Geometry of Periodic Modules over Tame Concealed and Tubular Algebras

Let A be a tame concealed or tubular algebra and d the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety mod _A (d) of all A-modules of dimension-vector d is a normal complete intersection. Moreover, we show that a module M in mod _A (d) is nonsingular if and only if Ext _A ²(M,M)=0.     

On the normalizers of the unitary subgroup in an integral group ring

In this paper, we investigate the properties of the normalizers of the unitary subgroup u_f(ZG) in an integral group rings. One of our main results is Theorem 2.6 which character¬izes the second normalizer of the unitary subgroup. As a conse¬quence of this theorem, we prove that the second normalizer of u_f(ZG) coincides with the first normalizer when G is a periodic group. Among other results, we give necessary and sufficient conditions for which the unitary subgroup is normal in the unit group when G is periodic and also characterize when all bicyclic units are nontrivial and elements of the normalizer of the unitary subgroup.     

A partial order in the knot table II

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.     

Gauge-periodic point perturbations on the Lobachevsky plane

We study periodic point perturbations of the Shrödinger operator with a uniform magnetic field on the Lobachevsky plane. We prove that the spectrum gaps of the perturbed operator are labeled by the elements of the K₀ group of aC ^* algebra associated with the operator. In particular, if theC ^* algebra has the Kadison property, then the operator spectrum has a band structure.     

The Conley index, cup-length and bifurcation

A module structure of the cohomology Conley index is used to define a relative cup-length. This invariant is applied then to prove a multiplicity theorem for periodic solutions to Hamiltonian systems.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

相对遗传模

本文首先利用正向极限的相对内射性和直和的相对内射性刻划了Noether环。然后,给出了弱相对遗传模、弱相对余遗传模和相对遗传模的定义,且得到了一些性质,特别讨论了自遗传模的自同态环。     

On M-coidempotent elements and fully coidempotent modules

Abstract

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.     

The periodic groups saturated by finitely many finite simple groups

Denote by $\mathfrak{M}$ the set whose elements are the simple 3-dimensional unitary groups U ₃(q) and the linear groups L ₃(q) over finite fields. We prove that every periodic group, saturated by the groups of a finite subset of $\mathfrak{M}$ , is finite.     

Localization for random perturbations of anisotropic periodic media

We prove localization for random perturbations of periodic divergence form operators of the form ∇ · a_ω · ∇ near the band edges. Here a_ω is a matrix function which results from an Anderson type perturbation of a periodic matrix function.     

On periodic Teichmüller disks of modular transformations

We study the periodic Teichmüller disks of modular transformations. Especially, we prove that a parabolic modular transformation has either no periodic Teichmullüller disk or infinitely many periodic Teichmüller disks which can be chosen to cover infinitely many arithmetic Teichmüller curves in the Riemann moduli space M _g . Some related topics are also discussed.     

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P₁P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rej_p-torsion parts, and the elements of L′ by their Tr_p-torsion-free parts.     

Modules of Differential Operators of Order 2 on Coxeter Arrangements

The collection of reflection hyperplanes of a finite reflection group is called a Coxeter arrangement. A Coxeter arrangement is known to be free. K. Saito has constructed a basis consisting of invariant elements for the module of derivations on a Coxeter arrangement. We study the module of \(\mathcal{A}\) -differential operators as a generalization of the study of the module of \(\mathcal{A}\) -derivations. In this article, we prove that the modules of differential operators of order 2 on Coxeter arrangements of types A, B and D are free, by exhibiting their bases. We also prove that the modules cannot have bases consisting of only invariant elements. Two keys for the proof of freeness are the “Cauchy-Sylvester theorem on compound determinants” and the “Saito-Holm criterion for freeness.”     

Steinberg modules and Donkin pairs

First we prove that in characteristicp>0 a module with good filtration for a group of type E₆ restricts to a module with good filtration for a group of type F₄. Thus we confirm a conjecture of Brundan for one more case. Our method relies on the canonical Frobenius splittings of Mathieu. Next we settle the remaining cases, with a computer-aided variation on the old method of Donkin.     

A chain of evolution algebras

We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman-Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time t_c such that the chain has only two idempotent elements if time t?t_c and it has four idempotent elements if time t<t_c.     

Enumeration of Mapping Classes for the Torus

We enumerate periodic, reducible, and Anosov elements of the mapping class group of the torus, and determine their growth function. Then we prove that almost all elements of the mapping class group of the torus are Anosov.     

一种广义周期Besov类的多元周期多项式样条逼近

本文证明多元多项式周期样条空间是某些多元周期光滑函数类的关于Kolmogorov n-宽度的弱渐近极子空间．给出了广义周期Besov类的一种推广，得到了空间元素的一种表示定理，不仅给出了一种多元周期多项式样条算子．而且证明了所得的结果．     

利用图的完形刻画wsr1条件

本文证明了：对于拟投射右R-模M,End_R（M）满足wsr1条件（即弱的stable range one条件）当且仅当M_R是T-投射模;wsr1条件是左右对称的;对于von Neumann正则环R,R满足wsr1条件当且仅当R为单侧幺正则环．     

Wavelets from trigonometric spline approach

The explicit form for the orthonormal periodic trigonometric spline wavelet is given. We also give the decomposition and reconstruction equations. Each of the two equations involves only two terms. We prove that the family of periodic trigonometric spline wavelets is dense in L₂ ([0,2π]). This work is pastially supported by NNSFC and the Foundation of Zhongshan University Advanced Research Centre.     

Approximation dans un Problème de Viscoélasticité Périodique

We are concerned with equations which derive from a quasistatic periodic problem of viscoelasticity. We give a condition which yields to existence and uniqueness of a periodic solution. Then we prove a finite element method based on equilibrium elements for the space approximation and on the explicit Euler scheme for the time approximation.