Search

In this paper we describe a method for teaching students to prove some mathematical statements independently, by using specially designed auxiliary assignments. The assignments are designed as homework problems and can be adapted for online learning. We illustrate our method using examples from calculus and differential equations.

In this article, a general notion of common diagonal Lyapunov matrix is formulated for a collection of n × n matrices A 1, … , A s , and cones k 1, … , k s in ℝ n . Necessary and sufficient conditions are derived for the existence of a common diagonal Lyapunov matrix in this setting. The conditions are similar to and extend the well-known criteria for the case s = 1, k 1 = ℝ n .

In this article we describe the process of studying the assumptions and the conclusion of a theorem. We tried to provide the students with exercises and problems where we discuss the following questions: What are the assumptions of a theorem and what are the conclusions? What is the geometrical meaning of a theorem? What happens when one or more of the theorem's assumptions are not fulfiled? What assumptions are necessary and which are sufficient? According to the students' opinion and the test results the developed approach was successful.

Let A be an n × n real matrix. A is called {0,1}-cp if it can be factorized as A = BB T with bij =0 or 1. The smallest possible number of columns of B in such a factorization is called the {0,1}-rank of A. A {0,1}-cp matrix A is called minimal if for every nonzero nonnegative n × n diagonal matrix D, A-D is not {0,1}-cp, and r-uniform if it can be factorized as A=BB T, where B is a (0, 1) matrix with r 1s in each column. In this article, we first present a necessary condition for a nonsingular matrix to be {0,1}-cp. Then we characterize r-uniform {0,1}-cp matrices. We also obtain some necessary conditions and sufficient conditions for a matrix to be minimal {0,1}-cp, and present some bounds for {0,1}-ranks.

This note discusses some of the shortcomings of multiple choice tests in Mathematics given to undergraduate engineering students. Examples are presented, where the disadvantages of a multiple choice test are pointed out and suggestions how to overcome the difficulties are given.

Several examples of test problems are presented in which students are asked to identify mistakes in given wrong proofs.

Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.

This note proves a conjecture of Merris that the minimal value of entries of the doubly stochastic matrix of the degree antiregular graph En of order n ≥ 3 is equal to (l/2(n + l)).

This note considers some examples of how multiple choice examinations can be used to check the understanding of engineering students of some theoretical parts of their course.

The paper consists of several remarks on complete positivity. The main result is that if the comparison matrix of an n × n completely positive matrix A is an M-matrix, then A has a support 3 rank 1 representation with at most terms.

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.

It is well known that a matrix, all of whose principal minors are positive, satisfies the Fischer-Hadamard inequalities if and only if it is weakly sign symmetric. In this paper we consider the general case of matrices whose principal minors may be nonpositive. Necessary conditions and a sufficient condition for the Fischer-Hadamard inequalities to hold are given in the general case.

Let λ be any subset of {1,2, 3, 4} containing 1 and consider the matrix equations (1) and AXA=A, (2) XAX=X, (3) AX=(AX) T (4) XA=(XA) T where A is an m×n real matrix. By a λ-inverse of A is meant a solution of the equations (i)ieλ. A result on binary relations is given and is used to obtain a characterization of all nonnegative matrices A having a nonnegative λ-inverse, for all possible λ.