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.

     

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).

     

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

     

Convergence of subdiagonal Padé approximations of C 0-semigroups

Let ${(r_{n})_{n \in \mathbb{N}}}$ be the sequence of subdiagonal Padé approximations of the exponential function. We prove that for ?A the generator of a uniformly bounded C ₀-semigroup T on a Banach space X, the sequence ${(r_{n}(-t A))_{n \in \mathbb{N}}}$ converges strongly to T(t) on D(A ^α ) for ${\alpha>\frac{1}{2}}$ . Local uniform convergence in t and explicit convergence rates in n are established. For specific classes of semigroups, such as bounded analytic or exponentially γ -stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.     

Boolean approach to planar embeddings of a graph

The purpose of this paper which is a sequel of “ Boolean planarity characterization of graphs ” [9] is to show the following results.

1. Both of the problems of testing the planarity of graphs and embedding a planar graph into the plane are equivalent to finding a spanning tree in another graph whose order and size are bounded by a linear function of the order and the size of the original graph, respectively.
2. The number of topologically non-equivalent planar embeddings of a Hamiltonian planar graphG is τ(G)=2 ^c(H)?1, wherec (H) is the number of the components of the graphH which is related toG.

     

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 ⁺⁺.

     

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.

     

Complex Banach spaces whose duals areL 1-spaces

We prove that for a complex Banach spaceA the following properties are equivalent:

1. A ^* is isometric to anL ₁(μ)-space;
2. every family of 4 balls inA with the weak intersection property has a non-empty intersection;
3. every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

     

Patch ideals and Peterson varieties

Patch ideals encode neighbourhoods of a variety in GL _n /B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen–Macaulay and Gorenstein. Consequently, we

— combinatorially describe the singular locus of the Peterson variety;

— give an explicit equivariant K-theory localization formula; and

— extend some results of [B. Kostant ‘96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties.

We conjecture that the tangent cones are Cohen–Macaulay, and that their h-polynomials are nonnegative and upper-semicontinuous. Similarly, we use patch ideals to briey analyze other examples of torus invariant subvarieties of GL _n /B, including Richardson varieties and Springer fibers.     

Convex rearrangements of stable processes

The asymptotic behavior of convex rearrangements for smooth approximations of random processes is considered. The main results are.

- the relations between the convergence of convex rearrangements of absolutely continuous on [0, 1] functions and the weak convergence of its derivatives considered as random variables on the probability space {[0, 1], ß_{[0, 1]}, λ} are established:

- a strong law of large numbers for convex rearrangements of polygonal approximations of stable processes with the exponent α, 1<α≦2, is proved:

- the relations with the results by M. Wshebor (see references) on oscillations of the Wiener process and with the results by Yu. Davydov and A. M. Vershik (see references) on convex rearrangements of random walks are discussed.

     

Flat modules and rings finitely generated as modules over their center

A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:

1. A is a right or left distributive semiprime ring;
2. for any maximal idealM of a subringR central inA, the ring of quotientsA _M is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;
3. all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.

     

Rational approximations by implicit Runge-Kutta schemes

Ehle [3] has pointed out that then-stage implicit Runge-Kutta (IRK) methods due to Butcher [1] areA-stable in the definition of Dahlquist [2] because they effect the operationR(Ah) whereR(μ) is the diagonal Padé approximation toe ^µ. The purpose of this note is to point out that ifR(μ)=P(μ)/Q(μ) is a rational polynomial whosen poles are distinct and nonzero, and if degreeP(μ)≦degreeQ(μ)=n, then ann-stage IRK method applied toy=A y can be used for the operation $$y^{n + 1} = R(Ah)y^n $$ This will no longer be of order 2n, nor necessarily the same order as the approximation ofR(Ah) toe ^Ah. However, if any particularly useful integration formsR can be found, they can be performed by the IRK method.     

Regular sets of matrices and applications

SupposeA ₁,...,A _s are (1, - 1) matrices of order m satisfying 1 $$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$ 2 $$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$ 3 $$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m } $$ 4 $$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$ Call A₁,…,A _s ,a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in “New Hadamard matrices and conference matrices obtained via Mathon’s construction”, Graphs and Combinatorics, 4(1988), 355-377. In this paper, we prove that

1. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
2. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
3. if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
4. an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
5. Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
6. anOD(4mp;ms₁,…,ms_u whenever anOD (4p;s₁,…,s_u)exists and s = 2p
7. a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 2c

This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.     

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.     

On the completion of excellent rings in characteristicp>0

Si considera il seguente problema posto da Grothendieck (E.G.A.): SeA è un anello eccellente edm un ideale diA, (A, m) ^{^}=m-adico completamento diA è eccellente? Si mostra che la risposta è positiva nei seguenti casi:

1. A=algebra di tipo finito su un DVR completo di caratteristicap>0;
2. A=algebra di tipo finito su un DVRC contenente un corpok di caratteristicap>0 e finito suk [C ^p ] oppure tale che:

1. per ogni sottocampok′ dik contenentek ^p tale che [k:k′]<∞, il modulo universale finito dei differenzialiD _{k′ (C)} esiste;
2. il corpo residuoK diC soddisfa rank _KK ? _K/k <∞
3. C ha una Der-base.

     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

Limit linear series: Basic theory

In this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type. The technically much simpler special case of 1-dimensional series was developed by Beauville [2], Knudsen [21–23], Harris and Mumford [17], in the guise of “admissible covers”. It has proved very useful for studying the Moduli space of curves (the above papers and Harris [16]) and the simplest sorts of Weierstrass points (Diaz [4]). With our extended tools we are able to prove, for example, that:

1. The Moduli spaceM _g of curves of genusg has general type forg≧24, and has Kodaira dimension ≧1 forg=23, extending and simplifying the work of Harris and Mumford [17] and Harris [16].
2. Given a Weierstrass semigroup Γ of genusg and weightw≦g/2 (and in a somewhat more general case) there exists at least one component of the subvariety ofM _g of curves possessing a Weierstrass point of semigroup Γ which has the “expected” dimension 3g-2?w (and in particular, this set is not empty).
3. The fundamental group of the space of smooth genusg curves having distinct “ordinary” Weierstrass points acts on the Weierstrass points by monodromy as the full symmetric group.
4. Ifr andd are chosen so that $$\rho : = g - (r + 1)(g - d + r) = 0,$$ then the general curve of genusg has a certain finite number ofg _d ^r’ s [15, 20]. We show that the family of all these, allowing the curve to vary among general curves, is irreducible, so that the monodromy of this family acts transitively. If4=1, we show further that the monodromy acts as the full symmetric group.
5. Ifr andd are chosen so that $$\rho = - 1,$$ then the subvariety ofM _g consisting of curves posessing ag _d ^r has exactly one irreducible component of codimension 1.
6. For anyr, g, d such that ρ≦0, the subvariety ofM _g consisting of curves possessing ag _d ^r has at least one irreducible component of codimension—ρ so long as $$\rho \geqq \left\{ \begin{gathered} - g + r + 3 (r odd) \hfill \\ - \frac{r}{{r + 2}}g + r + 3 (r even). \hfill \\ \end{gathered} \right.$$

In this paper we present the basic theory of “limit linear series” necessary for proving these results. The results themselves will be taken up in our forthcoming papers [8-12]. Simpler applications, not requiring the tools developed in this paper but perhaps clarified by them, have already been given in our papers [5-7].     

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).     

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.     

Flow of a weakly ionized gas between parallel cold walls with parabolic velocity profile

Ambipolar diffusion between flat cold insulating walls of a weakly ionized gas which flows in the direction parallel to the walls with parabolic velocity profile is investigated theoretically. It has been found that:

1. the patched velocity of linear and nonlinear regions tends to 1/√2 of the thermal velocity;
2. the thickness of the nonlinear region with parabolic velocity profile is found to be less than that of Shioda who considered uniform streaming (J. Phys. Soc. Japan, 1969,29, 197); and
3. the number density and the electric potential approximations in the sheath edge do not depend uponx, the coordinate in the streaming direction.