Sulle superficie di uno spazio euclideo an dimensioni aventi un campo tangente concircolare

Letx:M ²→E ⁿ be the immersion of a surfaceM ² in ann-dimensional Euclidean space. Letj and μ be the canonical isomorphism defined by the metricg ofM ² and by the canonical volume element ofM ², respectively. IfM ² carries a concircular tangent vector fieldX. then the following properties are proved:

1. The gaussian curvatureK ofM ² is identically zero.
2. X defines an infinitesimal homothety onM ².
3. The vector field (j ^?1 o μ) (X) is a Killing vector field.

Further ifX is alsoconcurrent (in the sense of K. Yano and B. Y. Chen [3]), then the immersion is degenerate (all the Killing-Lipchitz curvatures associated withx vanish).     

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.

     

Derivationen und Automorphismen von Algebren,in Denen Die Gleichung X2=0 Nur Trivial Lösbar Ist

Let A be a finite-dimensional algebra over a (commutative) field K of characteristic O, assume that x∈A and x²=0 implies x=0. We shall prove among others:

- The derivations and automorphisms of A are semisimple.

- If K is algebraically closed, then Der A=0 and |Aut A|<∞.

- If K=?, then Aut A (and hence Der A) is compact.

     

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

     

Elementary Abelianp-extensions of algebraic function fields

LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderp ⁿ.

1. E can be generated overF by some elementy whose minimal polynomial has the specific formT ^pn?T?z.
2. A formula for the genus ofE is given.
3. IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞).
4. We present a new example of a function fieldE/K whose gap numbers are nonclassical.

     

E-free objects inE-varieties of inverse rings

We show that for any regular ring (R, +, -), the following conditions are equivalent:

1. (R, -) is inverse.
2. (R, -) isE-solid.
3. (R, -) is locally inverse.
4. (R, -) is locallyE-solid.

We also show that there is ane-free object in eache-variety of inverse rings.     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

A Remark on non-separable solutions to the Schroeder-Bernstein Problem for Banach spaces

We show that the geometric structure of Banach spaces which are solutions to the Schroeder-Bernstein Problem is very complex. More precisely, we prove that there exists a non-separable solution E to this problem such that

1. E is isomorphic to each one of its finite codimensional subspaces.
2. E has no complemented Hereditarily Indecomposable subspace.
3. E has no complemented subspace isomorphic to its square.
4. E has no non-trivial divisor.

     

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.

     

Analysis of bounded sets in a topological tensor product

The aim of this paper is to investigate the nature of bounded sets in a topological ∈-tensor product EX_∈* F of any two locally convex topological vector spaces E and F over the same scalar field K. Next, we apply the results of this investigation to the study of each of the following:

1. Totally summable families in EX_∈*F;
2. ∈-tensor product of DF-spaces;
3. Topological nature of the dual of E X_∈*F, where E and F are strong duals of Banach spaces;
4. Properties of bounded sets in an ∈-tensor product of metrizable spaces.

Forπ-tensor product, the result corresponding to (b) is well known (see Grothendieck¹) that if E and F are DF-spaces then EX_π* F and EX_π* F are DF-spaces and that the strong topology on the topological dual (EX_π*F)′, which equals the space of continuous bilinear forms on EXF, coincides with the bibounded topology. We study each of the problems from (a) to (d) for ∈-tensor products. For terminology, notations and the well-known results in the theory of topological vector spaces and the topological tensor products we refer to [1–11]. However, for convenience in presentation of the results of our investigation we give a brief survey of notations and fundamental theorems which are needed throughout this paper.     

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 ).} $

     

Invariants of Lie superalgebras acting on associative algebras

LetR be a semiprime algebra over a fieldK acted on by a finite-dimensional Lie superalgebraL. The purpose of this paper is to prove a series of going-up results showing how the structure of the subalgebra of invariantsR ^Lis related to that ofR. Combining several of our main results we have: Theorem: Let R be a semiprime K-algebra acted on by a finite-dimensional nilpotent Lie superalgebra L such that if characteristic K=p then L is restricted and if characteristic, K=0 then L acts on R as algebraic derivations and algebraic superderivations.

1. If R^L is right Noetherian, then R is a Noetherian right R^L-module. In particular, R is right Noetherian and is a finitely generated right R^L-module.
2. If R^L is right Artinian, then R is an Artinian right R^L-module. In particular, R is right Artinian and is a finitely generated right R^L-module.
3. If R^L is finite-dimensional over K then R is also finite-dimensional over K.
4. If R^L has finite Goldie dimension as a right R^L-module, then R has finite Goldie dimension as a right R-module.
5. If R^L has Krull dimension α as a right R^L-module, then R has Krull dimension α as a right R^L-module. Thus R has Krull dimension at most α as a right R-module.
6. If R is prime and R^L is central, then R satisfies a polynomial identity.
7. If L is a Lie algebra and R^L is central, then R satisfies a polynomial identity.

We also provide counterexamples to many questions which arise in view of the results in this paper.     

On the number of non-isomorphic subgraphs

Let $\mathcal{K}$ be the family of graphs on ω₁ without cliques or independent subsets of sizew ₁. We prove that

1. it is consistent with CH that everyGε $\mathcal{K}$ has 2^ω many pairwise non-isomorphic subgraphs,
2. the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω₁ either G?G[A] orG?G[B],
3. the failure of (*) is consistent with ZFC.

     

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.

     

Constructing a perfect matching is in random NC

We show that the problem of constructing a perfect matching in a graph is in the complexity class Random NC; i.e., the problem is solvable in polylog time by a randomized parallel algorithm using a polynomial-bounded number of processors. We also show that several related problems lie in Random NC. These include:

1. Constructing a perfect matching of maximum weight in a graph whose edge weights are given in unary notation;
2. Constructing a maximum-cardinality matching;
3. Constructing a matching covering a set of vertices of maximum weight in a graph whose vertex weights are given in binary;
4. Constructing a maximums-t flow in a directed graph whose edge weights are given in unary.

     

Coherence classes of ideals of normal lattices with applications to C(X)

Given a topological space X, Jenkins and McKnight have shown how ideals of the ring C(X) are partitioned into equivalence classes — called coherence classes — defined by declaring ideals to be equivalent if their pure parts are identical. In this paper we consider a similar partitioning of the lattice of ideals of a normal bounded distributive lattice. We then apply results obtained herein to augment some of those of Jenkins and McKnight. In particular, for Tychonoff spaces, new results include the following:

1. all members of any coherence class have the same annihilator
2. every ideal is alone in its coherence class if and only if the space is a P-space.

     

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.     

On projective planes admitting elations and homologies

We consider projective planes Π of ordern with abelian collineation group Γ of ordern(n?1) which is generated by (A, m)-elations and (B, l)-homologies wherem =AB andA εl. We prove

1. Ifn is even thenn=2^e and the Sylow 2-subgroup of Γ is elementary abelian.
2. Ifn is odd then the Sylow 2-subgroup of Γ is cyclic.
3. Ifn is a prime then Π is Desarguesian.
4. Ifn is not a square thenn is a prime power.

     

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.     

Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node $v$ equal to $2\pi /d(v)$ . We show:

1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area.
2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution.
3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area.

Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.