Sufficient conditions for the existence of a left quotient ring of a ring decomposed into a direct sum of left ideals

Let R=P₁⊕P₂⊕…⊕P_n be a decomposition of a ring into a direct sum of indecomposable left ideals. Assume that these ideals possess the following properties: (1) any nonzero homomorphisms ϕ: P_i→P_j is a monomorphism; (2) if subideals Q₁, Q₂ of the ideal P_j are isomorphic to the ideal P_i, then there exists a subideal Q₃⊆Q₁⊎Q₂, which is also isomorphic to P_i. It is proved that, under these asumptions, a left quotient ring of the ring R exists. This left quotient ring inherits properties(1), (2) and satisfies condition (3): any nonzero homomorphism ϕ: P_i→P_i is an automorphism of the ideal P_i. Bibliography: 2 titles. Translated fromZapiski Nauchnyk Seminarov POMI, Vol. 227, 1995, pp. 9–14.     

Unitary similarity of algebras generated by pairs of orthoprojectors

It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors { P₁, Q₁} and {P₂,Q₂} can be verified by comparing the traces of P₁, Q₁, and (P₁Q₁)ⁱ, i = 1, 2, …, n, with those of P₂, Q₂, and (P₂Q₂)ⁱ. The conditions of the unitary similarity of two matrices with quadratic minimal polynomials presented in [A. George and Kh. D. Ikramov, Unitary similarity of matrices with quadratic minimal polynomials, Linear Algebra Appl., 349, 11–16 (2002)] are refined. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 5–14.     

Equivariant plateau problems

Let (M, Q) be a compact, three dimensional manifold of strictly negative sectional curvature. Let (Σ, P) be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let θ : π₁(Σ, P) → π₁(M, Q) be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and sufficient conditions for the existence of complex projective structures with specified holonomy to manifolds of non-constant negative curvature, we obtain necessary conditions on θ for the existence of a so called θ-equivariant Plateau problem over Σ, which is equivalent to the existence of a strictly convex immersion i : Σ → M which realises θ (i.e. such that θ = i _*).      

The construction of SL(2, 3)-polynomials

Octic polynomials over $\text{Z}$ with Galois group SL(2, 3) are constructed. This is done via suited quartic totally real polynomials with group A₄ over $\text{Q}$. A table of the cycle patterns of the imprimitive transitive permutation groups of degree 8 is included.     

A presentation for the unipotent group over F2

We define Unipn(F₂) to be the group of invertible upper-triangular matrices over F₂, the field of 2 elements. Let s _i for i = 1,2,…,n - 1 denote the matrix whosne diagonal entries are all 1, and whose only other nonzero entry is in the ith row and (i + 1)st column. Then it is easy to see that the s _i generate Unipn(F2). Reiner [4] gave relations among the s _i, which he conjectured gave a presentation for Unip_n(F₂). We show that a subset of these relations suffice to present the group     

The perpendicular bisector construction

LetQ ₁,Q ₂,Q ₃ be plane quadrilaterals such that the sides ofQ _i are the perpendicular bisectors of the sides ofQ _i–1 (i=2, 3). This note gives a simple trigonometrical solution of the long outstanding problem of showing thatQ ₁ andQ ₃ are similar.     

A Family of Geometries Related to the Suzuki Tower

ABSTRACT

Michio Suzuki constructed a sequence of five simple groups G _i , with i = 0,…, 4, and five graphs Δ _i , with i = 0,…, 4, such that Δ _i appears as a subgraph of Δ _i+1 for i = 0,…, 3 and G _i is an automorphism group of Δ _i for i = 0,…, 4. The largest group G ₄ was a new sporadic group of order 448 345 497 600. It is now called the Suzuki group Suz. These groups and graphs form what Jacques Tits calls the Suzuki tower. In a previous work, we constructed a rank four geometry Γ(HJ) on which the Hall-Janko sporadic simple group acts flag-transitively and residually weakly primitively. In this article, we show that Γ(HJ) belongs to a family of five geometries in bijection with the Suzuki tower. The largest of them is a geometry of rank six, on which the Suzuki sporadic group acts flag-transitively and residually weakly primitively.     

On Cayley digraphs on nonisomorphic 2‐groups

A necessary and sufficient condition is given for two Cayley digraphs X₁ = Cay(G₁, S₁) and X₂ = Cay(G₂, S₂) to be isomorphic, where the groups G_i are nonisomorphic abelian 2‐groups, and the digraphs X_i have a regular cyclic group of automorphisms. Our result extends that of Morris [J Graph Theory 3 (1999), 345–362] concerning p‐groups G_i, where p is an odd prime. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

On minimal,strongly proximal actions of locally compact groups

Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie groupG on homogeneous spacesG/Q, whereQ ⊂G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note we show that ifX is a compact manifold with a faithful boundary action of some locally compact groupH, then (under some mild regularity assumption) the groupH, the spaceX, and the action split into a direct product of a semi-simple Lie groupG acting onG/Q and a boundary action of a discrete countable group. The author was partially supported by NSF grants DMS-0049069, 0094245 and GIF grant G-454-213.06/95.     

Resolutions of PG(5, 2) with point‐cyclic automorphism group

A t‐(υ, k, λ) design is a set of υ points together with a collection of its k‐subsets called blocks so that all subsets of t points are contained in exactly λ blocks. The d‐dimensional projective geometry over GF(q), PG(d, q), is a 2‐(q^d + q^d−1 + … + q + 1, q + 1, 1) design when we take its points as the points of the design and its lines as the blocks of the design. A 2‐(υ, k, 1) design is said to be resolvable if the blocks can be partitioned as ℛ = {R₁, R₂, …, R_s}, where s = (υ − 1)/(k−1) and each R_i consists of υ/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length υ on the points and ℛ^σ = ℛ, then the design is said to be point‐cyclically resolvable. The design associated with PG(5, 2) is known to be resolvable and in this paper, it is shown to be point‐cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G = 〈σ〉 where σ is a cycle of length 63. These resolutions are the only resolutions which admit a point‐transitive automorphism group. Furthermore, some necessary conditions for the point‐cyclic resolvability of 2‐(υ, k, 1) designs are also given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 2–14, 2000     

Sharply k-Transitive Permutation Groups Viewed as Galois Groups

The classification of finite sharply k-transitive groups was achieved by the efforts of Jordan (1873), Dickson (1905), and Zassenhaus (1936). Likewise for other families of finite groups, one expects that they are realizable as Galois groups over the field of rational numbers \mathbbQ{\mathbb{Q}}. In this article, we study some properties of the polynomials f ? \mathbbQ[x]{f \in \mathbb{Q}[x]} such that the Galois group Gal(f) acts sharply k-transitively on its roots.     

The minimum increment of f-divergences given total variation distances

Let (P _i , Q _i ), i = 0, 1, be two pairs of probability measures defined on measurable spaces (Ω _i ,F _i ) respectively. Assume that the pair (P₁, Q₁) is more informative than (P₀,Q₀) for testing problems. This amounts to say that If (P₁,Q₁) ≥ If (P₀,Q₀), where If (·, ·) is an arbitrary f-divergence. We find a precise lower bound for the increment of f-divergences I_f(P₁,Q₁) ? I_f(P₀,Q₀) provided that the total variation distances ||Q₁ ? P₁|| and ||Q₀ ? P₀|| are given. This optimization problem can be reduced to the case where P₁ and Q₁ are defined on the space consisting of four points, and P₀ and Q₀ are obtained from P₁ and Q₁ respectively by merging two of these four points. The result includes the well-known lower and upper bounds for I_f(P,Q) given ||Q ? P||.     

Periodic Groups Saturated with the Linear Groups of Degree 2 and the Unitary Groups of Degree 3 over Finite Fields of Odd Characteristic

Let M denote the set of the simple 3-dimensional unitary groups U₃ and the simple linear groups L₂ over finite fields of odd characteristic.We prove that each periodic group saturated with groups in M is locally finite and isomorphic to either U₃(Q) or L₂(Q) for a suitable locally finite field Q of odd characteristic.     

Conservative loads on shells

Generalized load vectorsp and edge load vectorsF are denned in terms of the body force and surface on a shell. Necessary and sufficient conditions are derived forp andF, and therefore the body force and the surface force, to be conservative. It is shown for example thatp must satisfyp _i=P _ijk q _j,1 q _k,2+Q _ij ¹ q _j,2–Q _ij ² q _j,1+R _i whereq is the generalized position vector andP _ijk, Q_i,j ¹ and Q_ij ² are skew tensors.The case of hydrostatic pressure is examined in detail.This work was supported in part by NSF grant MSM 8618657.     

Characterization of the association schemes of Hermitian forms over GF(22)

It is shown that an association schemeY=(X, {R _i }_0⩽i⩽d ), in which the parameters coincide with those of the scheme Her(d, 2) of Hermitian forms ind-dimensional space over GF(2²), is isomorphic to Her(d, 2). A principal role in the proof is played by a theorem by P. Terwilliger on the number of 4-vertex configurations of a given type in (P andQ)-polynomial association schemes. Some partial results are obtained in the case of an arbitrary finite field.     

Dimensions of Formal Fibers of Height One Prime Ideals

Let T be a complete local (Noetherian) ring with maximal ideal M, P a nonmaximal ideal of T, and C = {Q ₁, Q ₂,…} a (nonempty) finite or countable set of nonmaximal prime ideals of T. Let {p ₁, p ₂,…} be a set of nonzero regular elements of T, whose cardinality is the same as that of C. Suppose that p _i  ∈ Q _j if and only if i = j. We give conditions that ensure there is an excellent local unique factorization domain A such that A is a subring of T, the maximal ideal of A is M ∩ A, the (M ∩ A)-adic completion of A is T, and so that the following three conditions hold: (1) p _i  ∈ A for every i; (2) A ∩ P = (0), and if J is a prime ideal of T with J ∩ A = (0), then J ? P or J ? Q _i for some i; (3) for each i, p _i A is a prime ideal of A, Q _i ∩ A = p _i A, and if J is a prime ideal of T with J ? Q _i , then J ∩ A ≠ p _i A.     

Positive vector measures with given marginals

Suppose E is an ordered locally convex space, X ₁ and X ₂ Hausdorff completely regular spaces and Q a uniformly bounded, convex and closed subset of M _t ⁺ (X ₁ × X ₂, E). For i = 1, 2, let μ _i   M _t ⁺ (X _i , E). Then, under some topological and order conditions on E, necessary and sufficient conditions are established for the existence of an element in Q, having marginals μ ₁ and μ ₂.     

The Weierstrass gap sets for quadruples II

Let M be a compact Riemann surface of genus g. Let P _i (i = 1, ..., 4) be 4 distinct points on M. We denote G(P ₁, P ₂, P ₃, P ₄) the Weierstrass gap set. In this paper, we prove that, for large g, the upper bound of #G(P ₁, P ₂, P ₃, P ₄) is attained if and only if M is hyperelliptic and |2P _i | = g ₂¹.     

Localization and Catenarity in Iterated Differential Operator Rings

ABSTRACT

Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ₁, δ₁] ··· [Θ _n , δ _n ] an iterated differential operator k-algebra over R such that δ _j (Θ _i ) ∈ R[Θ₁, δ₁] ··· [Θ _i?1, δ _i?1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A _P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ _i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U(g) the enveloping algebra of g. Then the crossed product of R by U(g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.     

On the structure of cluster point sets of iteratively generated sequences

LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {x _i} be a sequence generated iteratively bya(·) fromx ₀ inX, i.e.,x _i+1 =a(x _i),i=0, 1, 2, ...; and letQ(x ₀) be the cluster point set of {x _i}. In this paper, we prove that, if there exists a pointz inQ(x ₀) such that (i)z is isolated with respect toQ(x ₀), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x ₀) containsp points, (v) the sequence {x _i} is contained in a sequentially compact set, and (vi) every point inQ(x ₀) possesses properties (i) and (ii). The application of the preceding results to the caseF=E ⁿ leads to the following: (vii) ifQ(x ₀) contains one and only one point, then {x _i} converges; (viii) ifQ(x ₀) contains a finite number of points, then {x _i} is bounded; and (ix) ifQ(x ₀) containsp points, then every point inQ(x ₀) is a periodic point ofa(·) of periodp.