The property of being equationally Noetherian for some soluble groups

Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A₁ ⩾ A₂ ⩾ … A_n ⩾ … such that ⋂A_n = 1 and all factors A_n/A_n+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 46–59, January–February, 2007.     

Irreducible algebraic sets over divisible decomposed rigid groups

A soluble group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > …> G _p > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion-free when treated as right ℤ[G/G _i ]-modules. Free soluble groups are important examples of rigid groups. A rigid group G is divisible if elements of a quotient G _i /G _i+1 are divisible by nonzero elements of a ring ℤ[G/G _i ], or, in other words, G _i /G _i+1 is a vector space over a division ring Q(G/G _i ) of quotients of that ring. A rigid group G is decomposed if it splits into a semidirect product A ₁ A ₂…A _p of Abelian groups A _i ≅ G _i /G _i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α _i of bases of suitable vector spaces A _i , and we denote it by M(α₁,…, α _p ). The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR], ], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [11], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group M(α₁,…, α _p ). Our present goal is to derive important information directly about algebraic geometry over M(α₁,… α _p ). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over M(α₁,…, α _p ) using the language of equations.     

Universal theories for rigid soluble groups

A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G ₁ > G ₂ > … > G _p  > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion free when treated as \mathbbZ \mathbb{Z} [G/G _i ]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions A(F) ê A(G) ê A(W) \mathcal{A}(F) \supseteq \mathcal{A}(G) \supseteq \mathcal{A}(W) . We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α₁,…, α _p ). The latter group factors into a semidirect product of Abelian groups A ₁ A ₂…A _p , in which case every quotient M _i /M _i+1 of its rigid series is isomorphic to A _i and is a divisible module of rank α_i over a ring \mathbbZ \mathbb{Z} [M/M _i ]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable.     

Asymptotic behavior of the ratio of tail probabilities of sum and maximum of independent random variables

For a fixed integer n ≥ 2, let X ₁ ,…, X _n be independent random variables (r.v.s) with distributions F ₁,…,F _n , respectively. Let Y be another random variable with distribution G belonging to the intersection of the longtailed distribution class and the O-subexponential distribution class. When each tail of F _i , i = 1,…,n, is asymptotically less than or equal to the tail of G, we derive asymptotic lower and upper bounds for the ratio of the tail probabilities of the sum X ₁ + ⋯ + X _n and Y. By taking different G’s, we obtain general forms of some existing results.     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

Divisible rigid groups

A soluble group G is rigid if it contains a normal series of the form G = G₁ > G₂ > … > G_p > G_p+1 = 1, whose quotients G_i/G_i+1 are Abelian and are torsion-free as right ℤ[G/G_i]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients G_i/G_i+1 are divisible by any elements of respective groups rings Z[G/G_i]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1). Translated from Algebra i Logika, Vol. 47, No. 6, pp. 762–776, November–December, 2008.     

The bieri-strebel invariant and homological finiteness conditions for metabelian groups

A group Γ has type F P_n if a trivial ℤΓ-module ℤ has a projective resolution P:…P_n → … → P₁ → P₀ → ℤ in which ℤΓ-module P_n,…P₁, P₀ are finitely generated. Let the finitely generated group Γ be a split extension of the Abelian group M by an Abelian group Q, suppose M is torsion free, and assume Γ∈F P_m, m≥2. Then the invariant ∑ _c ^M is m-tame. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 194–218, March–April, 1997.     

Approximations simultanées de nombres algébriques de Q p par des rationnels

 In this paper, we prove that if β₁,…, β _n are p-adic numbers belonging to an algebraic number field K of degree n + 1 over Q such that 1, β₁,…,β _n are linearly independent over Z, there exist infinitely many sets of integers (q ₀,…, q _n ), with q ₀ ≠ 0 and

Coproducts of rigid groups

Let ε = (ε ₁, . . . , ε _m ) be a tuple consisting of zeros and ones. Suppose that a group G has a normal series of the form G = G ₁ ≥ G ₂ ≥ . . . ≥ G _m ≥ G _m+1 = 1, in which G _i > G _i+1 for ε _i = 1, G _i = G _i+1 for ε _i = 0, and all factors G _i /G _i+1 of the series are Abelian and are torsion free as right ℤ[G/G _i ]-modules. Such a series, if it exists, is defined by the group G and by the tuple ε uniquely. We call G with the specified series a rigid m-graded group with grading ε. In a free solvable group of derived length m, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid m-graded groups. It is proved that the category of rigid m-graded groups contains coproducts, and we show how to construct a coproduct G◦H of two given rigid m-graded groups. Also it is stated that if G is a rigid m-graded group with grading (1, 1, . . . , 1), and F is a free solvable group of derived length m with basis {x ₁, . . . , x _n }, then G◦F is the coordinate group of an affine space G ⁿ in variables x ₁, . . . , x _n and this space is irreducible in the Zariski topology.     

A Freiheitssatz for products of groups

Denote byB a class of solvable groups having a finite normal series with torsion-free Abelian factors, and by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaWefv3ySLgzgjxyRrxDYbqehuuDJXwAKbIrYf2A0vNC% aGGbaiqb-fa8czaaraaaaa!475E!\[\bar \mathfrak{B}\] a class of groups every finitely generated subgroup of which is approximated by {ie193-3}. We prove that if {ie193-4} is a free product with relations of groups A₁,…,A_n in the class {ie193-5}, where n>m and all relations are taken from the Cartesian subgroups, then there exist distinct indices i₁,…,i_n-m such that gp(A_i1,…,Ai_n-m)=A_i1 *…* Ai_n-m. A similar fact is established for solvable products with relations. Supported by RFFR grant No. 99-01-00567. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 354–367, May–June, 1999.     

Disjoint homotopic paths and trees in a planar graph

In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ², a subset {I ₁, …,I _p} of the faces ofG, and pathsC ₁, …,C _k inG, with endpoints on the boundary ofI ₁ ∪ … ∪I _p; find: pairwise disjoint simple pathsP ₁, …,P _k inG so that, for eachi=1, …,k, P _i is homotopic toC _i in the space ℝ²\(I ₁ ∪ … ∪I _p). Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ² and pairwise disjoint setsW ₁, …,W _k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT ₁, …,T _k ofG whereT _i (i=1, …, k).     

Equations and algebraic geometry over profinite groups

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.     

On Cycle Sequences

. For each vertex v in a graph G, the maximum length of a cycle which passes through v is called the cycle number of v, denoted by c(v). A sequence a ₁,a ₂,…,a _n of nonnegative integers is called a cycle sequence of a graph G if the vertices of G can be labeled as v ₁,v ₂,…,v _n such that a _i =c(v _i ) for 1≤i≤n. We give some sufficient and necessary conditions for a sequence to be a cycle sequence. We can thereby derive a polynomial time procedure for recognizing cycle sequences. Received: July 14, 1997 Final version received: June 15, 1998     

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     

A note on the coefficient rings of polynomial rings

LetA andB be two reduced commutative rings with finitely many minimal prime ideals. If the polynomial algebrasA[X ₁ …X _n ]=B[Y ₁ …Y _n ] whereX _i ,Y _iF are variables overA andB respectively, then there exists an injective ring homomorphism ϕ:A→B such thatB is finitely generated over ϕ(A).     

A finiteness condition for semigroups which generalizes Green-Rees' theorem

LetS be a finitely generated semigroup. ThenS is finite if every finitely generated subgroup ofS is finite and, for some integerm≥1, for everym-tuplex ₁,x ₂,…x _m of elements ofS there exist an integeri: 1≤i≤m and an integer ρ>1 such that:x _i +1…x _m (x ₁ x ₂…x _m )^ρ=x _i +1…x _m x ₁…x _m . The proof of the result is a direct generalisation of the original one by Green and Rees for the casem=1.     

Degree Sums and Path-Factors in Graphs

 Let G be a connected graph of order n and suppose that n=∑ _{i =1} ^k n _i , where n _i ≥2 are integers. In this paper we give some sufficient conditions in terms of degree sums to ensure that G contains a spanning subgraph consisting of vertex disjoint paths of orders n ₁,n ₂,…,n _k . Received: June 30, 1999 Final version received: July 31, 2000     

A class of problems for which cyclic relaxation converges linearly

The method of cyclic relaxation for the minimization of a function depending on several variables cyclically updates the value of each of the variables to its optimum subject to the condition that the remaining variables are fixed. We present a simple and transparent proof for the fact that cyclic relaxation converges linearly to an optimum solution when applied to the minimization of functions of the form for a _i,j ,b _i ,c _i ∈ℝ_≥0 with max {min {b ₁,b ₂,…,b _n },min {c ₁,c ₂,…,c _n }}>0 over the n-dimensional interval [l ₁,u ₁]×[l ₂,u ₂]×⋅⋅⋅×[l _n ,u _n ] with 0<l _i <u _i for 1≤i≤n. Our result generalizes several convergence results that have been observed for algorithms applied to gate- and wire-sizing problems that arise in chip design.     

An optimal version of an inequality involving the third symmetric means

Let (GA) _n ^[k](a), A _n (a), G _n (a) be the third symmetric mean of k degree, the arithmetic and geometric means of a ₁, …, a _n (a _i > 0, i = 1, …, n), respectively. By means of descending dimension method, we prove that the maximum of p is k−1/n−1 and the minimum of q is n/n−1(k−1/k) ^k/n so that the inequalities {fx505-1} hold.     

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom _R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H _{(x1, …,xn)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).