A Generalization of Divided Domains and Its Connection to Weak Baer Going-Down Rings

Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)?(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× _K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ? T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR _(P) is a going-down ring if and only if R/P and R _P are going-down rings. The weak Baer going-down rings R such that Spec(R)?Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.     

Maximal regular subsemigroups of certain semigroups of transformations

Let T _n and P _n be the full and partial transformation semigroups on a finite set of order n respectively. The properties of the subsemigroups of T _n and P _n have been widely studied. But the maximal regular subsemigroups of T _n and P _n seem to be unknown. In this note, we determine all the maximal regular subsemigroups of all ideals of T _n and P _n . April 7, 1999     

Dual linear connections on fitted manifolds of a space with projective connection

This paper presents an investigation of dual linear connections (projective and affine), induced by different fittings of a space with a projective connection P_n,n, a regular hypersurface V_n-1P _n,n , and a regular hyperbandH _m P _n,n .Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 25–46, 1977.     

Constructing infinite families of regular incidence (quasi-)polytopes

Given a regular incidence (quasi-)polytopeP of type {a ₁,a ₂, ...,a _n–1} and a function on its directed edges satisfying certain conditions, we construct for everym 2 a regular incidence (quasi-)polytope of type {ma ₁,a ₂, ...,a _n–1} with the same vertex figure asP.     

P4‐decompositions of regular graphs

In this article, we show that every simple r‐regular graph G admits a balanced P₄‐decomposition if r ≡ 0(mod 3) and G has no cut‐edge when r is odd. We also show that a connected 4‐regular graph G admits a P₄‐decomposition if and only if |E(G)| ≡ 0(mod 3) by characterizing graphs of maximum degree 4 that admit a triangle‐free Eulerian tour. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 135–143, 1999     

Sublattices of the Lattices of Strong P-Congruences on P-Inversive Semigroups

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C_P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C_P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C_P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C_P(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of C_P(S) and the greatest elements of these sublattices are given.     

On the Perturbations of Regular Linear Systems and Linear Systems with State and Output Delays

This paper is concerned with perturbation problems of regularity linear systems. Two types of perturbation results are proved: (i) the perturbed system (A + P, B, C) generates a regular linear system provided both (A, B, C) and (A, B, P) generate regular linear systems; and (ii) the perturbed system ((A_-1+DA)|_X,B,C^A_L){((A_{-1}+\Delta A)|_X,B,C^A_\Lambda)} generates a regular linear system if both (A, B, C) and (A, ΔA, C) generate regular linear systems. These allow us to establish a new variation of constants formula of the control system (A + P, B). Moreover, these results are applied to the linear systems with state and output delays. The regularity and the mild expressibility is deduced, and a necessary and sufficient condition for stabilizability of the delayed systems is proved.     

Extraction of smooth solutions of a class of regular equations in a rectangle

This work is devoted to the study of two-dimensional, regular, almost hypoelliptic operators P(D) = P(D ₂, D ₂) with regular Newton polyhedrons. It is proved that all generalized (weak) solutions of the equation P(D)u = f from a several weighted Sobolev space are infinitely differentiable functions in the rectangle {x ∈ E ²: −a < x ₁ < a, −b < x ₂ < b} in the variable x ₂, in which the function f is infinitely differentiable.     

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by A_T the localised ring and let M be a primitive ideal of A_T Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QR_Q is generated by a regular g-centralising set of elements, then

(1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (R_Q 0 + ∣X_A (P)∣

(2)d gldim(A_T ) = Kdim(A_T ) = ht(M) = ht(P).     

p-Congruences onp-regular semigroups

A pair (S, P) of a regular semigroupsS and a subsetP ofE _s whereE _s is the set of all idempotent elements ofS is called aP-regular semigroupS(P) if it satisfies the following:

Four dimensional regular algebras with point scheme,anonsingular quadric in P3

In [22], a class of four-dimensional, quadratic, Artin-Schelter regular algebras was introduced, whose point scheme is the graph of an automorphism of a nonsingular quadric in P³. These algebras are the first examples of quadratic Artin-Schelter regular algebras whose defining relations are not determined by the point scheme and, hence, not determined by the algebraic data obtained from the point modules. In this paper, we study these algebras via their line modules. In particular, the set of lines in P³ that correspond to left line modules is not the set of lines in P³ that correspond to right line modules. Our analysis focuses on a distinguished member R _λ of this class of algebras, where R _λ is a twist by a twisting system of the other algebras. We prove that R _λ is a finite module over its center and that its central Proj is a smooth quadric inP⁴.     

Principal Weak Flatness and Regularity of Diagonal Acts

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ? left P(P) ? weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 _X , D(S) is regular.     

Asymptotic behaviour of time evolutions of infinite particle systems

Summary Motions of one-dimensional infinite particle systems are considered where the dynamics is given by systems of ordinary differential equations of first order. The aim of the paper is to show that under certain assumptions about the system of differential equations the distribution law P _tof the particle system at time t becomes more and more regular under the influence of such an interaction. Moreover, P _tis tending weakly toward a distribution describing a random particle system with equal successive spacings.     

A note onQ-matrices

This paper concerns three classes of matrices that are relevant to the linear complementarity problem. We prove that within the class ofP ₀-matrices, theQ-matrices are precisely the regular matrices.Research supported by Department of Energy, Contract EY-76-S-03-0326 PA # 18.     

Symmetry groups of vertex-transitive polytopes

Groups which are not isomorphic to the symmetry group of any vertextransitive polytope (of any dimension) are characterized as generalized dicyclic, or abelian groups but not elementary 2-groups. The same class of groupsG is also characterized by the existence of a permutation groupP acting onG, containingG* (the regular representation ofG) as a proper subgroup, such that the members of the stabilizerP _u of the unitu ε G take everyg ε G tog ^±1.     

Estimates of global dimension

In this note we show that for a *ⁿ-module, in particular, an almost n-tilting module, P over a ring R with A = End_R P such that P _A has finite flat dimension, the upper bound of the global dimension of A can be estimated by the global dimension of R and hence generalize the corresponding results in tilting theory and the ones in the theory of *-modules. As an application, we show that for a finitely generated projective module over a VN regular ring R, the global dimension of its endomorphism ring is not more than the global dimension of R.     

Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

For a given convex body K in \Bbb R³{\Bbb R}^3 with C ² boundary, let P ^c _n be the circumscribed polytope of minimal volume with at most n edges, and let P ⁱ _n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of P ^c _n and P ⁱ _n are asymptotically regular triangles and squares, respectively, in a suitable sense.     

A partition property of a mixed type for Pκ(λ)

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on P_κ(λ) that satisfy the square brackets partition relation , where μ is a cardinal ≥2. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Orbits of linear maps and regular languages

The equivalence is established of the problem of hitting a polyhedral set by the orbit of a linear map and the intersection of a regular language and a language of permutations of binary words ( P_\mathbbBP_\mathbb{B}-realizability problem). The decidability of the both problems is presently unknown, and the first one is a straightforward generalization of the famous Skolem problem and the nonnegativity problem in the theory of linear recurrent sequences.     

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.