Cauty?s space enhanced

The title refers to Cauty?s example (Cauty, 1994 [3]) of a metric vector space which is not an absolute retract. It is shown that Cauty?s space can be refined to the effect that the completion of the refined space can be isomorphically embedded as a subspace of an F-space which itself is an absolute retract.     

Corrigendum and addendum to « Multifunctions on abstract measurable spaces and applications to stochastic decision theory »

Summary This note points out that the ? only if ? part of Theorem 1′ (i) of the above paper [Annali di Matematica pura ed applicata, (IV) 101 (1974), pp. 229–236] is false as is part (ii) of Theorem 2. Counterexamples are given and a weak form of the ? only if ? part of Theorem 1′ (i) is established. Entrata in Redazione il 16 novembre 1977. Partially supported by National Science Foundation Grant MCS 76-24436.     

Reflexive domains and fixed points

In a recent paper in this journal, J. Soto-Andrade and F. J. Varela draw attention to the fact that ifR is a retract of a reflexive domain in a suitable category thenR has the fixed point property. They suggest [1], pp. 1 and 18, that conversely every structure with the fixed point property is a retract of a reflexive domain. In this note it is shown that ifR is a retract of a reflexive domain thenR ^R has the fixed point property. This leads to counterexamples to the suggestion of Soto-Andrade and Varela in the categoryPo of partially ordered sets and monotone maps.     

Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities [Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4].     

Minimal extensions of graphs to absolute retracts

A graph H is an absolute retract if for every isometric embedding h of, into a graph G an edge-preserving map g from G to H exists such that g · h is the identity map on H. A vertex v is embeddable in a graph G if G ? v is a retract of G. An absolute retract is uniquely determined by its set of embeddable vertices. We may regard this set as a metric space. We also prove that a graph (finite metric space with integral distance) can be isometrically embedded into only one smallest absolute retract (injective hull). All graphs in this paper are finite, connected, and without multiple edges.     

An axiomatic analysis of the Droz-Farny Line Theorem

We analyze an elementary theorem of Euclidean geometry, the Droz-Farny Line Theorem, from the point of view of the foundations of geometry. We start with an elementary synthetic proof which is based on simple properties of the group of motions. The proof reveals that the Droz-Farny Line Theorem is a special case of the Theorem of Goormatigh which is, in turn, a special case of the Counterpairing Theorem of Hessenberg. An axiomatic analysis in the sense of Hilbert [14] and Bachmann [2] leads to a study of different versions of the theorems (e.g., of a dual version or of an absolute version, which is valid in absolute geometry) and to a new axiom system for the associated very general plane absolute geometry (the geometry of pencils and lines). In the last section the role of the theorems in the foundations of geometry is discussed.     

A topological property of the common fixed points set of two multivalued operators

Let (X,d)$(X,d)$ be a complete metric space and absolute retract for metric spaces. We prove that the common fixed points set of two multivalued operators defined on X$X$, which have the selection property and satisfy a contraction type condition, is an absolute retract for metric spaces.     

Pointwise Remez- and Nikolskii-type inequalities for exponential sums

Let So is the collection of all n + 1 term exponential sums with constant first term. We prove the following two theorems. Theorem 1 (Remez-type inequality for $E_n$ at 0). Let $s \in \left( 0, \frac 12 \right]\,.$ There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that where the supremum is taken for all $f \in E_n$ satisfying Theorem 2 (Nikolskii-type inequality for $E_n$ ). There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that for every $a < y < b$ and $q > 0\,.$ It is quite remarkable that, in the above Remez- and Nikolskii-type inequalities, behaves like , where denotes the collection of all algebraic polynomials of degree at most n with real coefficients. Received: 4 November 1998 / in final form: 2 March 1999     

On geometric properties of functors of positive-homogenous and semiadditive functionals

We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH(X) is an absolute retract if and only if X is an open-generated compactum, and OS(X) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω₁. We investigate the softness of mappings of multiplication of monads generated by these functors.     

关于凸子集和（绝对）收缩核

讨论了线性度量空间中凸子集在什么情况下为该空间的收缩核，以及在什么情况下为绝对收缩核。     

A remark on the incomparability of two criteria for a uniform convergence of probability measures

Summary This paper connects with Theorem 3 of the author’s paper [1], in which two criteria for type (B) _d convergence ([3]) are shown to be incomparable to each other by presenting two examples. However, the statement of the theorem is not complete. In the present paper, we shall modify the statement of the theorem and give a proof by presenting a new example.     

Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space is an absolute retract if and only if X is an absolute retract of weight ?ω₁. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω₁ is homeomorphic to Iω₁. An analogous result is also proved for the cone over Iω₁. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.     

Locally free sheaves on complex supermanifolds

A classification of locally free sheaves $ \mathcal{E} $ of $ \mathcal{O} $ -modules which have a given retract gr $ \mathcal{E} $ in the terms of non-abelian 1-cohomology is given. In the case of $ \mathbb{C}{{\mathbb{P}}^{1|m }} $ , m > 0, we show that the Birkhoff–Grothendieck Theorem does not hold true. We obtain a result similar to the Barth–Van de Ven–Tyurin Theorem for projective superspaces. Furthermore, a spectral sequence which connects the cohomology with values in a locally free sheaf $ \mathcal{E} $ to the cohomology with values in its retract gr $ \mathcal{E} $ is constructed.     

Fields with continuous local elementary properties. II

This article is a direct continuation of [1], a familiarity with the results of which is desirable. We will be concerned with a proof of an analog of Theorem 3 from [1] to extend it to the case of fields in RC _E ^* (with absolutely unramified valuation rings) possessing a “large” absolute Galois Δ-group. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 262-273, May-June, 1995.     

Analytically heavy spaces: Analytic Cantor and Analytic Baire Theorems

Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and K-analyticity, thereby adding to the ‘Baire space recognition literature’ (cf. Aarts and Lutzer (1974) [1], Haworth and McCoy (1977) [43]). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak α-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by K-analytic (in particular analytic) sets that are ‘heavy’ (everywhere large in the sense of some σ-ideal). From this perspective fine-topology versions are derived, allowing a unified view of the Baire Theorem which embraces classical as well as generalized Gandy-Harrington topologies (including the Ellentuck topology), and also various separation theorems. A multiple-target form of the Choquet Banach-Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in K-analytic form.     

On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069     

On theLU factorization ofM-matrices

Summary In this paper, we give in Theorem 1 a characterization, based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices, whereL is a nonsingular lower triangularM-matrix andU is an upper triangularM-matrix. This result generalizes earlier factorization results of Fiedler and Pták (1962) and Kuo (1977). As a consequence of Theorem 1, we show in Theorem 3 that the conditionx ^T A0 ^T for somex>0, for anM-matrixA, is both necessary and sufficient forPAP ^T to admit such anLU factorization for everyn×n permutation matrixP. This latter result extends recent work of Funderlic and Plemmons (1981). Finally, Theorem 1 is extended in Theorem 5 to give a characterization, similarly based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices.Dedicated to Professor Ky Fan on his sixty-seventh birthday, September 19, 1981.Research supported in part by the Air Force Office of Scientific Research, and by the Department of Energy     

The General Solution of Abel-Type Functional Equations

The general measurable solution of (A) was found by Stamate [8]. Aczél [3] and Lajkô [6] proved that the general solution of (A) for unknown functions ψ, g, h: ? → ? are (1), (2) and (3), respectively. Filipescu [5] found the general measurable solution of (B). We establish an elementary prof for the general solution of equation (A) (Theorem 1.). Our method is suitable for finding the general solution of (B) (Theorem 2.).     

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems known to the author for the integral lattice Z² are reduced to some special cases of this generalization. Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions.AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35.