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Let Mφ be a surface bundle over a circle with monodromy φ: S → S. We study deformations of certain reducible representations of π1(Mφ)intoSL(n, C), obtained by composing a reducible representation into SL(2, C) with the irreducible representation SL(2, C) → SL(n, C). In particular, we show that under certain conditions on the eigenvalues of φ∗, the reducible representation is contained in a (n + 1 + k)(n − 1) dimensional component of the representation variety, where k is the number of components of ∂Mφ . This result applies to mapping tori of pseudo-Anosov maps with orientable invariant foliations whenever 1 is not an eigenvalue of the induced map on homology, where the reducible representation is also a limit of irreducible representations. © 2022, Osaka University. All rights reserved.

This paper characterizes quasi-pure projective (q.p.p.) and quasi-pure injective (q.p.i.) p-groups, and hence characterizes all such (abelian) torsion groups. A p-group is q.p.i. if and only if it is the direct sum of a divisible group and a torsion complete group. A nonreduced p-group is q.p.p. if and only if it is the direct sum of a divisible group and a bounded group; a reduced p-group is q.p.p. if and only if it is a direct sum of cyclic groups. © 1977 American Mathematical Society.

H-spaces are examined by studying left translations, actions and a homotopy version of left translations to be called homolations. If (F, m) is an H-space, the map s: F-→FF given by s(x) = Lx, i.e. s(x) is left translation by x, is a homomorphism if and only if m is associative. In general, s is an An-map if and only if (F, m) is an An+1 space. The action r: FF × F → F is given by r(φ, x) = φ(x). The map s respects the action only of left translations. In general, s respects the action of homolations up to higherorder homotopies. Each homolation generates a family of maps to be called a homolation family. Denoting the set of all homolation families by H∞(F), s: F -→ FF factors through F → H∞(F) and this latter map is a homotopy equivalence. © 1971 Pacific Journal of Mathematics.

Let q be a prime number. The number of subgroups of order qk in an abelian group G of order qn and type λ is a polynomial in q, [kλ′]q. In 1987, Lynne Butler showed that the first difference, [kλ′] - [k - 1λ′], has nonnegative coefficients as a polynomial in q, when 2k ≤ |λ|. We generalize the first difference to the rth difference, and give conditions for the nonnegativity of its coefficients. © 1995.

Let X and Y be varieties over a field k; π : X → Y is a quadruple cover of Y if π*script O signX is a locally free, rank 4 script O signY-algebra. If char k ≠ 2, we see that π*script O signX splits as script O signY ⊕ε where ε is a locally free rank 3 sheaf over script O signY, which is locally the "trace zero" module. For each y ∈ Y, we therefore have a rank 4 associative, commutative algebra over script O signY,y. We find that these algebras are parametrized by an affine cone over the Grassmanian G(2, 6) with vertex corresponding to the algebra k[x, y, z]/(x, y, z)2. We then show that a quadruple cover with trace zero module ε over a variety Y is determined by a totally decomposable section η ∈ H0(∧2S2 E* ⊗ ∧3 E). We then examine the case in which the section η has no zeros. Here, each rank 4 algebra may be associated to a pencil of conics. As a special case of this, we look at the work of G. Casnati and T. Ekedahl on Gorenstein covers, and we show that their analysis is the subcase where the pencil of conics has length 4 base locus. Finally, we study the case in which the trace zero module ε is split. In this context, Galois covers, which are covers induced by the action of a group of order 4 on the covering variety X, are also studied.

We prove two art gallery theorems in which the guards must guard one another in addition to the gallery. A set G of points (the guards) in a simple closed polygon (the art gallery) is a guarded guard set provided (i) every point in the polygon is visible to some point in G; and (ii) every point in G is visible to some other point in G. We prove that a polygon with n sides always has a guarded guard set of cardinality ⌊(3n-1)/7⌋ and that this bound is sharp (n5); our result corrects an erroneous formula in the literature. We also use a coloring argument to give an entirely new proof that the corresponding sharp function for orthogonal polygons is ⌊n/3⌋ for n≤6; this result was originally established by induction by Hernández-Peñalver. © 2003 Elsevier B.V.

The *-polynomial identities of minimal degree of Mn(F) are determined for n = 2, 4, * the symplectic involution. Copyright © 2004 by Marcel Dekker, Inc.

Contractedness of m-primary integrally closed ideals played a central role in the development of Zariski's theory of integrally closed ideals in two-dimensional regular local rings (R, m). In such rings, the contracted m-primary ideals are known to be characterized by the property that I: m = I: x for some x ∈ m\m2. We call the ideals with this property full ideals and compare this class of ideals with the classes of m-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The m-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be m-full. We also show the equivalence of the properties full, m-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full. © Taylor & Francis Group, LLC.

When solving an algorithmic problem involving a polyhedron in R 3, it is common to start by partitioning the given polyhedron into simplier ones. The most common process is called triangulation and it refers to partitioning a polyhedron into tetrahedra in a face-to-face manner. In this paper instead of triangulations we will consider tilings by tetrahedra. In a tiling the tetrahedra are not required to be attached to each other along common faces. We will construct several polyhedra which can not be triangulated but can be tiled by tetrahedra. We will also revisit a nontriangulatable polyhedron of Rambau and a give a new proof for his theorem. Finally we will identify new families of non-tilable, and thus non-triangulable polyhedra.

Introduction: Although there is some common pedagogical ground for all teachers of mathematics, there is a fundamental difference in the topics and depth of content knowledge required for students preparing to teach elementary, middle, or high school mathematics. The middle school teacher preparation program at Western Oregon University (WOU) seeks to develop foundational content for middle school teachers while exploring best practices such as active learning, appropriate use of technology, and hands-on exploration. WOU offers many courses specifically for middle school teachers that are designed to develop mathematical maturity and content knowledge while connecting the subject matter to the middle school curriculum and standards. This article describes the structure of WOU's middle school mathematics program and the courses designed specifically for middle school mathematics teachers. We point out the difference in the mathematical preparation and requirements for middle school mathematics teachers compared to elementary teachers and high school mathematics teachers and explain the licensure requirements for middle school mathematics teachers in Oregon. Background and Philosophy of the Program: Western Oregon University's math curriculum for K–8 teacher preparation was among ten programs singled out as meeting critical coursework needs by the National Council on Teacher Quality [5]. The teacher preparation program as a whole was named the 2010 recipient of the Christa McAuliffe Award for Excellence in Teacher Education by the American Association of State Colleges and Universities (AASCU). © 2013 by The Mathematical Association of America (Incorporated).

At Western Oregon University, we offer a College Algebra for Teachers course using the Visual Algebra for College Students [3] materials that we wrote and class tested. Visual Algebra is designed to help middle school teachers gain a deep understanding of basic algebraic skills. Students visually show algebra using a concrete model (algebra pieces), verbally describe the meaning of each algebra piece move, symbolically connect the ideas to standard algebraic algorithms and procedures, and graphically connect the ideas of the visual models and symbolic work. Overall, students think deeply about topics, do not rely on rote memorization or rote rules, and understand ideas so well that they can easily describe, model, and teach core algebraic ideas to their middle grade students in a variety of ways. This allows our future teachers to meet the needs of the different learning styles in their classrooms. Visual Algebra takes students through modeling integer operations with black and red tiles to modeling linear and quadratic patterns with tiles and variable algebra pieces, looking at the general forms of the patterns and then connecting all the ideas to symbolic manipulation, creating data sets, graphing, and finding intercepts and points of intersection. Ideas are then extended to higher order polynomial functions and then to modeling complex number operations with black, red, yellow, and green tiles. Throughout the course, students are able to relate what they learned using visual methods to the standard methods and algorithms they will see every day in their classroom. For example, after factoring quadratic equations using visual methods, students learn factoring by grouping and factoring using the “ac” method. © 2013 by The Mathematical Association of America (Incorporated).

We study almost complete intersection ideals whose Rees algebras are extremal in the sense that some of their fundamental metrics depth or relation type have maximal or minimal values in the class. The focus is on those ideals that lead to almost Cohen-Macaulay algebras, and our treatment is wholly concentrated on the nonlinear relations of the algebras. Several classes of such algebras are presented, some of a combinatorial origin. We offer a different prism to look at these questions with accompanying techniques. The main results are effective methods to calculate the invariants of these algebras. © 2013 Rocky Mountain Mathematics Consortium.

If you are ever faced with an oncoming truck, this paper could save your life. We investigate the optimal path that you should take from the middle of the road to the curb in order to avoid being hit by an oncoming truck. Although your instincts may tell you to run directly toward the curb, it turns out that this path, although the shortest, is not generally the safest.

The art gallery theorem asserts that any polygon with n vertices can be protected by at most [n/3] stationary guards. The original proof by Chvátal uses a nonroutine and nonintuitive induction. We give a simple inductive proof of a new, more general result, the constrained art gallery theorem: If V∗and E∗are specified sets of vertices and edges that must contain guards, then the polygon can be protected by at most [(n + 2|V∗| + |E∗|) /3] guards. Our result reduces to Chvátal's art gallery theorem when V∗and E∗are empty. We give a second short proof of this generalization in the spirit of Fisk's proof of the art gallery theorem using graph colorings. © THE MATHEMATICAL ASSOCIATION OF AMERICA.

OBJECTIVE To characterize and compare injuries found in dogs involved in spontaneously occurring dogfights with those of dogs used in illegal organized dogfighting. DESIGN Retrospective case-control study. ANIMALS 36 medium-sized dogs evaluated following spontaneous fights with a dog of the same sex and similar weight (medium dog–medium dog [MDMD] fights), 160 small dogs examined following spontaneous fights with a larger dog (big dog–little dog [BDLD] fights), and 62 dogs evaluated after being seized in connection with dogfighting law enforcement raids. PROCEDURES Demographic characteristics and injuries were recorded from medical records. Prevalence of soft tissue injuries in predetermined body surface zones, as well as dental or skeletal injuries, was determined for dogs grouped by involvement in BDLD, MDMD, and organized dogfights. The extent of injuries in each location was scored and compared among groups by 1-factor ANOVA. Patterns of injuries commonly incurred by each group were determined by use of prevalence data. RESULTS Mean extent of injury scores differed significantly among groups for all body surface zones except the eye and periorbital region. Mean scores for dental injuries and rib fractures also differed significantly among groups. Organized fighting dogs more commonly had multiple injuries, particularly of the thoracic limbs, dorsal and lateral aspects of the head and muzzle or oral mucosa, dorsal and lateral aspects of the neck, and ventral neck and thoracic region. CONCLUSIONS AND CLINICAL RELEVANCE To the authors’ knowledge, this was the first study to compare injuries incurred during spontaneous and organized dogfighting. Establishing evidence-based patterns of injury will help clinicians identify dogs injured by organized dogfighting and aid in the prosecution of this crime. © 2017, American Veterinary Medical Association. All rights reserved.

Let D be any of the 10 digraphs obtained by orienting the edges of K4 - e. We establish necessary and sufficient conditions for the existence of a (K∗ n,D)-design for 8 of these digraphs. Partial results as well as some nonexistence results are established for the remaining 2 digraphs. © 2019 Ryan C. Bunge et al., published by Sciendo 2019.