Partition relations for uncountable ordinals

Partition relations of the form α→(α,m)², where α is an ordinal andm is a positive integer, are considered. Let κ be a cardinal. The following are proved: If κ is singular and 2^K=K ⁺ then (K⁺)²?((K⁺)²,3)². If κ is a strong limit cardinal, then², iff ((cfκ)²→((cfκ)²,m)². If κ is regular and K²→(K²,3)², then the κ-Souslin hypothesis holds. If K^ω<α+ and cfα=cfκ>ω, then α?(α,3)².     

Further results on the reverse-order law

An explicit expression is obtained for a pair of generalized inverses (B^?,A^?) such that B^?A^?=(AB)⁺_MN, and a class of pairs (B^?,A^? of this property is shown. A necessary and sufficient condition for (AB)^? to have the expression B^?A^? is also given.     

On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L¹ or the real Hardy spaces defined on ℝⁿ, where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H¹(ℝ) into L¹(ℝ) and from L¹(ℝ) into weak –L¹(ℝ). We also prove that the double maximal Riemann operator is bounded from the hybrid Hardy spaces H^(1,0)(ℝIsup2), H^(0,1)(ℝ²⁾ into weak –L¹(ℝ²). Hence pointwise Riemann summability of Fourier integrals of functions in H^(1,0) ∪ H^(0,1)(ℝ²) follows almost everywhere.The maximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.     

New examples of Cantor sets in <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> that are not <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-minimal

Although every Cantor subset of the circle (S¹) is the minimal set of some homeomorphism of S¹, not every such set is minimal for a C¹ diffeomorphism of S¹. In this work, we construct new examples of Cantor sets in S¹ that are not minimal for any C¹-diffeomorphim of S¹.     

Weak Covering at Large Cardinals

We show that weakly compact cardinals are the smallest large cardinals k where k⁺ < k⁺ is impossible provided 0^# does not exist. We also show that if k^+Kc < k⁺ for some k being weakly compact (where K^c is the countably complete core model below one strong cardinal), then there is a transitive set M with M ? ZFC + “there is a strong cardinal”.     

On pseudoinverses of operator products

The analytical structure of the Moore-Penrose pseudoinverse of the product ab of any two operators over finite-dimensional unitary spaces is studied. The existence of the unique representation of the form (ab)⁺=b⁺(h+g)a⁺ is proved. Here h:= (a⁺abb⁺)⁺ is an (oblique) projector and g is an operator with a number of special properties. In particular, h+g is a projector, g is orthogonal to h in some metric, and g³=0. A necessary and sufficient condition for the case (ab)⁺=b⁺ha⁺ is established. This case contains the classical one (ab)⁺=b⁺a⁺ (the reverse-order law). For the latter a new necessary and sufficient condition is given.     

Derivations On a C∞-ring

The main purpose of this paper is to provide several results on objects lying between differential geometry and algebraic geometry such as C^∞-rings and derivations on a C^∞-ring. A C^∞-ring is defined as a set with operations by C^∞-functions on Euclidean spaces. A derivation on a C^∞-ring is defined in two method, an ?-derivation and a C^∞-derivation. The main result of this paper is to show that any ?-derivation is a C^∞-derivation for some classes of C^∞-rings (Theorems 3.1, 3.2).     

A new upper bound on the cyclic chromatic number

A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number χ^c. Let Δ^* be the maximum face degree of a graph. There exist plane graphs with χ^c = ?3/2 Δ^*?. Ore and Plummer [ 5 ] proved that χ^c ≤ 2, Δ^*, which bound was improved to ?9/5, Δ^*? by Borodin, Sanders, and Zhao [ 1 ], and to ?5/3,Δ^*? by Sanders and Zhao [ 7 ]. We introduce a new parameter k^*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that χ^c ≤ max {Δ^* + 3,k^* + 2, Δ^* + 14, 3, k^* + 6, 18}, and if Δ^* ≥ 4 and k^* ≥ 4, then χ^c ≤ Δ^* + 3,k^* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Comparison of Two Generalized Trigonometric Integrals

The P ²-integral of James is compared with the T ²-integral. A trigonometric series convergent almost everywhere to a function which is T ²-integrable but not P ²-integrable is constructed.     

THE FUNDAMENTAL GROUP IN FIBRATIONS

A generalized Mayer-Vietoris sequence involving crossed homomorphisms is established and the construction is applied to the homotopy sequence of the CW-pair (X.X¹) to relate the homotopy sequences of (X.X¹) and the fibre bundle F → E → X in low dimensions. If there is a partial cross-section of E → X over X², the classical form, π₁ E ～ π₁ [xtilde] π₁ F as a semidirect product, results. In case there is no extension over X² of any cross-section of the restricted bundle χ:π₂ (x², x¹) → X¹ the corresponding obstruction map X_E:π₂(x²,x¹) → π₁F is non-trivial and in case F → E → X is an SO(n)-bundle (n ≥ 3), χ_E maps into a subgroup of the centre, Z(π₁ F), of order at most 2.     

Internal conversion coefficients of E2 transitions

An analysis of K-conversion coefficients of E2 transitions of the type 2⁺→0⁺, 0⁺→2⁺, 4⁺→2⁺, 6⁺→4⁺, and 8⁺→6⁺ together with their methods of determination has been carried out. The ratios of experimental to theoreticalα _k values are plotted against the atomic weights and it is seen that there is no similar behaviour among these transitions.     

Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

In this paper, we consider complete hypersurfaces in R ⁿ⁺¹ with constant mean curvature H and prove that M ⁿ is a hyperplane if the L ² norm curvature of M ⁿ satisfies some growth condition and M ⁿ is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M ⁿ is a hyperplane (or a round sphere) under the condition that M ⁿ is strongly stable (or weakly stable) and has some finite L ^p norm curvature. Received: 14 July 2007     

Transformation de Fourier des distributions de type positif sur un groupe de Lie unimodulaire

Call a locally compact group G, C¹-unique, if L¹(G) has exactly one (separating) C¹-norm. It is easy to see that a ¹-regular group G is C¹-unique and that a C¹-unique group is amenable. For connected groups G it is proved that G is C¹-unique, if the interior R(G)⁰ of a certain part R(G) of Prim(G), called the regular part of Prim(G), is dense in Prim(G), and that C¹-uniqueness of G implies the density of R(G) in Prim(G). From this it is derived that a connected group of type I is C¹-unique if and only if R(G)⁰ is dense in Prim(G). For exponential G, a quite explicit version of this result in terms of the Lie algebra of G is given. As an easy consequence, examples of amenable groups, which are not C¹-unique, and C¹-unique groups, which are not ¹-regular are obtained. Furthermore it is shown that a connected locally compact group G is amenable if and only if L¹(G) has exactly one C¹-norm, which is invariant under the isometric ¹-automorphisms of L¹(G).     

Associative Cones and Integrable Systems

Abstract We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S⁶. It is known that a cone over a surface M in S⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ are the equation for primitive maps associated to the 6-symmetric space G₂=T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0529756.     

Cylindricity of tangent bundles of strongly parabolic metrics and strongly parabolic surfaces

It is proved that if the intrinsic zero-index of the Sasaki metric of a tangent bundleTM ⁿ isk, thenk is even andM ⁿ is the metric product of a Riemannian manifoldM ^n–k/2 by a Euclidean spaceE ^k/2, whileTM ⁿ is the metric product ofTM ^n–k/2 byE ^k . An expression is obtained for the second fundamental forms of the imbeddingTF ^l TM ⁿ in terms of the second fundamental forms of the imbeddingF ^l M ⁿ and the curvature tensor ofM ⁿ . It is proved thatTF ^l is totally geodesic inTM ⁿ if and only ifF ^l is totally geodesic inM ⁿ .Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 12–32.     

Semigroups satisfying (xy)m = xmym

In this paper we characterize Archimedean semigroups with idempotents satisfying (xy)^m = x^my^m as exactly those semigroups which are a retract extension of a completely simple semigroup satisfying (xy)^m = x^my^m by a nil semigroup satisfying (xy)^m = x^my^m. Regular semigroups satisfying (xy)² = x²y² are exactly those semigroups which are a spined product of a band and a semigroup which is a semilattice of Abelian groups. A semigroup which is a nil extension of a regular semigroup satisfies (xy)² = x²y² if and only if it is a retract extension of a regular semigroup satisfying (xy)² = x²y² by a nil semigroup satisfying (xy)² = x²y²     

On the hypersurfacex+x 2y+z 2+t 3=0 in ℂ4 or a ℂ4-like threefold which is not C3

In this note it will be proved that the threefold in ?⁴ which is given byx+x ² y+z ²+t ³=0 is not isomorphic to ?³. Here ? is the field of complex numbers.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

The de Rham complex on unbounded domains with Sobolev space topology

This paper studies the operator dd^∗+d^∗d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d^∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d^∗ is determined and an expression for d^∗ is obtained. The operator dd^∗+d^∗d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators.