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Chin, C. (2023). Everything I Learned, I Learned in a Chinese Restaurant. Little Brown and Company
No full textFrog Olympics
No full textGroups of three students make a small, medium, and large frog, using Cricut Design Space to size correctly and personalize their frogs. Once the frogs are folded, they are entered into five Olympic events: Long jump, same-side flipper, backwards flipper, high jump, and fastest faller. Students keep track of their measurements. During each event, the student can make three attempts and keep the best measurement. After each team member has completed all the events, students will calculate the mean, median, and mode for the three frogs for each event. As a class, they will put all the data on the board and decide if the mean, median, or mode best represent the data fairly to decide which team will get first place
Exploring the Volume of a Cylinder
No full textDuring this lesson, students create cylinders of different radii and heights using the Cricut Maker 3. They then test the volume of their own cylinder and compare it to the volume they calculate from the formula. They observe the differences in volume for cylinders with various radii or heights.
This lesson is intended for 8th grade math, but it could be modified for different grades by changing the shape students work with. For example, 5th graders could use rectangular prisms. Other shapes could include square and triangular pyramids, triangular or hexagonal prisms, and dodecahedrons
The nonnegative inverse eigenvalue problem with prescribed zero patterns in dimension three
No full textThe nonnegative inverse eigenvalue problem is considered in this paper with the additional restriction of fixed zero patterns in the matrix. A full analysis of the case is given. Some remarks on the four-dimensional case are made
Characterizations of complex P-matrices
No full textA P-matrix is a square matrix all of whose principal minors are positive. The characterization of real P-matrices as matrices that do not reverse the sign of any nonzero real vector is generalized to complex P-matrices by associating them with the reflection of complex vectors. This prompts the extension of other P-matrix properties and related real matrix classes to the complex field. In particular, semipositivity of real P-matrices is generalized to complex P-matrices. Principal pivot transforms and Cayley transforms of complex P-matrices are also considered
Kemeny’s constant and the Lemoine point of a simplex
No full textKemeny's constant is an invariant of discrete-time Markov chains, equal to the expected number of steps between two states sampled from the stationary distribution. It appears in applications as a concise characterization of the mixing properties of a Markov chain and has many alternative definitions. In this short article, we derive a new geometric expression for Kemeny's constant, which involves the distance between two points in a simplex associated with the Markov chain: the circumcenter and the Lemoine point. Our proof uses an expression due to Wang, Dubbeldam, and Van Mieghem of Kemeny's constant in terms of effective resistances and Fiedler's interpretation of effective resistances as edge lengths of a simplex
Affine subspaces of matrices with rank in a range
No full textThe problem of finding the maximal dimension of linear or affine subspaces of matrices whose rank is constant, or bounded below, or bounded above, has attracted many mathematicians from the sixties to the present day. The problem has caught also the attention of algebraic geometers, since vector spaces of matrices of constant rank give rise to vector bundle maps whose images are vector bundles of rank . Moreover, there is a link with the so-called "rank metric codes," since a constant rank subspace of can be viewed as a constant weight rank metric code; it can be interesting to study also the maximal dimension of the subspaces of whose elements have rank in a range , since such subspaces obviously give rank metric codes with weights in . In this paper, with the main purpose to get an organic result including the ones on spaces of matrices with constant rank, the ones on spaces of matrices with rank bounded below and the ones on spaces of matrices with rank bounded above and to generalize a previous result on real matrices with constant rank to matrices on a more general field, we study the maximal dimension of affine subspaces of matrices whose rank is between two numbers under mild assumptions on the field. We get also a result on antisymmetric matrices and on matrices in row echelon form
