Free Module Research Articles

Applications of Local Algebras of Differentiable Manifolds

This paper is devoted to some applications of local algebras in geometry. We recall some properties of free finite-dimensional modules over local algebras of a certain type, so-called A-spaces in the sense of Macdonald, where A denotes the algebra considered. Using these properties, we study bilinear, special symmetric, and symplectic forms on A-spaces and obtain some their invariants. Properties of these spaces are used in the study of projective Klingenberg spaces over the ring A. We present fundamental notions of points and subspaces of projective Klingenberg spaces. We examine the neighborship property of points and homologies. We obtain a criterion of projective equivalence of quadrics on these spaces.

Σ‐algebraically compact modules and ‐compact cardinals

We prove that the property characterizes Σ‐algebraically compact modules if is not ω‐measurable. Moreover, under a large cardinal assumption, we show that over any ring R where is not ω‐measurable, any free module M of ω‐measurable rank satisfies , hence the assumption on cannot be dropped in general (e.g., over small non‐right perfect rings). In this way, we extend results from a recent paper by Simion Breaz .

F-zips with additional structure

An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "$F$-zips with a $G$-structure", for an arbitrary reductive linear algebraic group $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F$-zips over $S$. Locally any such functor has a type $\chi$, which is a cocharacter of $G$. The other incarnation is a certain $G$-torsor analogue of the notion of $F$-zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form $[E_{G,\chi}\backslash G_k]$ that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over $k$. Using our earlier results we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others. For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary $F$-zips. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) and to truncated Barsotti-Tate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl-Oort stratification of the special fibers of arbitrary Shimura varieties.

An algorithm to compute a primary decomposition of modules in polynomial rings over the integers

We present an algorithm to compute the primary decomposition of a submodule N of the free module ℤ[x1,...,xn]m. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of N, i.e. the minimal associated primes of the ideal Ann (ℤ[x1,...,xn]m/N) in ℤ[x1,...,xn] and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in Singular.

Projective geometry in characteristic one and the epicyclic category

AbstractWe show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield ofmax-plus integersℤmax. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ℤmax. The associated projective spaces arefiniteand provide a mathematically consistent interpretation of Tits's original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

The Critical Choice of PEDOT:PSS Additives for Long Term Stability of Roll‐to‐Roll Processed OPVs

The impact of additives mixed with poly(3,4‐ethylenedioxythiophene):polysty­renesulfonate (PEDOT:PSS) on the stability of organic photovoltaic modules is investigated for fully ambient roll‐to‐roll (R2R) processed indium tin oxide free modules. Four different PEDOT:PSS inks from two different suppliers are used. The modules are manufactured directly on barrier foil without a UV filter to accelerate degradation and enable completion of the study in a reasonable time span. The modules are subjected to stability testing following well‐established protocols developed by the international summit on organic photovoltaic stability (ISOS). For the harsh indoor test (ISOS‐L‐3) only a slight difference in stability is observed between the different modules. During both ISOS‐L‐3 and ISOS‐D‐3 one new failure mode is observed as a result of tiny air inclusions in the barrier foil and a R2R method is developed to detect and quantify these. During outdoor operation (ISOS‐O‐1) the use of ethylene glycol (EG) as an additive is found to drastically increase the operational stability of the modules as compared to dimethylsulfoxide (DMSO) and a new failure mode specific to modules with DMSO as the additive is identified. The data are extended in an ongoing experiment where DMSO is used as additive for long‐term outdoor testing in a solar park.

Layout flexibility for sheet-to-sheet produced flexible ITO-free organic solar modules with organic functional layers slot die coated under ambient atmospheric conditions

Four different module layouts for indium tin oxide (ITO)-free flexible organic solar modules with device performance varying from 1.8% to 2.7% are reported in this work. Module 1, with a device efficiency of 2.5% on an active area of 88cm2, demonstrates the up scaling of small area cells to larger area modules. Module 2, with a device efficiency of 1.8% and 106.4V on an active area of 32.6cm2, demonstrates high voltage on a small area. Module 3, with a device efficiency of 2% on an active area of 60.8cm2 with a hexagonal layout, demonstrates the flexibility in module layout. Module 4 built as a reference module device on small active area of 6.1cm2 had a device efficiency of 2.7%. PEDOT:PSS together with a metal grid is the transparent hole contact replacing ITO. The reference modules were also built without the metal grid (only PEDOT:PSS has a hole contact) and measured under 2000 and 500lx resulting in the modules without grid outperforming the ones with grid. Therefore for indoor applications modules can be processed even without the top metal grid for current collection which not only reduces the material cost and the number of processing step but also eliminates the major concern of depositing metal electrodes on top of organic layers.

Finite domination and Novikov rings: Laurent polynomial rings in two variables

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗L R〚x, y〛[(xy)-1] and C ⊗L R[x, x-1]〚y〛[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

The spectrum of an Adelic Markov operator

With the help of the representation of SL(2,Z) on the rank two free module over the integer adeles, we define the transition operator of a Markov chain. The real component of its spectrum exhibits a gap, whereas the non-real component forms a circle of radius 1/\sqrt{2}.

Free divisors in a pencil of curves

A plane curve on a the projective space over a field of characteristic zero is free if its associated sheaf T of tangent vector fields tangent is a free module. Relatively few free curves are known. Here we prove that a divisor consisting of a union of curves of a pencil of plane projective curves with the same degree and with a smooth base locus is a free divisor if and only this union contains all the singular members of the pencil and its Jacobian ideal is locally a complete intersection.

A comparative study of the physical and elastic properties of new generation elastomeric ligatures with conventional elastomeric ligatures

Context: One of the most common methods of securing arch wires to orthodontic brackets is the application of elastomeric ligatures. New generation elastomeric ligatures were introduced to overcome the disadvantages of force decay and staining in conventional ligatures. Aims: The aim was to compare the physical and elastic properties of new generation elastomeric ligatures with the conventional elastomeric ligatures. Materials and Methods: Four groups of clear elastomeric modules (conventional polyurethane based modules, latex free elastomeric modules, alastik easy to tie modules, silver and silicone impregnated super ligatures) were tested for tensile properties, frictional resistance, and color stability in as received condition and following 1-week, 2 weeks, and 1-month of intra-oral exposure. A universal testing machine was used for testing tensile properties. The color stability was tested by measuring the color change using scanned images of the modules at different time periods. Statistical Analysis: Two-way ANOVA, post-hoc multiple comparisons Bonferroni-Dunn test and the paired t-test were used. Results: The alastik group showed the highest mean tensile strength and super ligatures showed the lowest tensile strength values. There was a statistically significant decrease in the mean tensile strength in all groups following intra-oral exposure (P Conclusions: There are significant differences in physical and elastic properties of different brands of ligatures, which should be considered during the selection of these products.

A Hopf Algebra Freeness Result Revisited

The affine (Hopf) algebra of the special linear group SL(2ℓ, k) for ℓ ≥1 is not a free module over one of its Hopf subalgebras. In this paper we give a very simple proof of this fact over a field k of characteristic zero. The proof boils down to showing that a certain module is not cyclic.

The action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of Molien functions

The present article completes the mathematical description initiated in the paper by Dhont and Zhilinskií (2013 The action of the orthogonal group on planar vectors: invariants, covariants and syzygies J. Phys. A: Math. Theor. 46 455202) of the algebraic structures that emerge from the symmetry-adapted polynomials in the coordinates of n planar vectors under the action of the SO(2) group. The set of -covariant polynomials contains all the polynomials that transform according to the weight of SO(2) and is a free module for but a non-free module for . The sum of the rational functions of the Molien function for -covariants describes the decomposition of the ring of invariants or the module of -covariants as a direct sum of submodules. A method for extracting the generating function for -covariants from the comprehensive generating function for all polynomials is introduced. The approach allows the direct construction of the integrity basis for the module of -covariants decomposed as a direct sum of submodules and gives insight into the expressions for the Molien functions found in our earlier paper. In particular, a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed, where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed.

Loop Heisenberg-Virasoro Lie conformal algebra

Let HV be the loop Heisenberg-Virasoro Lie algebra over ℂ with basis {Lα,i, Hβ,j∣α, β, i, j ∈ ℤ} and brackets [Lα,i, Lβ,j] = (α − β) Lα+β,i+j, [Lα,i, Hβ,j] = − βHα+β,i+j, [Hα,i, Hβ,j] = 0. In this paper, a formal distribution Lie algebra of HV is constructed. Then, the associated conformal algebra CHV is studied, where CHV has a ℂ[∂]-basis {Li, Hi∣i ∈ ℤ} with λ-brackets [Li λ Lj] = (∂ + 2λ) Li+j, [Li λ Hj] = (∂ + λ) Hi+j, [Hi λ Lj] = λLi+j, and [Hi λ Hj] = 0. In particular, the conformal derivations of CHV are determined. Finally, rank one conformal modules and ℤ-graded free intermediate series modules over CHV are classified.

Two-quadratic modules and cofibration category

Abstract In this paper, totally free 2-quadratic modules are defined. It is shown that the category of totally free 2-quadratic complexes is a cofibration category in the sense of Baues.

Decomposing modular coinvariants

We consider the ring of coinvariants for a modular representation of a cyclic group of prime order p. We show that the classes of the terminal variables in the coinvariants have nilpotency degree p and that the coinvariants are a free module over the subalgebra generated by these classes. An incidental result we have is a description of a Gröbner basis for the Hilbert ideal and a decomposition of the corresponding monomial basis for the coinvariants with respect to the monomials in the terminal variables.

Flexible broadcasting and multicasting in 4λ × 10 Gb/s AOPR TWDM PON system

In this letter, we propose a new architecture of Time Wavelength Division Multiplexing Passive Optical Network (TWDM PON) system to support dynamic multi wavelength allocation (DMWA) in both upstream and downstream directions using an integrated semiconductor optical amplifier (SOA) and arrayed waveguide grating (AWG) with multi wavelength select continuous wave (CW) pump probe signal module. The significance of this architecture is the flexible routing function with the capability of multicasting and broadcasting between multiple optical line terminal (OLT) PON port with multiple optical distribution network (ODN) link using a new wavelength tuning free (WTF) OLT transmitter module to eliminate wavelength tuning delay in downstream signal utilizing multicasting Cross Gain Modulation (XGM) wavelength conversion. The experimental results show that 4λ×10-Gb/s TWDM PON system can be used to connect 4096 users with the conventional fixed wavelength OLT transceivers with 36dB link loss.

Bolstering the quality and integrity of online collaborative university- level courses via an open Sim standalone server in conjunction with sloodle

The contemporary era provides several challenges which extend from the reconstitution of an innovative knowledge domain and curricula to candidate learning platforms that support online course delivery methods. Educators and scholars on these demands have recently started to rethink alternative ways for the assimilation of the experiential knowledge in three-dimensional (3D) technologically advanced environments, like 3D multi-user virtual worlds. In spite the widespread dissemination and proliferation of novel educational implications by utilizing 3D multi-user virtual worlds combined with the 2D interface of LMS (Learning Management Systems) and the assessment of the effectiveness based on the online course delivery method in a long-term usability is still absent from the international academic literature. This study presents interoperability issues focused on the utilization of the virtual world Open Simulator (Open Sim) in conjunction with Sloodle (Simulation Linked Object Oriented Dynamic Learning Environment) as a free plug-in module. The evaluation of this online learning process according to an empirical research method became possible with the assistance of ninety-five (95) students by two different academic sectors that participated in this project and measured the capabilities and instructional affordances of this platform. The evaluation process was focused on four multi-dimensional parameters (psychological-pedagogical, technical-operational, organizational-financial, and socio-cultural). The study findings based on students’ experiences revealed that both platforms could be sufficiently connected as a unique platform which can increase the users’ learning abilities. This “hybrid” platform can adequately convert the ordinary multi-user virtual world of Open Sim and Sloodle in a common “incubator” of knowledge for online courses at university level. At the end, the instructional affordances and implications for future-driven directions are also discussed.

Simple-root bases for Shi arrangements

In his affirmative answer to the Edelman–Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, we know very little about the bases themselves except their existence. In this article, we prove the unique existence of two distinguished bases which we call the simple-root basis plus (SRB+) and the simple-root basis minus (SRB−). They are characterized by nice divisibility properties relative to the simple roots.

On special covariants in the exterior algebra of a simple Lie algebra

We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n -th symmetric power of the defining representation. As main result we prove that this subspace is a free module over the subalgebra of the exterior algebra generated by all primitive invariants except the one of highest degree.