and should be submitted in accordance with the instructions given on the inside front cover. An a... more and should be submitted in accordance with the instructions given on the inside front cover. An asterisk placed beside a
It is well known that mathematics students have to be able to understand and prove theorems. From... more It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using ...
Challenges and Strategies in Teaching Linear Algebra, 2018
We present examples of interesting problems that hopefully make the learning and the teaching of ... more We present examples of interesting problems that hopefully make the learning and the teaching of linear algebra enjoyable. The problems are on matrix multiplication, rank, determinants, eigenvalues and eigenvectors, and matrices and graphs. The problem solving strategies used include “look for invariants”, “check parity” and “define an energy function”.
ABSTRACT Many researchers consider problem posing tasks to be a powerful tool for assessing mathe... more ABSTRACT Many researchers consider problem posing tasks to be a powerful tool for assessing mathematical creativity. Frequently mentioned indicators of creativity in students' problem posing include flexibility, fluency and originality. In the presented study we characterize, in terms of these indicators, problem posing of 15 high achieving secondary school students. We then argue that the aforementioned indicators do not fully capture the essence of the students’ creativity and suggest that considerations of aptness that the students imply in the process of problem posing can serve as a useful indicator of their creativity in the chosen context.
We introduce a new variant of zero forcing - signed zero forcing. The classical zero forcing numb... more We introduce a new variant of zero forcing - signed zero forcing. The classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph (i.e. zero-nonzero pattern). Our new variant provides an analo- gous bound for the maximum nullity of a matrix with a given sign pattern. This allows us to compute, for instance, the maximum nullity of a Z-matrix whose graph is L(K_n), the line graph of a clique.
A block graph is a graph in which every block is a complete graph. Let G be a block graph and let... more A block graph is a graph in which every block is a complete graph. Let G be a block graph and let A(G) be its (0,1)-adjacency matrix. Graph G is called nonsingular (singular) if A(G) is nonsingular (singular). An interesting open problem, proposed in 2013 by Bapat and Roy, is to characterize nonsingular block graphs. In this article, we present some classes of nonsingular and singular block graphs and related conjectures.
Research Reports The goal of a research report is to present empirical or theoretical research in... more Research Reports The goal of a research report is to present empirical or theoretical research in the field of mathematical creativity and the education of gifted students. Projects and Ideas This type of presentations includes descriptions of the teaching practices, curricula, projects and programs aimed at the developing of students' mathematical creativity and promoting giftedness. Workshops This type of presentations requires the active involvement of the participants, and emphasize problem-solving or hands-on activities, and requires the involvement of the participants. Posters Exhibitions and Poster Presentations are available for those whose work are more suitably communicated in a pictorial or graphic form or as a demonstration (e.g., a computer program), rather than through an oral presentation.
... Page 4. 310 ABRAHAM BERMAN AND MICHAEL NEUMANN This means that (AY) is nonsingular. Decompose... more ... Page 4. 310 ABRAHAM BERMAN AND MICHAEL NEUMANN This means that (AY) is nonsingular. Decompose (2.13) (AY E ... Finally, by (2.10), (A Y)(B ?)( j)=D, and so, by (2.12), D is similar to (0 0). THEOREM 3. Let M, N, Ml 1 and N1 1 be given by (1.2) and (2.9). ...
We study the spectrum of a positive matrix that arises in the study of certain communication netw... more We study the spectrum of a positive matrix that arises in the study of certain communication networks. Bounds are given on the rate of convergence of the networks to the equilibrium condition.
The relationships between heuristic literacy development and mathematical achievements of middle ... more The relationships between heuristic literacy development and mathematical achievements of middle school students were explored during a five-month classroom experiment in two 8 th grade classes (N=37). By heuristic literacy we refer to an individual's capacity to use heuristic vocabulary in problemsolving discourse and to approach scholastic mathematical problems by using a variety of heuristics. During the experiment, the heuristic constituent of curriculum-determined topics in algebra and geometry was gradually revealed and promoted by means of incorporating heuristic vocabulary in classroom discourse and seizing opportunities to use the same heuristics in different mathematical contexts. Students' heuristic literacy development was indicated by means of individual thinking-aloud interviews and their mathematical achievement-by means of the Scholastic Aptitude Test. We found that heuristic literacy development and changes in mathematical achievements are correlated yet distributed unequally among the students. In particular, the same students, who progressed with respect to SAT scores, progressed also with respect to their heuristic literacy. Those students, who were weaker with respect to SAT scores at the beginning of the intervention, demonstrated more significant progress regarding both measures.
When a lot of books are written on a subject, one of two cases obtains. Either the subject is wel... more When a lot of books are written on a subject, one of two cases obtains. Either the subject is well understood, and the book is easy to write; such is the case with books on real variables, convexity, projective geometry in the plane, or compact orientable surfaces. Or else, the subject is of great importance, but nobody really understands what is going on; such is the case with quantum field theory, the distribution of primes, pattern recognition, and cluster analysis. The present book is an instance of the second one.
In this article we prove the strict monotonicity of the spectral radius of weakly irreducible non... more In this article we prove the strict monotonicity of the spectral radius of weakly irreducible nonnegative tensors. As an application, we give a necessary and sufficient condition for an interval hull of tensors to be contained in the set of all strong M-tensors. We also establish some properties of M-tensors. Finally, we consider some problems related to interval hull of positive (semi)definite tensors and P (P 0)-tensors.
We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the ... more We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the minimum rank of matrices described by Z 1 is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by Z 2 is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture in [AHKLR] about rational realization of minimum rank of sign patterns. Using Z 1 and Z 2 , we construct symmetric patterns, equivalent to graphs G 1 and G 2 , with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.
We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the ... more We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the minimum rank of matrices described by Z 1 is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by Z 2 is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture in [AHKLR] about rational realization of minimum rank of sign patterns. Using Z 1 and Z 2 , we construct symmetric patterns, equivalent to graphs G 1 and G 2 , with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.
and should be submitted in accordance with the instructions given on the inside front cover. An a... more and should be submitted in accordance with the instructions given on the inside front cover. An asterisk placed beside a
It is well known that mathematics students have to be able to understand and prove theorems. From... more It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using ...
Challenges and Strategies in Teaching Linear Algebra, 2018
We present examples of interesting problems that hopefully make the learning and the teaching of ... more We present examples of interesting problems that hopefully make the learning and the teaching of linear algebra enjoyable. The problems are on matrix multiplication, rank, determinants, eigenvalues and eigenvectors, and matrices and graphs. The problem solving strategies used include “look for invariants”, “check parity” and “define an energy function”.
ABSTRACT Many researchers consider problem posing tasks to be a powerful tool for assessing mathe... more ABSTRACT Many researchers consider problem posing tasks to be a powerful tool for assessing mathematical creativity. Frequently mentioned indicators of creativity in students' problem posing include flexibility, fluency and originality. In the presented study we characterize, in terms of these indicators, problem posing of 15 high achieving secondary school students. We then argue that the aforementioned indicators do not fully capture the essence of the students’ creativity and suggest that considerations of aptness that the students imply in the process of problem posing can serve as a useful indicator of their creativity in the chosen context.
We introduce a new variant of zero forcing - signed zero forcing. The classical zero forcing numb... more We introduce a new variant of zero forcing - signed zero forcing. The classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph (i.e. zero-nonzero pattern). Our new variant provides an analo- gous bound for the maximum nullity of a matrix with a given sign pattern. This allows us to compute, for instance, the maximum nullity of a Z-matrix whose graph is L(K_n), the line graph of a clique.
A block graph is a graph in which every block is a complete graph. Let G be a block graph and let... more A block graph is a graph in which every block is a complete graph. Let G be a block graph and let A(G) be its (0,1)-adjacency matrix. Graph G is called nonsingular (singular) if A(G) is nonsingular (singular). An interesting open problem, proposed in 2013 by Bapat and Roy, is to characterize nonsingular block graphs. In this article, we present some classes of nonsingular and singular block graphs and related conjectures.
Research Reports The goal of a research report is to present empirical or theoretical research in... more Research Reports The goal of a research report is to present empirical or theoretical research in the field of mathematical creativity and the education of gifted students. Projects and Ideas This type of presentations includes descriptions of the teaching practices, curricula, projects and programs aimed at the developing of students' mathematical creativity and promoting giftedness. Workshops This type of presentations requires the active involvement of the participants, and emphasize problem-solving or hands-on activities, and requires the involvement of the participants. Posters Exhibitions and Poster Presentations are available for those whose work are more suitably communicated in a pictorial or graphic form or as a demonstration (e.g., a computer program), rather than through an oral presentation.
... Page 4. 310 ABRAHAM BERMAN AND MICHAEL NEUMANN This means that (AY) is nonsingular. Decompose... more ... Page 4. 310 ABRAHAM BERMAN AND MICHAEL NEUMANN This means that (AY) is nonsingular. Decompose (2.13) (AY E ... Finally, by (2.10), (A Y)(B ?)( j)=D, and so, by (2.12), D is similar to (0 0). THEOREM 3. Let M, N, Ml 1 and N1 1 be given by (1.2) and (2.9). ...
We study the spectrum of a positive matrix that arises in the study of certain communication netw... more We study the spectrum of a positive matrix that arises in the study of certain communication networks. Bounds are given on the rate of convergence of the networks to the equilibrium condition.
The relationships between heuristic literacy development and mathematical achievements of middle ... more The relationships between heuristic literacy development and mathematical achievements of middle school students were explored during a five-month classroom experiment in two 8 th grade classes (N=37). By heuristic literacy we refer to an individual's capacity to use heuristic vocabulary in problemsolving discourse and to approach scholastic mathematical problems by using a variety of heuristics. During the experiment, the heuristic constituent of curriculum-determined topics in algebra and geometry was gradually revealed and promoted by means of incorporating heuristic vocabulary in classroom discourse and seizing opportunities to use the same heuristics in different mathematical contexts. Students' heuristic literacy development was indicated by means of individual thinking-aloud interviews and their mathematical achievement-by means of the Scholastic Aptitude Test. We found that heuristic literacy development and changes in mathematical achievements are correlated yet distributed unequally among the students. In particular, the same students, who progressed with respect to SAT scores, progressed also with respect to their heuristic literacy. Those students, who were weaker with respect to SAT scores at the beginning of the intervention, demonstrated more significant progress regarding both measures.
When a lot of books are written on a subject, one of two cases obtains. Either the subject is wel... more When a lot of books are written on a subject, one of two cases obtains. Either the subject is well understood, and the book is easy to write; such is the case with books on real variables, convexity, projective geometry in the plane, or compact orientable surfaces. Or else, the subject is of great importance, but nobody really understands what is going on; such is the case with quantum field theory, the distribution of primes, pattern recognition, and cluster analysis. The present book is an instance of the second one.
In this article we prove the strict monotonicity of the spectral radius of weakly irreducible non... more In this article we prove the strict monotonicity of the spectral radius of weakly irreducible nonnegative tensors. As an application, we give a necessary and sufficient condition for an interval hull of tensors to be contained in the set of all strong M-tensors. We also establish some properties of M-tensors. Finally, we consider some problems related to interval hull of positive (semi)definite tensors and P (P 0)-tensors.
We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the ... more We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the minimum rank of matrices described by Z 1 is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by Z 2 is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture in [AHKLR] about rational realization of minimum rank of sign patterns. Using Z 1 and Z 2 , we construct symmetric patterns, equivalent to graphs G 1 and G 2 , with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.
We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the ... more We use a technique based on matroids to construct two nonzero patterns Z 1 and Z 2 such that the minimum rank of matrices described by Z 1 is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by Z 2 is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture in [AHKLR] about rational realization of minimum rank of sign patterns. Using Z 1 and Z 2 , we construct symmetric patterns, equivalent to graphs G 1 and G 2 , with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.
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