This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is elementary and based on the effective Nullstellensatz. The existence of a common invariant subspace of $A_{1},\dots,A_{s}$ of dimension $d$ can be expressed in the first-order language of fields, and hence, the constructive version of Tarski's theorem yields the criterion. In addition, some applications of this criterion in quantum information theory are discussed.

A new kind of companion matrix is introduced, for polynomials of the form c(λ) = λa(λ)b(λ)+c_0, where upper Hessenberg companions are known for the polynomials a(λ) and b(λ). This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrence’s Mandelbrot companion matrix. The construction was motivated by use of Narayana-Mandelbrot polynomials, which are also new to this paper.

This paper explores how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. It demonstrates that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. A class of patterns is developed that are refined inertially arbitrary but not spectrally arbitrary, making use of the property of a properly signed nest. The paper includes a characterization of the inertially arbitrary and refined inertially arbitrary patterns of order three, as well as the patterns of order four with the least number of nonzero entries.

Student assessment in the classroom is necessary to support student growth and increase students’ content knowledge. Formative assessment, or assessment that helps guide instruction and learning, can take many forms. One widely-used form is student self-assessment, in which students assess their own learning and set goals to increase their understanding of a topic. While experts agree that the process of self-assessment is valuable, this value is dependent on the teaching methods, practice and support provided by teachers for students in the classroom throughout the school year. A number of research-based best practices were incorporated into a middle-school science self-assessment, which was used throughout the 2016-207 school year. Teacher observations are discussed and next steps for further student growth are identified.

Many states, including Wyoming, have adopted the Next Generation Science Standards (NGSS) or a very similar version of science standards. Research shows that in order to remain competitive in the field trip market, science museums need to align their curriculum to the same standards (Anderson, Kisiel, & Storksdieck, 2006). The major challenge for museums engaged in this process is the gaps in content introduced to various grade levels across performance expectations. I proposed that these gaps may be filled by creating a learning progression to inform the alignment process. Literature on cognitive development, informal learning, and previous learning progressions is used to create a physical science learning progression, scaffolded by foundational concepts and grade level. Next, I analyzed lessons from The Science Zone in Casper, WY to identify which foundational concept they addressed, then created a matrix to show which lessons were appropriate for each grade level. The final product is a suggested progression of physical science lessons for The Science Zone that is developmentally appropriate and aligned to the NGSS.

Teacher beliefs are personal constructs that develop over a lifetime and are influenced by a teacher’s personal experiences, experience with schooling and instruction, and experience with formal knowledge. A teacher’s belief system has a greater impact on their practice than their subject matter knowledge, which translates into the design of classroom materials. This study was conducted to explore the impact of my teacher beliefs and understanding of threedimensional learning on the development of a theme-based, integrated, standards aligned unit. Teacher beliefs can be changed through the implementation of authentic professional learning communities (PLCs). PLCs are defined as the ongoing collaboration of educators in recurring cycles of collective inquiry and action research to improve student achievement. Throughout the study, a reflective process was used to identify my core teacher beliefs: the use of high quality text, integration, collaboration, understanding by design, and place-based principles. Findings indicate that curriculum development and collaboration are challenging, but worthwhile, and deeper understanding of my beliefs was essential to my persistence throughout the process.

Gardens can be used as a valuable tool in schools to teach curriculum, improve student behavior and self-confidence, instill a sense of responsibility in participants, and strengthen connections between schools and the community. The benefits of school gardens are well documented, but several challenges impede the effective use of gardens in many youth programs due to the time and knowledge required in maintaining garden-based education programs. In response to this need, a business plan was developed for a garden-based education center that will provide schools and communities with educational resources and support for maintaining gardening programs.

The business plan is supported by in-depth research on garden-based education (GBE), beginning with a literature review that explores the successes and shortcomings of garden-based education programs. This research is additionally informed by educator surveys and interviews with existing GBE non-profit organizations. The information gathered and analyzed from these sources form the foundation of the business plan for Growing Real Opportunities for Wyoming (GROWyoming), an organization promoting sustainable living, educational innovation, and community partnerships through garden-based education programs.

Metabolic engineering can be used to alter or enhance various metabolic pathways in microorganisms for the purpose of producing fuels, chemicals, and pharmaceutical products. In this work, we aim to discover if certain genetic modifications can redirect carbon flux to increase product yields of isoprene, which is used to synthesize rubber. Here, Rubrivivax gelatinosus CBS is used as a bacterial platform for investigating the carbon flux from CO through terpenoid pathways. Methods for increasing feedstock consumption, introduction of foreign genes, and deletion of native genes are all utilized in an attempt to increase isoprene titers.

Pathways targeted for deletion are responsible for the production of polyhydroxyalkanoates, which are energy storage molecules. Portions of the mevalonate pathway are added along with an isoprene synthase gene to enable increased production of isoprene. Previous works have shown successful expression of the mevalonate pathway in cyanobacteria, as well as production of isoprene. But no work has been done to investigate redirecting carbon flux from energy storage pathways towards isoprene production. Also, previous works investigating the conversion of CO to isoprene have not used a photosynthetic microorganism to alleviate the energy constraints of growth on CO. We also investigate the transcriptional regulators tied to growth on carbon monoxide, which have not been fully elucidated in CBS.

Successful readers must be able to fluently decode words they are reading independently and with automaticity. Once they can decode words fluently, they are able to comprehend what they are reading more easily. This action research study describes how reformatting traditional fluency instruction paired with additional student practice time increases the reading ability of first grade students.

The highlighted research was conducted in a first grade classroom with six and seven year old students. Each student was given explicit fluency instruction within a small group setting in addition to whole group fluency instruction, at their specific reading level, with independent fluency practice. Fluency strategies were taught during whole group times, and reinforced in small groups. In addition, each student was given fluency passages or Readers’ Theater scripts at their individual reading level to practice at home nightly. This action research project shares the benefits of specific, focused fluency instruction within a small group setting and how fluency instruction in first grade can help to close the reading achievement gap.

Five reading assessments were used to determine the growth of each student before and after the instructional changes in fluency were put in place. The results show how the small group that participated in the reformatted fluency instruction, increased in words per minute, number of mastered sight words, increased comprehension, and overall reading level at twice the rate as that of students not in the group in the same class.

This paper considers the problem of computing the state reachable points (from the origin) of a linear constant coefficient second order descriptor system. A new method is proposed to compute the reachable set in a numerically stable way. The original descriptor system is transformed into a strangeness-free system within the behavioral framework followed by a projection that separates the system into different order differential and algebraic equations while keeping the original state variables. This reformulation is followed by a first order formulation that avoids all unnecessary smoothness requirements. For the resulting first order system, it is shown that the computation of the image space of two matrices, associated with the projected system, is enough to numerically compute the reachable set. Moreover, a characterization is presented of all the inputs by which one can reach an arbitrary point in the reachable set. These results are used to compute two different types of reachable sets for second order systems. The new approach is demonstrated through a numerical example.

The spectra of digraphs, unlike those of graphs, is a relatively unexplored territory. In a digraph, a separation is a pair of sets of vertices D and Y such that there are no arcs from D and Y. For a subclass of Eulerian digraphs, a bound on the size of a separation is obtained in terms of the eigenvalues of the Laplacian matrix. An infinite family of tournaments, namely, the Paley digraphs, where the bound holds with equality, is also given.

The enhanced principal rank characteristic sequence (epr-sequence) of an $n \times n$ symmetric matrix over a field $\F$ was recently defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either $\tt A$, $\tt S$, or $\tt N$ based on whether all, some (but not all), or none of the order-$k$ principal minors of the matrix are nonzero. Here, a complete characterization of the epr-sequences that are attainable by symmetric matrices over the field $\Z_2$, the integers modulo $2$, is established. Contrary to the attainable epr-sequences over a field of characteristic $0$, this characterization reveals that the attainable epr-sequences over $\Z_2$ possess very special structures. For more general fields of characteristic $2$, some restrictions on attainable epr-sequences are obtained.

Let S be a column stochastic matrix with at least one full row. Then S describes a Pagerank-like random walk since the computation of the Perron vector x of S can be tackled by solving a suitable M-matrix linear system Mx = y, where M = I − τ A, A is a column stochastic matrix and τ is a positive coefficient smaller than one. The Pagerank centrality index on graphs is a relevant example where these two formulations appear. Previous investigations have shown that the Euler- Richardson (ER) method can be considered in order to approach the Pagerank computation problem by means of preconditioning strategies. In this work, it is observed indeed that the classical power method can be embedded into the ER scheme, through a suitable simple preconditioner. Therefore, a new preconditioner is proposed based on fast Householder transformations and the concept of low complexity weakly stochastic algebras, which gives rise to an effective alternative to the power method for large-scale sparse problems. Detailed mathematical reasonings for this choice are given and the convergence properties discussed. Numerical tests performed on real-world datasets are presented, showing the advantages given by the use of the proposed Householder-Richardson method.