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We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.

The primary goals of this study are to determine if the datasets of positive COVID-19 test cases and CO2 emissions from Connecticut over the span of March 24th, 2020-October 31, 2021 are in any ways correlated. With climate change a prominent issue facing the entire world today, it is important to explore methods of providing records of past patterns of greenhouse gas emissions in order to inform decision making that could reduce future ones. Autoregressive integrated moving average (ARIMA) modeling is also implemented in this paper to provide forecasting based on CO2 emissions in CT starting from 2019. The most significant results from this paper are as follows: the CO2 emission data of transportation sectors including ground transportation, domestics aviation, and international aviation and weekly COVID-19 positive test cases data has a strong relationship during the first 28 weeks of the pandemic with a correlation of -86.34%. The CO2 emissions experienced on average a -22.96% change of pre-pandemic vs during initial quarantine conditions and at most a - 44.48% change when comparing the pre-pandemic mean to the during initial quarantine minimum value. Lastly, the ARIMA model found to have the lowest Akaike information criterion (AIC) was ARIMA (4,0,4). In conclusion, in the event of a collective global pandemic and lockdown conditions, less traveling resulting in a correlated decrease of CO2 emissions. This means that perhaps concentrated efforts on reducing unnecessary travel could help mitigate the levels of carbon dioxide emissions as a more long-term solution to climate change opposed to the pandemic’s short-term example.

Given commutative, unital rings $$\mathcal {A}$$and $$\mathcal {B}$$with a ring homomorphism $$\mathcal {A}\rightarrow \mathcal {B}$$making $$\mathcal {B}$$free of finite rank as an $$\mathcal {A}$$-module, we can ask for a “trace” or “norm” homomorphism taking algebraic data over $$\mathcal {B}$$to algebraic data over $$\mathcal {A}$$. In this paper we we construct a norm functor for the data of a quadratic algebra: given a locally-free rank-2 $$\mathcal {B}$$-algebra $$\mathcal {D}$$, we produce a locally-free rank-2 $$\mathcal {A}$$-algebra $$\textrm{Nm}_{\mathcal {B}/\mathcal {A}}(\mathcal {D})$$in a way that is compatible with other norm functors and which extends a known construction for étale quadratic algebras. We also conjecture a relationship between discriminant algebras and this new norm functor.

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $$\omega$$. Attached to each such ring R, when $$\omega$$is an ideal, there are integers–the type of R, the reduction number of $$\omega$$–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where $$\omega$$is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.

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.