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1.
Let H be a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group. Let G be any group with maximal condition. We show that there exists a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group and an epimorphism such that for any homomorphism ?:GH, it factors through , i.e., there exists a homomorphism such that . We show that this factorization property cannot be extended to any finitely generated group G. As an application of factorization, we give necessary and sufficient conditions for N(f,g)=R(f,g) to hold for maps f,g:XY between closed orientable n-manifolds where π1(X) has the maximal condition, Y is an infra-solvmanifold, N(f,g) and R(f,g) denote the Nielsen and Reidemeister coincidence numbers, respectively.  相似文献   

2.
A.R. Rao 《Discrete Mathematics》2006,306(14):1595-1600
For a digraph G, let R(G) (respectively, R(k)(G)) be the number of ordered pairs (u,v) of vertices of G such that uv and v is reachable from u (respectively, reachable from u by a path of length ?k). In this paper, we study the range Sn of R(G) and the range of R(k)(G) as G varies over all possible digraphs on n vertices. We give a sufficient condition and a necessary condition for an integer to belong to Sn. These determine the set Sn for all n?208. We also determine for k?4 and show that whenever n?k+(k+1)0.57+2, for arbitrary k.  相似文献   

3.
Integral quadratic forms q:ZnZ, with n≥1, and the sets Rq(d)={vZn;q(v)=d}, with dZ, of their integral roots are studied by means of mesh translation quivers defined by Z-bilinear morsifications bA:Zn×ZnZ of q, with Z-regular matrices AMn(Z). Mesh geometries of roots of positive definite quadratic forms q:ZnZ are studied in connection with root mesh quivers of forms associated to Dynkin diagrams An,Dn,E6,E7,E8 and the Auslander-Reiten quivers of the derived category Db(R) of path algebras R=KQ of Dynkin quivers Q. We introduce the concepts of a Z-morsification bA of a quadratic form q, a weighted ΦA-mesh of vectors in Zn, and a weighted ΦA-mesh orbit translation quiver Γ(Rq,ΦA) of vectors in Zn, where Rq?Rq(1) and ΦA:ZnZn is the Coxeter isomorphism defined by A. The existence of mesh geometries on Rq is discussed. It is shown that, under some assumptions on the morsification bA:Zn×ZnZ, the set admit a ΦA-orbit mesh quiver , where ΦA:ZnZn is the Coxeter isomorphism defined by A. Moreover, splits into three infinite connected components , , and , where are isomorphic to a translation quiver ZΔ, with Δ an extended Dynkin quiver, and has the shape of a sand-glass tube.  相似文献   

4.
We call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M≠0) generalized triangular matrix ring , for some rings R and S and some R-S-bimodule RMS. Examples of such rings include rings with only the trivial idempotents 0 and 1, as well as endomorphism rings of vector spaces, or more generally, semiprime indecomposable rings. We show that if R and S are strongly indecomposable rings, then the triangulation of the non-trivial generalized triangular matrix ring is unique up to isomorphism; to be more precise, if is an isomorphism, then there are isomorphisms ρ:RR and ψ:SS such that χ:=φM:MM is an R-S-bimodule isomorphism relative to ρ and ψ. In particular, this result describes the automorphism groups of such upper triangular matrix rings   相似文献   

5.
6.
Let be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over . Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier (Computational Perspectives on Number Theory (Chicago, IL, 1995), AMS/IP 7 (1998) 97-126), we define an inner product on for every . Suppose a system of orthogonal polynomials {Pn,ψ(x)}n=0 with respect to exists. We prove that if n is sufficiently large and ψ(x)Pn,ψ(x) is p-integral, then over . Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.  相似文献   

7.
A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor1(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor1(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.  相似文献   

8.
Let R be a local ring and M a finitely generated R-module. The complete intersection dimension of M-defined by Avramov, Gasharov and Peeva, and denoted -is a homological invariant whose finiteness implies that M is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger’s Gorenstein dimension by the inequalities .Using Blanco and Majadas’ version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms φ:RS and ψ:ST such that φ has finite Gorenstein dimension, if ψ has finite complete intersection dimension, then the composition ψ°φ has finite Gorenstein dimension. This follows from our result stating that, if M has finite complete intersection dimension, then M is C-reflexive and is in the Auslander class AC(R) for each semidualizing R-complex C.  相似文献   

9.
Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?f is right equivalent to f, i.e. there exists a diffeomorphism such that ?f=fh at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings MS1.  相似文献   

10.
Let H be a complex Hilbert space and let {Tn}n?1 be a sequence of commuting bounded operators on H such that . Let denote the space of all operators X in B(H) for which and suppose that . We will show that there exists a triple {K,Γ,{Un}n?1} where K is a Hilbert space, Γ:KH is a bounded operator and {Un}n?1B(K) is a sequence of commuting normal operators with such that TnΓ=ΓUn for n?1, and for which the mapping Y?ΓYΓ is a complete isometry from the commutant of {Un}n?1 onto the space . Moreover we show that the inverse of this mapping can be extended to a -homomorphism
  相似文献   

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