Free Module Research Articles

Integrating Million Hearts into nursing and interprofessional educational curricula and community settings: a key strategy for improving population health across the United States.

Million Hearts is a national initiative to prevent 1 million heart attacks and strokes by 2017 by screening and educating the public on the "ABCS" of cardiovascular health. Million Hearts is an innovative platform for educating nursing and health sciences students on the importance of population health and interprofessional teamwork. The National Interprofessional Education and Practice Consortium to Advance Million Hearts was created, and a free on-line educational module was developed to help health care professionals and health sciences faculty and students learn about the Million Hearts initiative, conduct community screenings, and refer people who screen positive to appropriate resources. After completion of the module, individuals receive certification as a Million Hearts Fellow. More than 2,500 individuals from 80 colleges across the United States have accessed the module. More than 20,000 people have been screened. The module and screenings have been incorporated into health sciences curricula and community activities. Academic institutions and health science professions partnering together as part of the National Interprofessional Education and Practice Consortium to Advance Million Hearts provide a unique opportunity to demonstrate the impact that a unified approach can have on improving population health through the use of screening, education, and prevention.

On the Zariski-Lipman Conjecture for normal algebraic surfaces

We consider the Zariski–Lipman Conjecture on a free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several classes of affine and projective surfaces.

The direct limit closure of perfect complexes

Every projective module is flat. Conversely, every flat module is a direct limit of finitely generated free modules; this was proved independently by Govorov and Lazard in the 1960s. In this paper we prove an analogous result for complexes of modules, and as applications we reprove some results due to Enochs and García Rozas and to Neeman.

Fixed points and homology of superelliptic Jacobians

Let $$\eta : C_{f,N}\rightarrow \mathbb {P}^1$$ be a cyclic cover of $$\mathbb {P}^1$$ of degree $$N$$ which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic covering group $${{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}$$ acting on the Jacobian $$J_N:={{\mathrm{Jac}}}(C_{f,N})$$ . For each prime $$\ell $$ distinct from the characteristic of the base field, the Tate module $$T_\ell J_N$$ is shown to be a free module over the ring $$\mathbb {Z}_\ell [T]/(\sum _{i=0}^{N-1}T^i)$$ . We also study the subvarieties of $$J_N$$ and calculate the degree of the induced polarization on the new part $$J_N^\mathrm {new}$$ of the Jacobian.

Supersymmetric analogues of the classical theorem on harmonic polynomials

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of the Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. In this paper, we first establish two-parameter ℤ2-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra gl(n | m). Then we extend the result to two-parameter ℤ-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras osp(2n | 2m) and osp(2n + 1 | 2m).

Twisted Witt Groups of Flag Varieties

AbstractCalmès and Fasel have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, we sharpen their result to an if-and-only-if statement. In particular, we show that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, we find that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading.Our proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their K-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.

Outdoor Operational Stability of Indium‐Free Flexible Polymer Solar Modules Over 1 Year Studied in India, Holland, and Denmark

ABSTRACTWe present an outdoor interlaboratory stability study of fully printed and coated indium‐tin‐oxide (ITO)‐free polymer solar cell modules in JNCASR Bangalore (India), ECN (Holland), and DTU (Denmark) carried over more than 1 year. The modules comprising a fully printed and coated stack (Ag grid/PEDOT:PSS/ZnO/P3HT:PCBM/PEDOT:PSS/Ag grid) were prepared in two successive generations and evaluated for outdoor operational stability according to the test protocols laid out by the International Summit on OPV stability (ISOS‐3). The modules (70–100 cm2 active area) were encapsulated between two sheets of low‐cost plastic barrier material with the use of a UV curing adhesive. The impact of differences in the climatic conditions on the performance of the modules is highlighted and the performance of the modules under storage conditions in parallel with the outdoor study is investigated. While all Gen‐I modules failed, the best devices of Gen‐II module in which simple improvement in the encapsulation scheme (Gen‐II modules) was carried out maintained 95% of the initial performance after 1 year of outdoor testing. We provide detailed insight into the failure mode and offer a discussion on the need for improvement in flexible encapsulation. Finally, recommendations on future encapsulation schemes are also presented.

On growth in minimal totally acyclic complexes

Given a commutative Noetherian local ring, we provide a criterion under which a minimal totally acyclic complex of free modules has symmetric growth. As a special case, we show that, whenever an image in the complex has finite complete intersection dimension, then the complex has symmetric polynomial growth.

Recent progress in an algebraic analysis approach to linear systems

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.

Relative determinant of a bilinear module

This note proposes a new definition of the determinant of a symmetric bilinear form on an finitely generated projective module (over a domain) which is not necessarily free. The notion of the determinant is fundamental to (linear) algebra. Its most basic variant is the determinant of matrices and endomorphisms of vector spaces. This generalizes naturally to the determinant of endomorphisms on free modules, since free modules (like vector spaces) have bases and associated coordinate systems. In 1960s, Goldman showed that it is possible to generalize the notion of the determinant to endomorphisms of projective modules, that in general lack coordinates. Another standard application of the determinant is in the theory of bilinear forms over fields. It is well known that the determinant of a bilinear form is non-zero if and only if the form is non-degenerate. This can be even taken as a definition of non-degeneracy over fields. Next fundamental property is that the determinant factors over orthogonal sums. Like in the case of endomorphisms, the notion of the determinant generalizes naturally to bilinear forms on free modules (see e.g. [Mar85]). However, to the best of our knowledge, the notion has not been generalized to forms on arbitrary projective modules. Some efforts in this direction may be found in [Pet08], but under quite a strong assumption, that the determinant bundle of the module in question is free. The notion of the determinant in [Pet08] is also implicitly relative to an isomorphism from the determinant bundle to the base ring. Such a relativity with respect to some reference object seems to be intrinsic to the notion of the determinant of forms on projective modules and is present also in our approach. In this paper, we propose another definition of the determinant of a bilinear form on a finitely generated projective module. Our strategy consists of three steps. First, we define a relative determinant of one form with respect to another (necessarily non-degenerate) “reference form” on the same module. In the second step, we show that, for non-degenerate forms, instead of having separate “reference forms” on each module, it suffices to have “reference isomorphism” only for line bundles (with rank ≤ 2 in the Picard group of R). In addition, if R is a domain, then all one needs is one global “reference object” in form of a certain semi-group homomorphism. Finally, in the last step, using the notion of the relative determinant, we extend the definition of the determinant to all forms, including degenerate ones. We also show that with such a definition, the determinant still has the two basic properties: the form is non-degenerate if and only if its determinant is invertible (see Corollary 16) and the determinant factors over orthogonal sums (see Proposition 17). The notation utilized throughout this note is conventional. For definitions of used terms, we refer the reader to standard textbooks like e.g. [Mar85, MH73] (for the theory of bilinear forms) and [Wei13] (for the terms from K-theory). In particular, all rings here are always commutative, associative and has 1. If M 2010 Mathematics Subject Classification. 11E39, 15A63, 13C10.

Loop Virasoro Lie conformal algebra

The Lie conformal algebra of loop Virasoro algebra, denoted by \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW, is introduced in this paper. Explicitly, \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW is a Lie conformal algebra with \documentclass[12pt]{minimal}\begin{document}$\mathbb {C}[\partial ]$\end{document}C[∂]-basis \documentclass[12pt]{minimal}\begin{document}$\lbrace L_i\,|\,i\in \mathbb {Z}\rbrace$\end{document}{Li|i∈Z} and λ-brackets [Li λ Lj] = (−∂−2λ)Li+j. Then conformal derivations of \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW are determined. Finally, rank one conformal modules and \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}$\end{document}Z-graded free intermediate series modules over \documentclass[12pt]{minimal}\begin{document}$\mathscr {CW}$\end{document}CW are classified.

Stably free cancellation for abelian group rings

We show that if Γ is a finitely generated abelian group, then every stably free module over Z[Γ] is free.

Self-Dual Normal Basis of a Galois Ring

LetR′=GR(ps,psml)andR=GR(ps,psm)be two Galois rings. In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis forR′overR, whereR′is considered as a free module overR. Moreover, we explain a way to construct self-dual normal basis using particular system of polynomials. Finally, we show the connection between self-dual normal basis forR′overRand the set of all invertible, circulant, and orthogonal matrices overR.

Base modules for parametrized iterativity

The concept of a base, that is a parametrized finitary monad, which we introduced earlier, followed the footsteps of Tarmo Uustalu in his attempt to formalize parametrized recursion. We proved that for every base free iterative algebras exist, and we called the corresponding monad the rational monad of the base. Here we introduce modules for a base, and we prove that the rational monad of a base gives rise to a canonical module, that is characterized as the free iterative module on the given base.This generalizes the classical, nonparametric case of iterative Σ-algebras whose rational monad is the monad of rational Σ-trees and that was characterized by Calvin Elgot et al. as the free iterative monad on Σ. A basic parametrized example is the base assigning to every parameter set X the monad A↦X⁎×A whose rational monad is the monad of all right-wellfounded rational binary trees; the rational module for this base is the natural transformation (X⁎×X)×A→X⁎×A given by parametrized concatenation.

Relation modules of infinite groups, II

Abstract Let F n denote the free group of rank n and d(G) the minimal number of generators of the finitely generated group G. Suppose that R ↪ F m ↠ G and S ↪ F m ↠ G are presentations of G and let $$\bar R$$ and $$\bar S$$ denote the associated relation modules of G. It is well known that $$\bar R \oplus (\mathbb{Z}G)^{d(G)} \cong \bar S \oplus (\mathbb{Z}G)^{d(G)}$$ even though it is quite possible that . However, to the best of the author’s knowledge no examples have appeared in the literature with the property that . Our purpose here is to exhibit, for each integer k ≥ 1, a group G that has presentations as above such that . Our approach depends on the existence of nonfree stably free modules over certain commutative rings and, in particular, on the existence of certain Hurwitz-Radon systems of matrices with integer entries discovered by Geramita and Pullman. This approach was motivated by results of Adams concerning the number of orthonormal (continuous) vector fields on spheres.

Relative parametrization of linear multidimensional systems

In the last chapter of his book “The Algebraic Theory of Modular Systems” published in 1916, F. S. Macaulay developped specific techniques for dealing with “unmixed polynomial ideals” by introducing what he called “inverse systems”. The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential or partial differential equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients. The first and main idea is to replace unmixed polynomial ideals by pure differential modules. The second idea is to notice that a module is \(0\)-pure if and only if it is torsion-free and thus if and only if it admits an “absolute parametrization” by means of arbitrary potential like functions, or, equivalently, if it can be embedded into a free module by means of an “absolute localization”. The third idea is to refer to a difficult theorem of algebraic analysis saying that an \(r\)-pure module can be embedded into a module of projective dimension \(r\), that is a module admitting a projective resolution with exactly \(r\) operators. The fourth and final idea is to establish a link between the use of extension modules for such a purpose and specific formal properties of the underlying multidimensional system through the use of “involution” and a ‘relative localization” leading to a “relative parametrization”. The paper is written in a rather effective self-contained way and we provide many explicit examples that should become test examples for a future use of computer algebra.

The action of the orthogonal group on planar vectors: invariants, covariants and syzygies

The construction of invariant and covariant polynomials from the x, y components of n planar vectors under the SO(2) and O(2) orthogonal groups is addressed. Molien functions determined under the SO(2) symmetry group are used as a guide to propose integrity bases for the algebra of invariants and the modules of covariants. The Molien functions that describe the structure of the algebra of invariants and the free modules of (m)-covariants, m ⩽ n − 1, are written as a ratio of a numerator in λ with positive coefficients over a (1 − λ2)2n − 1 denominator. This form of single rational function is standard in invariant theory and has a clear symbolic interpretation. However, its usefulness is lost for the non-free modules of (m)-covariants, m ⩾ n, due to negative coefficients in the numerator. We propose for these non-free modules a new representation of the Molien function as a sum of n rational functions with positive coefficients in the numerators and different numbers of terms in the denominators. This non-standard form is symbolically interpreted in terms of a generalized integrity basis. Integrity bases are explicitly given for n = 2, 3, 4 planar vectors and m ranging from 0 to 5. The integrity bases obtained under the SO(2) symmetry group are subsequently extended to the O(2) group.

Tensor complexes: multilinear free resolutions constructed from higher tensors

The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1 -tensor), and the Eagon–Northcott and Buchsbaum–Rim complexes, which are constructed from a matrix (i.e., a 2 -tensor). The subject of this paper is a multilinear analogue of these complexes, which we construct from an arbitrary higher tensor. Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on a wide variety of complexes including: the Eagon–Northcott, Buchsbaum–Rim and similar complexes, the Eisenbud–Schreyer pure resolutions, and the complexes used by Gelfand–Kapranov–Zelevinsky and Weyman to compute hyperdeterminants. In addition, we provide applications to the study of pure resolutions and Boij–Söderberg theory, including the construction of infinitely many new families of pure resolutions, and the first explicit description of the differentials of the Eisenbud–Schreyer pure resolutions.

Annihilators of Artinian modules compatible with a Frobenius map

In this paper we consider Artinian modules over power series rings endowed with a Frobenius map. We describe a method for finding the set of all prime annihilators of submodules which are preserved by the given Frobenius map and on which the Frobenius map is not nilpotent. This extends the algorithm by Karl Schwede and the first author, which solved this problem for submodules of the injective hull of the residue field.The Matlis dual of this problem asks for the radical annihilators of quotients of free modules by submodules preserved by a given Frobenius near-splitting, and the same method solves this dual problem in the F-finite case.

Tobacco control education in pediatric anesthesiology fellowships

Cigarette smoking and secondhand smoke exposure (SHS) increase the risk of perioperative complications. Traditionally, anesthesiologists have limited involvement in tobacco control. To develop and disseminate an educational curriculum that educates pediatric anesthesia fellows in tobacco control. After IRB approval, an online survey was disseminated to pediatric anesthesiology fellowship directors. Thirty-one surveys were completed. Most report that they ask pediatric patients about tobacco use. A majority advise their patients who smoke about the health effects of smoking, but only 40% advise children to quit, and the majority never provide educational materials to assist in smoking cessation. Half reported that they sometimes or always ask about SHS. Approximately one-third never advise about the ill effects of SHS, nearly half never advise parents to stop smoking, and the majority never provide educational material about quitting to parents. Two-thirds felt that it is their responsibility to advise pediatric patients not to smoke, but less than half felt the same sense of responsibility about advising parents not to smoke. Approximately two-thirds believe that fellowship programs should provide education about the effects of smoking in the perioperative period and the effects of SHS exposure, but few programs do. Almost all would implement a free teaching module about SHS exposure and tobacco control as part of fellowship education. Many pediatric anesthesiology fellowship directors agree that exposure to cigarette smoke adversely impacts patients in the perioperative period, but few participate in tobacco control, and issues germane to tobacco control are not consistently addressed.