Laguerre-Ebenen mit Symmetrien an Punktepaaren und Zykeln

Let ∞ be a point of a Laguerre plane, such that

1. For any cycle containing ∞ there exists an automorphism of order 2 whose set of fixed points is exactly z.
2. For any point X, not parallel to ∞, there exists an automorphism of order 2 whose set of fixed points is exactly {∞,X}.

Then the give Laguerre plane is a Miquelian one of characteristik ≠ 2.     

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.

     

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.     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $

     

Tilings whose members have finitely many neighbors

Let ? be a tiling of the plane such that each tile of ? meets at most finitely many other tiles. Then exactly one of the following must occur:

1. Uncountably many boundary points of ? belong to no nondegenerate edge of ?, hence ? has uncountably many singular points; or
2. Every boundary point of ? belongs to a nondegenerate edge of ?, moreover, ? has no singular points.

Furthermore, ifS is the set of singular points of ? andW={t:t∈bdry ? andt belongs to no nondegenerate edge of ?}, thenS=clW.     

Scalar-type spectral operators and holomorphic semigroups

We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold.

1. A generates a uniformly bounded holomorphic semigroup {e?^zA}_Re(z)≥0.
2. If \(F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr\) , then {‖F_N‖} _N=1 ^∞ is uniformly bounded on [0,∞) and, for all x in X, the sequence {F_N(s)x} _N=1 ^∞ converges pointwise on [0, ∞) to a vector-valued function of bounded variation.

The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup {e^?zA}_Re(z)≥0 generated by A, as follows. $$\frac{1}{2}(E\{ s\} )x + (E[0,s)) x = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {\frac{{\sin (sr)}}{r}} e^{irA} x\frac{{dr}}{\pi }$$ for any x in X.     

Supercritical Holes for the Doubling Map

For a map \({S : X \to X}\) and an open connected set (= a hole) \({H \subset X}\) we define \({\mathcal{J}_H(S)}\) to be the set of points in X whose S-orbit avoids H. We say that a hole H ₀ is supercritical if

1. for any hole H such that \({\overline{H}_0 \subset H}\) the set \({\mathcal{J}_H(S)}\) is either empty or contains only fixed points of S;

1. for any hole H such that \({\overline{H} \subset H_0}\) the Hausdorff dimension of \({\mathcal{J}_H(S)}\) is positive.

The purpose of this note is to completely characterize all supercritical holes for the doubling map Tx =  2x mod 1.     

Platitude d'une adherence schematique et lemme de Hironaka generalise

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied:

1. O _Y,y is reduced.
2. f is universally open at the generic points of the components of X_y which contain x.
3. For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to X_y, we have dim_x, (X_y')=dim_x(X_y)=d.
4. X_y is reduced at the generic points of the components of X_y which contain x and (X_y)_red is geometrically normal over K(y) in x.

Then there exist an open neighbourhood U of x in X and a subscheme U₀ of U which have the same underlying space as U such that f₀:U₀\arY is normal (i.e. f₀ is a flat morphism whose geometric fibers are normal).     

A new proof of a theorem of Dembowski

It was shown by P.Dembowski [1;Satz 3] that any finite semiaffine plane(=FSAP) is of the types:

1. A finite affine plane,
2. A finite projective plane with one line and all its points except one deleted,
3. A finite projective plane with one point deleted,
4. A finite projective plane.

This was established by using the results obtained for natural parallelisms of incidence structures. The purpose of this note is to give a new proof based on purely combinatorial arguments.     

On Generalized Affine Rotation Surfaces

From group-theoretical point of view, we discuss affine rotation surfaces in R³ and projective rotation surfaces in RP³. These pave a way toward generalized affine rotation surfaces in R³. We will follow closely the modern approach introduced by Nomizu in the study affine differential geometry [N]. In this paper, we have the following results on generalized affine rotation surfaces:

1. Many nice properties showing duality between x² + y² = g²(z) and x² ? y² = g²(z).
2. In this set, any surface with zero Pick invariant is a quadratic surface.
3. Excluding the Caley surface z = xy ? x³/3 and the surface z = xy + log x, any affine unimodular homogeneous surface belongs to this set.
4. In this set, the following surfaces are characterized by some affine invariants: $${\matrix{x^{2}+\epsilon\ y^{2}=z \cr \qquad \qquad \qquad \qquad \qquad \qquad \qquad y^{2}=z(x+\epsilon\ (z+a)^{2/3}(z+6a)),\cr \qquad \qquad \qquad \ \ \ y^{2}=z(x+\epsilon\ z^{3}),\cr \qquad \qquad \ \ \ \ \ \ y^{2}=x+\epsilon\ z^{2/3},\cr \qquad \qquad \ \ \ \ \ \ \ y^{2}=x+\epsilon\ z^{-2/3},\cr \qquad \qquad \ \ \ y^{2}=x+\epsilon\ {\rm log}z,\cr}} $$ where ? represents 1 or ?1 in this paper.

     

Topological conditions enabling use of harris methods in discrete and continuous time

This paper describes the role of continuous components in linking the topological and measuretheoretic (or regenerative) analysis of Markov chains and processes. Under Condition $\mathcal{T}$ below we show the following parallel results for both discrete and continuous time models:

1. when the model is open set irreducible it is ?-irreducible;
2. under (i), the measure-theoretic classification of the model as Harris recurrent or positive Harris recurrent is equivalent to a topological classification in terms of not leaving compact sets or of tightness of transition kernels;
3. under (i), the ‘global’ classification of the model as transient, recurrent or positive recurrent is given by a “local’ classification of any individual reachable point;
4. under (i), every compact set is a small set, so that through the Nummelin splitting there is pseudo-regeneration within compact sets, and compact sets are ‘test sets’ for stability;
5. even without irreducibility, there is always a Doeblin decomposition into a countable disjoint collection of Harris sets and a transient set. We conclude with a guide to verifying Condition $\mathcal{T}$ and indicate that it holds under very mild constraints for a wide range of specific models: in particular a ?-irreducible Feller chain satisfies Condition $\mathcal{T}$ provided only that the support of ? has nonempty interior.

     

Space of operators acting from one banach lattice to another

In this note we construct a pair of Banach lattices X and Y, which have the following properties:

1. X is not order isomorphic to an AL-space,
2. Y is not order isomorphic to an AM-space,
3. for any continuous linear operator T:X → Y there exists a modulus ¦T¦: X → Y.

This example refutes the conjecture of Cartwright-Lotz, saying that the negation of at least one of the conditions a) or b) is necessary for the validity of c).     

Standard completions for quasiordered sets

A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators:

1. V-distributive completions,
2. Completely distributive completions,
3. A-completions (i.e. standard completions which are completely distributive algebraic lattices),
4. Boolean completions.

Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ?, then Q must be a Boolean lattice and ? its MacNeille completion.     

Über Deformationen von analytischen Abbildungskeimen

The notion of deformations of germs of k-analytic mappings generalizes the one of deformations of germs of k-analytic spaces. Using algebraic terms, we prove:

1. The morphism f: A→B of analytic algebras is rigid, iff it is infinitesimally rigid. Moreover, this is equivalent to Ex_A (B,B)=0. This theorem generalizes a result of SCHUSTER [11].
2. Let A be a regular analytic algebra. Then f is rigid iff there exists a rigid analytic algebra B_o such that f is equivalent to the canonic injection A→A?B_o.
3. If f is “almost everywhere” rigid or smooth, then the injection Ext _B ^l (Ω_B|A, Bⁿ)→Ex_A(B, Bⁿ) is an isomorphism.

     

Cyclic algebras,complete fields,and crossed products

Theorem 1

Let q=char(k). Let M be a subfield of D which is Galois over K of degree m with Galois group H.

1. If q/m then H has a normal q-Sylow subgroup.
2. Iq q ? m then H is an abelian group with one or two generators, an extension of a cyclic group by a cyclic group of order e where k contains a primitive e-th root of unity.

Letk(X) be the generic division ring overk of indexn as defined by Amitsur.

Theorem 2

If n is divisible by the square of a prime p≠char(k) and k does not contain a primitive p-th root of unity, then k(X) is not a crossed product.     

Les F-reguliers a gauche

1. The concept of left F-regular semigroups was first defined by Batbedat at the Oberwolfach meeting in 1981. It generalizes the notion of F-regular semigroup introduced by Edwards [4], itself a generalization of the F-inverse semigroups defined by McFadden/O’Carroll [6]
2. In the present paper we generalize the results of [4] and [6] by defining two preorders and ? on a monoid S with a distinguished band E, as follows: iff x=ay for some a∈E xδy iff x=yb for some b∈E
3. When S is regular orthodox and E=E(S), is the preorder of [1] p. 29 and is the order of [1] p. 31 (the order of [4]): in fact is the natural partial order introduced by Nambooripad [7].
4. In b), we define the relation Σ on S: xΣy iff exe=eye for some e∈E Then we consider the congruence σ generated by Σ.
5. DEFINITION. S is left F_E-monoid if each σ-class contain a greatest element with respect to .
6. PARTICULAR CASES. When S is regular, S is left F_E-regular. When S is regular orthodox and E=E(S), S is left F-regular.
7. We describe the structure of left F-regular semigroups like in [1], [2], [4] and [6]. Note that every left F-regular semigroup is a gammasemigroup [3]
8. Particular Cases (gamma morphism) and applications (congruences).

     

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.

     

Analytische Fortsetzbarkeit von zufälligen Potenzreihen bezüglich abhängiger Zufallsvariabler

Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X _n ⁴ )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:

1. дляf _x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
2. для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
3. для по крайней мере од ного из рядовf _x′ илиf _x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa _n .

     

Über die Glattheit von Quotientenabbildungen

Let G be a semisimple algebraic group acting on a factorial Gorenstein algebra S. Let X:=Spec S, Y:=Spec S^G and π:X→Y be the quotient map. The main results are:

1. Let x be a smooth point of X whose orbit has maximal dimension and such that π(x) is a smooth point of Y. Then π is smooth at x.
2. Let S be positively graded and let χ_S(t) be its generating function which is a rational function. Then: deg χ_S≦deg \(X_{S^G } \) .

     

Certain classes of sets in Banach spaces and a topological characterization of operators of type RN

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:

1. T is an operator of type RN;
2. for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
3. for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.

We note that the implication I)?2), is apparently new even if the operator T is weakly compact.