Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

Two cardinal versions of diamond

Roughly speaking, ◇_K,λ asserts the existence of a sequence of size <κ sets that captures every subset ofλ on a stationary set. The paper is devoted to the study of ◇_K,λ and related principles, which are for instance obtained by considering sequences of larger sets, or by requesting the simultaneous capture of many subsets ofλ. Our main result is that ◇_K,λ holds in caseλ>2^<K .     

On the improperness sets of families of linear differential systems

We consider families of linear differential systems continuously depending on a real parameter with continuous (or piecewise continuous) coefficients on the half-line. The improperness set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are Lyapunov improper. We show that a subset of the real axis is the improperness set of some family if and only if it is a G _δσ -set. The result remains valid for families in which the matrices of the systems are bounded on the half-line. Almost the same result holds for families in which the parameter occurs only as a factor multiplying the system matrix: their improperness sets are the G _δσ -sets not containing zero. For families of the last kind with bounded coefficient matrix, we show that their improperness set is an arbitrary open subset of the real line.     

Sums of lengtht in Abelian groups

LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB _t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB _t (modB _t) forB _tσ and writeB _t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B _t (modK)|≧|B (modK)| for any finite subgroupK ofG.     

A ramsey theorem for trees,with an application to Banach spaces

LetS be the binary tree of all sequences of 0’s and 1’s. A chain ofS is any infinite linearly ordered subset. Letℋ be an analytic set of chains, we show that there exists a binary subtreeS’ ofS such that either all chains ofS’ lie inℋ or no chain ofS’ lies inℋ. As an application, we prove the following result on Banach spaces: If (x _s) _sɛs is a bounded sequence of elements in a Banach spaceE, there exists a subtreeS’ ofS such that for any chainβ ofS’ the sequence (x _s ) _{s ∈β} is either a weak Cauchy sequence or equivalent to the usuall ¹ basis.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceS _n of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS _n is viewed as a subset of the moduli spaceA ₂ of principally polarized abelian surfaces. On the other hand the Humbert surfaceH _Δ of invariant Δ is defined as a subvariety ofA ₂(C), the set of C-valued points ofA ₂. The purpose of this paper is to releaseS _n withH _Δ.     

Families of cuts with the MFMC-property

A family ℱ of cuts of an undirected graphG=(V, E) is known to have the weak MFMC-property if (i) ℱ is the set ofT-cuts for someT⊆V with |T| even, or (ii) ℱ is the set of two-commodity cuts ofG, i.e. cuts separating any two distinguished pairs of vertices ofG, or (iii) ℱ is the set of cuts induced (in a sense) by a ring of subsets of a setT⊆V. In the present work we consider a large class of families of cuts of complete graphs and prove that a family from this class has the MFMC-property if and only if it is one of (i), (ii), (iii).     

Admissibility spectra throughω 1

Jensen showed that any countable sequenceA ofA-admissibles is the initial part of the admissibility spectrum of a real. We considerω ₁-long sequences, to be realized byB ⊆ω ₁. The problem is similar to finding a club subset of a stationary set. We investigate when such aB can be forced and when one is already inV.     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

Major subsets ofα-recursively enumerable sets

Let s2cf(α), s2p(α) and ts2p(α) denote the Σ₂-confinality, Σ₂-projectum and the tame Σ₂-projectum of an admissible ordinalα. We show that if s2cf(α)<s2p(α), then noα-recursively enumerable set (α-r.e.) with complement of order type less than ts2p(α) can have a major subset. As a corollary, if s2cf(α)<s2p(α), then no hyperhypersimpleα-r.e. set can have a major subset.     

A combinatorial condition for the existence of polyhedral 2-manifolds

LetP denote a polyhedral 2-manifold, i.e. a 2-dimensional cell-complex inR ^d (d≧3) having convex facets, such that set (P) is homeomorphic to a closed 2-dimensional manifold. LetE be any subset of odd valent vertices ofP, andc _E its cardinality. Then for the numberc _P(E) of facets containing a vertex ofE the inequality 2c _P(E)≧c_E+1 is proved. This local combinatorial condition shows that several combinatorially possible types of polyhedral 2-manifolds cannot exist.     

A theorem on level lines of continuous functions

A property of a continuous functionf(x), x ∈ E ₂, similar to the classical intermediate value property is established. Namely, let a Jordan compactJ ⊂ E ₂ be the domain of definition off. Then, for each parametrizationx(t), 0≦t≦T,x(0)=x(T), of the boundary Fr(J) ofJ there exists a unique real λ and a unique connected component Φ of the level set {x ∈ J: f(x)=λ} with the following property: any connected subset Ω ofJ containing “opposite” points of Fr(J) (i.e. pointsx(t′) andx(t″) such thatt″−t′=T/2) has a common element with Φ.     

Mappings of Baire spaces into function spaces and Kadeč renorming

Assuming that there exists in the unit interval [0, 1] a coanalytic set of continuum cardinality without any perfect subset, we show the existence of a scattered compact Hausdorff spaceK with the following properties: (i) For each continuous mapf on a Baire spaceB into (C(K), pointwise), the set of points of continuity of the mapf: B → (C(K), norm) is a denseG _δ subset ofB, and (ii)C(K) does not admit a Kadeč norm that is equivalent to the supremum norm. This answers the question of Deville, Godefroy and Haydon under the set theoretic assumption stated above.     

Splitting recursively enumerable subalgebras in recursive Boolean algebras

A Boolean algebraB= is recursive ifB is a recursive subset of ω and the operations Λ, v and ┌ are partial recursive. A subalgebraC ofB is recursive an (r.e.) ifC is a recursive (r.e.) subset of B. Given an r.e. subalgebraA, we sayA can be split into two r.e. subalgebrasA ₁ andA ₂ if (A ₁ ∪A ₂)^*=A andA ₁ ∩A ₂={0, 1}. In this paper we show that any nonrecursive r.e. subalgebra ofB can be split into two nonrecursive r.e. subalgebras ofB. This is a natural analogue of the Friedberg's splitting theorem in ω recursion theory.     

Integrability and closest approximation representations

Summary Suppose U is a set,F is a field of subsets of U, andp _AB is the set of all real-valued bounded finitely additive functions defined onF. This paper consists of two main parts. In the first, a previously given (seeRiv. Math. Univ. Parma, (3),2 (1973), pp. 251–276) notion of a subset ofp _AB defined by certain closure properties and called a C-set, is considered, and those C-sets that are linear spaces are characterized. Now, suppose γ is a function whose domain isF and whose range is a collection of number sets with bounded union. The set,J _γ, of all elements ofp _AB with respect to which γ is integrable, for refinements of subdivisions, is a C-set and a linear space (seeRend. Sem. Mat. Univ. Padova,52 (1974), pp. 1–24). The second part of this paper concerns, for μ inp _AB and nonnegative-valued, a representation of the element ofJ _γ closest to μ with respect to variation norm. Entrata in Redazione il 14 giugno 1977.     

Criteria for selection of response variables and the asymptotic properties in a multivariate calibration

Summary Let a set ofp responsesy=(y ₁,...y _p )′ has a multivariate linear regression on a set ofq explanatory variablesx=(x ₁,...x _q )′. Our aim is to select the most informative subset of responses for making inferences about an unknownx from an observedy. Under normality ony, two selection methods, based on the asymptotic mean squared error and on the Akaike's information criterion, are proposed by Fujikoshi and Nishii (1986,Hiroshima Math. J.,16, 269–277). In this paper, under a mild condition we will derive the cross-validation criterion and obtain the asymptotic properties of the three procedures.     

A fixed point theorem for a class of star-shaped sets inc 0

A subsetK ofc ₀ is coordinatewise star-shaped (c.s.s.) if there exists a center pointx ∈K such that fory ∈K andz ∈c ₀, ifz is coordinatewise betweenx andy thenz ∈K. We prove that a weakly compact c.s.s. subset ofc ₀ has the fixed point property for nonexpansive mappings and that a fixed point for such a mapping can be obtained in a constructive manner. Research of the first two authors was partially supported by NSF Grant MCS78-01344 and of the last author by MCS78-01501.     

Boundaries of random walks on graphs and groups with infinitely many ends

Consider an irreducible random walk {Z _n} on a locally finite graphG with infinitely many ends, and assume that its transition probabilities are invariant under a closed group Γ of automorphisms ofG which acts transitively on the vertex set. We study the limiting behaviour of {Z _n} on the spaceΩ of ends ofG. With the exception of a degenerate case,Ω always constitutes a boundary of Γ in the sense of Furstenberg, and {Z _n} converges a.s. to a random end. In this case, the Dirichlet problem for harmonic functions is solvable with respect toΩ. The degenerate case may arise when Γ is amenable; it then fixes a unique end, and it may happen that {Z _n} converges to this end. If {Z _n} is symmetric and has finite range, this may be excluded. A decomposition theorem forΩ, which may also be of some purely graph-theoretical interest, is derived and applied to show thatΩ can be identified with the Poisson boundary, if the random walk has finite range. Under this assumption, the ends with finite diameter constitute a dense subset in the minimal Martin boundary. These results are then applied to random walks on discrete groups with infinitely many ends.     

On dentability and the Bishop-Phelps property

It is shown that for Banach spaces the Radon-Nikodym property and the Bishop-Phelps property are equivalent. Using similar techniques, we prove that ifC is a bounded, closed and convex subset of a Banach space such that every nonempty subset ofC is dentable, then the strongly exposing functionals ofC form a denseG _δ-subset of the dual.