Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

On an Ulam's problem

The existence and the uniqueness (with respect to a filtration-equivalence) of a vector flowX on ? ⁿ ,n≥3, such that:

1. X has not any stationary points on ? ⁿ ;
2. all orbits ofX are bounded;
3. there exists a filtration forX are proved in the present note.

     

Verallgemeinerungen eines Satzes von H. Steinhaus

The following result is due to H. Steinhaus [20]: “If A,B?R are sets of positive inner Lebesgue measure and if the function f: R x R→R is defined by f(x,y):=x+y (x,y?R), then the interior of f(A x B) is non void”. In this note there is proved, that the theorem of H. Steinhaus remains valid, if

1. R is replaced by certain topological measure spaces X, Y and a Hausdorff space Z,
2. f is a continuous function from an open set T?X x Y into Z and satisfies a special local (respectively global) solvability condition in T,
3. A?X is a set of positive outer measure, B?Y contains a set of positive measure and A x B?T.

     

Die Endlichen Polardreiseitgeometrien

LetG be a finite group which is generated by a subsetS of involutions satisfying the theorem of the three reflections: Ifa,b,x,y,z ∈ S, ab ≠ 1 and ifabx,aby,abz are involutions, thenxyz ∈ S. Assume thatS contains three elements which generate a four-group. IfS contains four elements of which no three have a product of order two, then one of the following occurs.

1. G?PGL(2,n), n≡1 (mod 2).
2. G?PSL(2,n), n≡1 (mod 2) and n≥5.
3. G?PSU(3,16).
4. G/Z(G)?PSL(2,9) with ¦Z(G)¦=3.

     

On groups with isomorphic holomorphs

Groups with isomorphic holomorphs are said to be holomorphically isomorphic. The following problems are treated.

1. What Abelian groups have the property that their holomorphic isomorphism implies the isomorphism of the groups themselves?
2. Does holomorphic isomorphism of two groups, one of which is Abelian, imply the commutativity of the other group?

Classes of groups are selected for which the answers to these questions are positive.     

Mean dimension of $${\mathbb{Z}^k}$$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

1. (1)
   When is X isomorphic to the inverse limit of finite entropy systems?
    
2. (2)
   Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
    
3. (3)
   When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
    

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.     

Multidimensional symmetric stable processes

This paper surveys recent remarkable progress in the study of potential theory for symmetric stable processes. It also contains new results on the two-sided estimates for Green functions, Poisson kernels and Martin kernels of discontinuous symmetric α-stable process in boundedC ^1,1 open sets. The new results give explicit information on how the comparing constants depend on parameter α and consequently recover the Green function and Poisson kernel estimates for Brownian motion by passing α ↑ 2. In addition to these new estimates, this paper surveys recent progress in the study of notions of harmonicity, integral representation of harmonic functions, boundary Harnack inequality, conditional gauge and intrinsic ultracontractivity for symmetric stable processes. Here is a table of contents.

1. Introduction
2. Green function and Poisson kernel estimates

1. Estimates on balls
2. Estimates on boundedC ^1,1 domains
3. Estimates on boundedC ^1,1 open sets

1. Harmonic functions and integral representation
2. Two notions of harmonicity
3. Martin kernel and Martin boundary
4. Integral representation and uniqueness
5. Boundary Harnack principle
6. Conditional process and its limiting behavior
7. Conditional gauge and intrinsic ultracontractivity

     

On a conjecture about automorphisms of finite p-groups

Let G be a nonabelian finite p-group. A longstanding conjecture asserts that G admits a noninner automorphism of order p. In this paper, we prove that if G satisfies one of the following conditions

1. ${\mathrm{rank}(G'\cap Z(G))\neq \mathrm{rank}(Z(G))}$
2. ${\frac{Z_{2}(G)}{Z(G)}}$ is cyclic
3. C _G (Z(Φ(G))) = Φ(G) and ${\frac{Z_{2}(G)\cap Z(\Phi(G))}{Z(G)} }$ is not elementary abelian of rank rs, where r = d(G) and s = rank (Z(G)),

then G has a noninner central automorphism of order p which fixes Φ(G) elementwise.     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

Normalizers,Centralizers and Action Representability in Semi-Abelian Categories

We introduce the notion of normalizer as motivated by the classical notion in the category of groups. We show for a semi-abelian category ? that the following conditions are equivalent:

1. ? is action representable and normalizers exist in ?;
2. the category Mono(?) of monomorphisms in ? is action representable;
3. the category ?² of morphisms in ? is action representable;
4. for each category \(\mathbb {D}\) with a finite number of morphisms the category \({\mathbb {C}} ^{\mathbb {D}}\) is action representable.

Moreover, when in addition ? is locally well-presentable, we show that these conditions are further equivalent to:

1. ? satisfies the amalgamation property for protosplit normal monomorphism and ? satisfies the axiom of normality of unions;
2. for each small category \(\mathbb {D}\) , the category \({\mathbb {C}} ^{\mathbb {D}}\) is action representable.

We also show that if ? is homological, action accessible, and normalizers exist in ?, then ? is fiberwise algebraically cartesian closed.     

Δ2 degrees without Σ1 induction

In this paper, we study the structure of Turing degrees below 0′ in the theory that is a fragment of Peano arithmetic without Σ₁ induction, with special focus on proper d-r.e. degrees and non-r.e. degrees. We prove:

1. P ^? + BΣ₁ + Exp ? There is a proper d-r.e. degree.
2. P ^? +BΣ₁+ Exp ? IΣ₁ ? There is a proper d-r.e. degree below 0′.
3. P ^? + BΣ₁ + Exp ? There is a non-r.e. degree below 0′.

     

Maximal independent, analytic sets in Abelian polish groups

The paper deals with the questions:

1. whether a topological module admits maximal linearly independent subsets that are analytic
2. whether an Abelian topological group admits maximal independent subsets that are analytic
3. whether a topological field extension admits transcendence bases that are analytic.

     

Some results on the nonstationary ideal

The strength of precipitousness, presaturatedness and saturatedness of NS_κ and NS _κ ^λ is studied. In particular, it is shown that:

1. The exact strength of “ $NS_{\mu ^ + }^\lambda $ for a regularμ > max(λ, ?₁)” is a (ω,μ)-repeat point.
2. The exact strength of “NS_κ is presaturated over inaccessible κ” is an up-repeat point.
3. “NS_κ is saturated over inaccessible κ” implies an inner model with ?αo(α) =α ⁺⁺.

     

Packing seagulls

A seagull in a graph is an induced three-vertex path. When does a graph G have k pairwise vertex-disjoint seagulls? This is NP-complete in general, but for graphs with no stable set of size three we give a complete solution. This case is of special interest because of a connection with Hadwiger’s conjecture which was the motivation for this research; and we deduce a unification and strengthening of two theorems of Blasiak [2] concerned with Hadwiger’s conjecture. Our main result is that a graph G (different from the five-wheel) with no three-vertex stable set contains k disjoint seagulls if and only if

1. |V (G)|≥3k
2. G is k-connected
3. for every clique C of G, if D denotes the set of vertices in V (G)\C that have both a neighbour and a non-neighbour in C then |D|+|V (G)\C|≥2k, and
4. the complement graph of G has a matching with k edges.

We also address the analogous fractional and half-integral packing questions, and give a polynomial time algorithm to test whether there are k disjoint seagulls.     

Analoga der steinerschen konstruktion in einer absoluten geometrie

We consider an absolute geometry with the following base of axioms: Hilbert's plane axioms of incidence, order and congruence and a circle axiom. Thus no parallelism and not much continuity is involved. In this geometry the metric cannot be determined by Steiner's basic structure “fixed circle with centre”. In this work it will be proved that the following basic figures are suitable for such an absolute geometry in the sense that, after tracing any one of them, all constructions of second order can be done only with a ruler:

1. Two non-concentric circles, one of them with centre.
2. A unit-turner and a non-concentric circle without centre.
3. A circle with centreO and a line segmentA B with midpointM, the linesA B andO M being not orthogonal.
4. A circle with centre and two orthogonal lines, none of them passing through the centre.
5. A circle with centre and a distance-line (with their two branches).

In the basic structures 1, 3, 4, 5, instead of a circle with centre, a finite arc of a circle with centre or two concentric circles without centre may be taken.     

A compactness criterion for translation invariant Banach spaces of functions

Установлено, что для и нвариантных относит ельно сдвига банаховых про странствВ (классов) измеримых фу нкций на локально ком пактной группе, удовлетворяю щих некоторым дополнительным усло виям, справедлив след ующий критерий компактнос ти. Замкнутое подмножествоМ?В яв ляется компактным вВ тогда и только тогда, к огда оно удовлетворя ет условиям:

1. М ограничено вВ;
2. для каждого ?>0 сущест вуетkε?(G) такое, что ∥k*f-f∥_BfεМ;
3. для каждого ?0 существ уетh??(G) такое, что ∥hf - f ∥_BfεМ.

     

A translation plane of order 25

In this paper a new translation plane of order 25 is constructed. It has a collineation group acting on the line at infinity as a permutation group Z of order 48 with the properties:

1. Z contains a normal subgroup 1/2M of order 3 such that Z/1/2M is the direct product of an involution with a dihedral group of order 8.
2. The orbits of Z have lengths 2, 12, 12.

     

Cosmic ray nuclei of chargeZ⩾3 during the period of low solar activity

A detailed study of the composition and energy spectra of heavy nuclei of charge Z?3 in the primary cosmic rays has been made during the period of low solar activity, using two stacks of nuclear emulsions exposed in balloon flights from Fort Churchill, Canada, in June 1963. Each of the stacks was composed of 120 nuclear emulsions of three different sensitivities and was exposed at about 3·5 g. cm.^?2 of residual air for about 11.1 hr. Reliable resolution of charges of nuclei from lithium to oxygen was obtained; for heavier nuclei, charge groups were determined. From the analysis of 793 tracks of nuclei with Z?3, results on the following aspects were obtained:

1. The differential energy spectra of L (Z=3–5), M (Z=6–9) and H (Z=10–28) nuclei were measured in the energy intervel 150–600 MeV/nucleon; integral fluxes were obtained for energy >600 MeV/nucleon;
2. The energy dependence of the L/M ratio at the top of the atmosphere was determined; the ratios were obtained as 0·45 ± 0·06 and 0·29 ± 0·03 in the energy intervals of 200–575, and >575 MeV/nucleon respectively;
3. Relative abundances of individual nuclei of Li, Be, B, C, N and O at the top of the atmosphere were determined as 36, 29, 55, 100, 60 and 106 respectively in the energy interval 150–600 MeV/nucleon; corresponding values were also obtained for energy >600 MeV/nucleon.
4. The differential fluxes of multiply charged nuclei measured by us and by other investigators were used to determine the solar modulation between solar maximum to solar minimum. It was found that solar modulation of the fluxes of M and He nuclei were consistent with Rβ dependence and that the modulation parameter Δη between 1965 and 1957 was about 1·1.

The implications of these results are discussed.     

Nonhomogeneous Markov processes

We present the foundations of the theory of nonhomogeneous Markov processes in general state spaces and we give a survey of the fundamental papers in this topic. We consider the following questions:

1. The existence of transition functions for a Markov process.
2. The construction of regularization of processes.
3. The properties of right and left processes: the strict Markov property, the behavior of excessive functions, etc.
4. The relation of right and left processes with dual homogeneous processes and the application of the results of the nonhomogeneous theory to dual homogeneous processes, etc.