Session A-3: Three-Act Math Tasks, 2017 Illinois Mathematics and Science Academy

#### Session A-3: Three-Act Math Tasks, Lindsey Herlehy

*Professional Learning Day*

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.

Lorentz Transformation From An Elementary Point Of View, 2017 Laboratoire de Physique Th\'{e}orique, Universit\'{e} de Toulouse III \& CNRS

#### Lorentz Transformation From An Elementary Point Of View, Arkadiusz Jadczyk, Jerzy Szulga

*Electronic Journal of Linear Algebra*

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.

An Introduction To Boolean Algebras, 2016 California State University - San Bernardino

#### An Introduction To Boolean Algebras, Amy Schardijn

*Electronic Theses, Projects, and Dissertations*

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures ...

Mathematics Education From A Mathematicians Point Of View, 2016 University of Tennessee, Knoxville

#### Mathematics Education From A Mathematicians Point Of View, Nan Woodson Simpson

*Masters Theses*

This study has been written to illustrate the development from early mathematical learning (grades 3-8) to secondary education regarding the Fundamental Theorem of Arithmetic and the Fundamental Theorem of Algebra. It investigates the progression of the mathematics presented to the students by the current curriculum adopted by the Rhea County School System and the mathematics academic standards set forth by the State of Tennessee.

Controllability And Nonsingular Solutions Of Sylvester Equations, 2016 Poly-disciplinary Faculty of Safi

#### Controllability And Nonsingular Solutions Of Sylvester Equations, Hamid Maarouf

*Electronic Journal of Linear Algebra*

The singularity problem of the solutions of some particular Sylvester equations is studied. As a consequence of this study, a good choice of a Sylvester equation which is associated to a linear continuous time system can be made such that its solution is nonsingular. This solution is then used to solve an eigenstructure assignment problem for this system. From a practical point view, this study can also be applied to automatic control when the system is subject to input constraints.

Teaching Functions: The Good, The Bad, And The Many Ways To Do Better, 2016 Murray State University

#### Teaching Functions: The Good, The Bad, And The Many Ways To Do Better, Melanie Bayens

*Honors College Theses*

The way functions are taught in school needs improvement. Many times when students are introduced to functions in Algebra 2, the definition is glossed over, the instruction is lacking, and deeper understanding of the concept is lost. This causes problems when students are required to use this knowledge of functions in later classes, particularly, in Precalculus and Calculus. First, this paper will give the definition of a function and its role in mathematics. Second, this paper will delve into the problems of teaching functions the standard way. Finally, it will present multiple alternative methods for teaching functions. Specifically, it will ...

Group Inverse Extensions Of Certain $M$-Matrix Properties, 2016 National Institute of Technology Warangal

#### Group Inverse Extensions Of Certain $M$-Matrix Properties, Appi Reddy K., Kurmayya T., K. C. Sivakumar

*Electronic Journal of Linear Algebra*

In this article, generalizations of certain $M$-matrix properties are proved for the group generalized inverse. The proofs use the notion of proper splittings of one type or the other. In deriving certain results, we make use of a recently introduced notion of a $B_{\#}$-splitting. Applications in obtaining comparison results for the spectral radii of matrices are presented.

On The Characterization And Parametrization Of Strong Linearizations Of Polynomial Matrices, 2016 Alexander Technological Education Institute of Thessaloniki

#### On The Characterization And Parametrization Of Strong Linearizations Of Polynomial Matrices, Efstathios Antoniou, Stavros Vologiannidis

*Electronic Journal of Linear Algebra*

In the present note, a new characterization of strong linearizations, corresponding to a given regular polynomial matrix, is presented. A linearization of a regular polynomial matrix is a matrix pencil which captures the finite spectral structure of the original matrix, while a strong linearization is one incorporating its structure at infinity along with the finite one. In this respect, linearizations serve as a tool for the study of spectral problems where polynomial matrices are involved. In view of their applications, many linearization techniques have been developed by several authors in the recent years. In this note, a unifying approach is ...

The Inverse Along An Element In Rings, 2016 Universidad Politecnica de Valencia

#### The Inverse Along An Element In Rings, Julio Benitez, Enrico Boasso

*Electronic Journal of Linear Algebra*

Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.

Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, 2016 University of Houston

#### Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta

*STAR (STEM Teacher and Researcher) Presentations*

With advances in technology in brighter sources and larger and faster detectors, the amount of data generated at national user facilities such as SLAC is increasing exponentially. Humans have a superb ability to recognize patterns in complex and noisy data and therefore, data is still curated and analyzed by humans. However, a human brain is unable to keep up with the accelerated pace of data generation, and as a consequence, the rate of new discoveries hasn't kept pace with the rate of data creation. Therefore, new procedures to quickly assess and analyze the data are needed. Machine learning approaches ...

Some 2-Categorical Aspects In Physics, 2016 The Graduate Center, City University of New York

#### Some 2-Categorical Aspects In Physics, Arthur Parzygnat

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description ...

On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, 2016 The Graduate Center, City University of New York

#### On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form ...

Generalizations Of The Cauchy And Fujiwara Bounds For Products Of Zeros Of A Polynomial, 2016 University of Guelph

#### Generalizations Of The Cauchy And Fujiwara Bounds For Products Of Zeros Of A Polynomial, Rajesh Pereira, Mohammad Ali Vali

*Electronic Journal of Linear Algebra*

The Cauchy bound is one of the best known upper bounds for the modulus of the zeros of a polynomial. The Fujiwara bound is another useful upper bound for the modulus of the zeros of a polynomial. In this paper, compound matrices are used to derive a generalization of both the Cauchy bound and the Fujiwara bound. This generalization yields upper bounds for the modulus of the product of $m$ zeros of the polynomial.

Cycle Structures Of Orthomorphisms Extending Partial Orthomorphisms Of Boolean Groups, 2016 Portland State University

#### Cycle Structures Of Orthomorphisms Extending Partial Orthomorphisms Of Boolean Groups, Nichole Louise Schimanski, John S. Caughman Iv

*Mathematics and Statistics Faculty Publications and Presentations*

A partial orthomorphism of a group GG (with additive notation) is an injection π:S→G for some S⊆G such that π(x)−x ≠ π(y) for all distinct x,y∈S. We refer to |S| as the size of π, and if S=G, then π is an orthomorphism. Despite receiving a fair amount of attention in the research literature, many basic questions remain concerning the number of orthomorphisms of a given group, and what cycle types these permutations have.

It is known that conjugation by automorphisms of G forms a group action on the set of orthomorphisms ...

A Survey Of Graphs Of Minimum Order With Given Automorphism Group, 2016 University of Texas at Tyler

#### A Survey Of Graphs Of Minimum Order With Given Automorphism Group, Jessica Alyse Woodruff

*Math Theses*

We survey vertex minimal graphs with prescribed automorphism group. Whenever possible, we also investigate the construction of such minimal graphs, confirm minimality, and prove a given graph has the correct automorphism group.

Homological Characterizations Of Quasi-Complete Intersections, 2016 University of Nebraska - Lincoln

#### Homological Characterizations Of Quasi-Complete Intersections, Jason M. Lutz

*Dissertations, Theses, and Student Research Papers in Mathematics*

Let R be a commutative ring, (**f**) an ideal of R, and E = K(**f**; R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T, the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.

Advisers: Luchezar L. Avramov and Srikanth B. Iyengar

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, 2016 Rose-Hulman Institute of Technology

#### Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

Two Riemann surfaces *S*_{1} and *S*_{2} with conformal *G*-actions have topologically equivalent actions if there is a homeomorphism *h :* *S _{1} -> S_{2} *which intertwines the actions. A weaker equivalence may be defined by comparing the representations of

*G*on the spaces of holomorphic

*q-*differentials

*H*and

^{q}(S_{1})*H*In this note we study the differences between topological equivalence and

^{q}(S_{2}).*H*equivalence of prime cyclic actions, where

^{q}*S*and

_{1}/G*S*have genus zero.

_{2}/GKemeny's Constant And An Analogue Of Braess' Paradox For Trees, 2016 University of Manitoba

#### Kemeny's Constant And An Analogue Of Braess' Paradox For Trees, Steve Kirkland, Ze Zeng

*Electronic Journal of Linear Algebra*

Given an irreducible stochastic matrix M, Kemeny’s constant K(M) measures the expected time for the corresponding Markov chain to transition from any given initial state to a randomly chosen final state. A combinatorially based expression for K(M) is provided in terms of the weights of certain directed forests in a directed graph associated with M, yielding a particularly simple expression in the special case that M is the transition matrix for a random walk on a tree. An analogue of Braess’ paradox is investigated, whereby inserting an edge into an undirected graph can increase the value of ...

Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, 2016 College of Saint Benedict/Saint John's University

#### Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

*Mathematics Faculty Publications*

Anderson and Harary introduced two impartial games on finite groups. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.

Richard Dedekind And The Creation Of An Ideal: Early Developments In Ring Theory, 2016 Colorado State University-Pueblo

#### Richard Dedekind And The Creation Of An Ideal: Early Developments In Ring Theory, Janet Heine Barnett

*Abstract Algebra*

No abstract provided.