Bases normales,unités et conjecture faible de leopoldt

Letk be a number field,p an odd prime,R _k the ring ofp-integers ofk. We use Iwasawa theory to study theZ _p -moduleG(R _k ,Z _p ) (resp.NB (R _k ,Z _p )) ofclasses ofZ _p -extensions (resp.Z _p -extensions having a normal basis overR _k ) ofR _k . The rank ofG(G _k ,Z _p ) (resp.NB(R _k ,Z _p )) is related to Leopoldt's conjecture (resp. weak Leopoldt's conjecture) fork andp.      

Observations on primitive,normal, and subnormal elements of field extensions

Let B ₁ and B ₂ be disjoint separable algebraic extensions of a field F, and let B = B ₁ B ₂ be their composite. Let α ₁ be an element of B ₁ and α ₂ be an element of B ₂. Suppose α ₁ and α ₂ are primitive (resp. normal, resp. subnormal). We investigate the question of when α ₁ + α ₂ and α ₁ α ₂ are necessarily primitive (resp. normal, resp. subnormal) elements of B. (A normal element of a Galois extension is defined to be one that is part of a normal basis, and a subnormal element is defined analogously for a non-Galois extension).     

Dependence of the wave-front speeds on the propagation direction

The dependence of the characteristic speeds of quasilinear hyperbolic systems on the propagation directionn is investigated. It is proved that any non-vanishing characteristic speedc(n) is the sum of a homogeneous functionc ₁ (n) and a positively homogeneous functionc ₂ (n). As a further result, ifc ₂ (n) is non-vanishing, then bothc ₂ (n)±c ₂ (n) are characteristic speeds.

---

Sommario Nel lavoro si analizza la dipendenza dalla direzione di propagazionen delle velocità caratteristiche associate ad un sistema iperbolico quasi lineare. Si prova che ogni velocità caratteristicac(n) non nulla è somma di una funzione omogeneac ₁ (n) e di una funzione positivamente omegeneac ₂(n). Come ulteriore risultato si ha che, sec ₂(n) è non nulla, allora entrambe le funzionic ₁(n)±c ₂(n) sono velocità caratteristiche.

     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

Quasi-Frobenius Corings and Quasi-Frobenius Extensions

In this article, the notions of a Frobenius pair of functors and Frobenius corings are generalized to an l-QF pair of functors and l-QF corings. We prove that an extension ι:B → A is left quasi-Frobenius if and only if (F ₁,G ₁) is an l-QF pair of functors, where F ₁: _A ? →  _B ? is the restriction of scalars functors, and G ₁ = A? _{B ^?} : _B ? →  _A ? is the induction functor. For an A-coring , we prove that  is an l-QF coring if and only if A → ? is an l-QF extension and _A  is a finitely generated projective modules if and only if (G ₂,F ₂) is an l-QF pair of functors, where G ₂ =  ? _{A ^?} : _A ? →  ? is the induction functor, F ₂:  ? →  _A ? is the forgetful functor, the result of Brzezinski is generalized.     

On the Grigorchuk–Kurchanov conjecture

 Let denote the free group of rank 2g. An automorphism φ? Aut(F _{2 g} ) is generating if N _a φ (N _b ) = F _{2 g} , where N _a is the normal closure of and N _b is defined analogously. We present a characterization of generating automorphisms in Aut(F ₂) and observe that there exists a unique (up to equivalence) epimorphism F ₂→Z×Z: this is a particular case of the Grigorchuk–Kurchanov conjecture. This leads to further investigations for splitting homomorphisms for the pairs (F _{2 g} , F _g) and (G _g, F _g) where G _g denotes the fundamental group of a closed orientable surface of genus g and a reformulation of the Poincaré and Grigorchuk–Kurchanov conjectures is derived. Received: 1 October 2001     

Structure Theorems for Pure Semisimple Grothendieck Locally PI-Categories

Let Q be a nondegenerate quadratic form over the finite field F _q and O _n (F _q ) be the associated orthogonal group. Let O _n (F _q ) act linearly on the polynomial ring F _q [x ₁, …, x _n ]. We find the invariant subring of O ₄(F _q ) with explicit generators.     

Additive Maps Preserving Moore–Penrose Inverses of Matrices on Symmetric Matrix Spaces

Suppose F is a field of characteristic not 2. Let M_n F and S_n F be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from S_n F to S_n F (respectively, M_n F) preserving Moore–Penrose inverses of matrices are characterized. We first characterize all additive Moore–Penrose inverse preserving maps from S_n F to M_n F, and thereby, all additive Moore–Penrose inverse preserving maps from S_n F to itself are characterized by restricting the range of these additive maps into the symmetric matrix space.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.     

Centrality in Rees-Sushkevich varieties

Denote by RS _n the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RS _n are called Rees-Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups are said to be exact. For each positive integer m, define C _m RS _n to be the class of all semigroups S in RS _n with the property that if the product of m idempotents of S belongs to some subgroup of S, then the product belongs to the center of that subgroup. The classes C _m RS _n constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C ₂ RS _n is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties over groups of exponent dividing n. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core with any subgroup, if nonempty, is contained in the center of that subgroup. The equational property of the varieties C _m RS _n is also addressed. For any fixed n ≥ 2, it is shown that although the varieties C _m RS _n , where m = 1, 2, ... , are all finitely based, their complete intersection (denoted by C _∞ RS _n ) is non-finitely based. Further, the variety C _∞ RS _n contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based. Received October 29, 2003; accepted in final form February 11, 2007.     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

Higher Power Residue Codes

For certain primeslandp, and characters χ:F*_p→F*_l², we construct codesWof lengthp+ 1 overF_l²which are linear overF_l, but not overF_l², and which are invariant under a monomial action of the group SL(2,p). We consider the cases of cubic and quartic characters in detail and use theWto construct linear codes overF_lin these cases.     

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G ₁ ?  G are complex linear simple Lie groups. Let  ₁ ?  be the corresponding pair of Lie algebras. For the Killing-orthogonal  of  ₁ in  we have a vector space direct sum  =  ₁⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G ₁) = (SL (4,?),Sp (2,?)); here  =  (4,?),  ₁ =  (2,?), and  = {X ?  | X ^? = X}, where X? X ^? is the symplectic involution. We prove that G  =  G ₁exp  ∪ i G ₁exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ^? = X, which is here a natural counterpart of the standard Pfaffian.     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

On nonsingularity of combinations of three group invertible matrices and three tripotent matrices

Let T?=?c ₁ T ₁?+?c ₂ T ₂?+?c ₃ T ₃???c ₄(T ₁ T ₂?+?T ₃ T ₁?+?T ₂ T ₃), where T ₁, T ₂, T ₃ are three n?×?n tripotent matrices and c ₁, c ₂, c ₃, c ₄ are complex numbers with c ₁, c ₂, c ₃ nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices.     

Generating Random Vectors in (ℤ/<Emphasis Type="Italic">p</Emphasis>ℤ)<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> via an Affine Random Process

This paper considers some random processes of the form X _n+1=T X _n +B _n (mod p) where B _n and X _n are random variables over (ℤ/pℤ) ^d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B _n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)² steps suffice to make X _n close to uniformly distributed where X ₀ is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p ^b steps are needed for X _n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X ₀ is the zero vector.     

Consistency of a nonparametric conditional quantile estimator for random fields

Given a stationary multidimensional spatial process (Z _i = (X _i , Y _i ) ∈ ℝ ^d × ℝ, i ∈ ℤ ^N ), we investigate a kernel estimate of the spatial conditional quantile function of the response variable Y _i given the explicative variable X _i . Almost complete convergence and consistency in L ^2r norm (r ∈ ℕ*) of the kernel estimate are obtained when the sample considered is an α-mixing sequence.     

On the Connection Between the Projection and the Extension of a Parallelotope

This paper presents a result concerning the connection between the parallel projection P _v,H of a parallelotope P along the direction v (into a transversal hyperplane H) and the extension P + S(v), meaning the Minkowski sum of P and the segment S(v) = {λv | −1 ≤ λ ≤ 1}. A sublattice L _v of the lattice of translations of P associated to the direction v is defined. It is proved that the extension P + S(v) is a parallelotope if and only if the parallel projection P _v,H is a parallelotope with respect to the lattice of translations L _v,H , which is the projection of the lattice L _v along the direction v into the hyperplane H.