, so long as it is healthy;
  4. All oysters are good-humoured;
  5. All spiders are healthy, except the green ones;
  6. Unsuccessful crabs, if good-humoured, are popular;
  7. Green crabs are always singing;
  8. The only simpletons, that are popular, are frogs;
  9. Rash young oysters are always unsuccessful;
  10. None but simpletons are good-humoured and yet despised;
  11. No old crabs are healthy;
  12. A rash spider is always despised.

Illustration by Gwynedd M. Hudson, courtesy of British Library.
This is one of the numerous Lewis Carroll's soriteses. Treat this and similar soriteses using modern tools - for example, present it as a system of Boolean polynomial equations, and solve it using the Gröbner bases method (possibly on computer). Compare modern solutions with those proposed by Carroll himself. Do Carroll's methods of solutions of soriteses described in his Symbolic Logic - method of separate syllogisms and method of underscoring - resemble some simple form of words composition in Gröbner bases method?

Created: Jan 19 2015
Last modified: Thu Aug 06 09:04:14 Central Europe Daylight Time 2020