Properties of a certain product of submodules

Let R be a commutative ring with identity, let M be an R-module, and let K ₁, . . . ,K _n be submodules of M: We construct an algebraic object called the product of K ₁, . . . ,K _n : This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M ⁿ   = M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M ⁿ is a projective (flat) R(M)-module.     

On inner dilatations of the mappings with unbounded characteristic

For the mappings f:D ? D￠,  D,  D￠ ì \mathbbRⁿ f:D \to D',\,\,D,\,\,D' \subset {\mathbb{R}^n} , n ≥ 2, satisfying certain geometric conditions in the fixed domain D, we have proved estimates of the form K _I (x, f) ≤ Q(x) almost everywhere, where K _I (x, f) is the inner dilatation of f at a point x, and Q(x) is a fixed real-valued function responsible for the “control” over a distortion of the families of curves in D at a mapping f.     

On the Diophantine Equation G n ( x )?= G m ( P ( x ))

 Let K be a field of characteristic 0 and let p, q, G ₀ , G ₁ , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G _n (x)) _n=0 ^∞ be defined by the second order linear recurring sequence

A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ , and let V(⋅) be the Euclidean volume in R ⁿ . The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim _{r → ∞} e^2r s(r) = 0, then (M, g) has to be isometric to ℍ ⁿ . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim _{r → ∞} r ² s(r) = 0, then (M, g) is isometric to ℝ ⁿ , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B _p (R) in M and K = a on ∂B _p (R), then K = a on B _p (R).     

Some Elements of Finite Order in K2Q

Let K2 be the Milnor functor and let Фn （x）∈ Q[X] be the n-th cyclotomic polynomial. Let Gn（Q） denote a subset consisting of elements of the form {a, Фn（a）}, where a ∈ Q^＊ and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn（Q） is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn（Q） is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h（Q（ζp）） is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.     

Elliptic equation with singular potential

We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R ⁿ , n ≥ 3:

Counting models in universal Horn classes

Definen _K (λ) to be either ω, or the number of non-isomorphic models inK having cardinality α, whichever cardinal is larger. This paper contains a proof that for a congruence modular variety ⋎ of algebras of countable similarity type, there are only six possible functionsn _⋎. It is also proved that ifn _K (λ)≠2^λ for some λ, andK is a universal Horn class of models for a countable language, thenK must satisfy two conditions, one of which is quite restrictive and requires that the members ofK are all in a certain sense Abelian. Presented by B. Jonsson.     

On the Classification of Twisting Maps Between Kn and Km

We define the notion of admissible pair for an algebra A, consisting on a couple (Γ, R), where Γ is a quiver and R a unital, splitted and factorizable representation of Γ, and prove that the set of admissible pairs for A is in one to one correspondence with the points of the variety of twisting maps T_Aⁿ:=T(Kⁿ,A)\mathcal{T}_A^n:=\mathcal{T}(K^n,A). We describe all these representations in the case A = K ^m .     

On the Diophantine Equation G n ( x )?= G m ( P ( x ))

 Let K be a field of characteristic 0 and let p, q, G ₀ , G ₁ , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G _n (x)) _n=0 ^∞ be defined by the second order linear recurring sequence

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let K = k( ⁿ?{d} ) K = k\left( {\sqrt[n]{d}} \right) be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σ_R denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ^R denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ_R) be the subgroup generated by all transvections from the net group G(σ_R). In the paper it is proved that the product TE(σ_R) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ_R,then TE(σ_R) ≤ H ≤ N(σ_R),where N(σ_R) is the normalizer of the elementary net group E(σ_R) in G. For the normalizer N(σ_R),the formula N(σ_R)= TG(σ^R) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.     

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

Annihilating power values of co-commutators with generalized derivations

Let R be a prime ring with its Utumi ring of quotient U, H and G be two generalized derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(H(u)u − uG(u)) ⁿ  = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then there exist b′,c′ ∈ U such that H(x) = b′x + xc′, G(x) = c′x for all x ∈ R with ab′ = 0, unless R satisfies s ₄, the standard identity in four variables.     

Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and _R M _S is an S-R-bimodule. In the main theorem of this paper we show that if T _S is a tilting S-module, then under certain homological conditions on the S-module M _S , one can extend T _S to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and T _S , and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M _S is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M _S has a finite projective resolution and Ext _S ⁿ (M _S , S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, Hom _S (M, S)).     

The Chow ring of a singular surface

We construct the Chow ringCH*(X) =CH ⁰ (X)⊕CH ¹ (X)⊕CH ² (X) of a reduced, quasi-projective surfaceX, together with Chern class mapsc _i :K ₀ (X) → CH ⁱ (X), with the usual properties. As a consequence, we show that the cycle mapCH ² (X)→ F ₀ K ₀ (X) is an isomorphism. Our treatment is greatly influenced by an unpublished 1983 preprint of Levine’s, but is much simpler, since we deal only with surfaces.     

IBN rings and orderings on grothendieck groups

LetR be a ring with an identity element.R∈IBN means thatR ^m⋟Rⁿ impliesm=n, R∈IBN ₁ means thatR ^m⋟Rⁿ⊕K impliesm≥n, andR∈IBN ₂ means thatR ^m⋟R^m⊕K impliesK=0. In this paper we give some characteristic properties ofIBN ₁ andIBN ₂, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN ₁ and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK ₀(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK ₀(R) of a ringR is a partial ordering, thenR∈IBN ₁ orK ₀(R)=0. Supported by National Nature Science Foundation of China.     

Generalized Ham-Sandwich Cuts

Bárány, Hubard, and Jerónimo recently showed that for given well-separated convex bodies S ₁,…,S _d in R ^d and constants β _i ∈[0,1], there exists a unique hyperplane h with the property that Vol (h ⁺∩S _i )=β _i ⋅Vol (S _i ); h ⁺ is the closed positive transversal halfspace of h, and h is a “generalized ham-sandwich cut.” We give a discrete analogue for a set S of n points in R ^d which are partitioned into a family S=P ₁∪⋅⋅⋅∪P _d of well-separated sets and are in weak general position. The combinatorial proof inspires an O(n(log n) ^d−3) algorithm which, given positive integers a _i ≤|P _i |, finds the unique hyperplane h incident with a point in each P _i and having |h ⁺∩P _i |=a _i . Finally we show two other consequences of the direct combinatorial proof: the first is a stronger result, namely that in the discrete case, the conditions assuring existence and uniqueness of generalized cuts are also necessary; the second is an alternative and simpler proof of the theorem in Bárány et al., and in addition, we strengthen the result via a partial converse.     

Elements of Small Orders in K2F Ⅱ

In "Elements of small orders in K2(F)" (Algebraic K-Theory, Lecture Notes in Math., 966, 1982, 1-6.), the author investigates elements of the form {a, Φn(a)} in the Milnor group K2F of a field F, where Φn(x) is the n-th cyclotomic polynomial. In this paper, these elements are generalized. Applying the explicit formulas of Rosset and Tate for the transfer homomorphism for K2, the author proves some new results on elements of small orders in K2F.     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573