Area-preserving isotopies of self-transverse immersions of S 1 in ℝ2

Let C and C′ be two smooth self-transverse immersions of S ¹ into ?². Both C and C′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C′ induces a one-to-one correspondence between the disks of C and C′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C′ is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C′. In this paper we show that this is also a sufficient condition.     

Brown-Booth-Tillotson theory for classes of exponentiable spaces

Brown, Booth and Tillotson introduced the C-product, or the BBT C-product, for any class C of topological spaces. It is proved that any topological space is exponentiable with respect to the BBT C-product if and only if C is a subclass of the class of exponentiable spaces. The topology of the function space is induced by a canonical manner making use of the exponential topology for the spaces in C. It is not the C-open topology in general. The function space defined by this method enjoys good properties for algebraic topology. A necessary and sufficient condition on the class C is obtained for the exponential function to be a homeomorphism with the BBT C-product.     

More on positive commutators

Let A and B be positive operators on a Banach lattice E such that the commutator C=AB−BA is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Bra?i? et al. (in press) [3]. If the sum A+B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. If we assume that the operator A is compact and the commutator AC−CA is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected.     

Collectionwise weak continuity duals

A function f: (X, τ) → (Y, σ) is weakly collectionwise continuous if for some C ? 2 ^X with τ ? C we have f ^?1(V) ∈ C for each V ∈ σ. In this case, f is said to be C-continuous. If also τ ? C* ? 2 ^X , C*-continuity is a dual to C-continuity if C?C* = τ and then the pair (C-continuity, C*-continuity) is a decomposition of continuity. In this paper, two natural topological methods are found for construction of a dual to any collectionwise weak continuity. Some known decompositions are improved.     

The convenient setting for non-quasianalytic Denjoy-Carleman differentiable mappings

For Denjoy-Carleman differentiable function classes CM where the weight sequence M=(Mk) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is CM if it maps CM-curves to CM-curves. The category of CM-mappings is cartesian closed in the sense that CM(E,CM(F,G))≅CM(E×F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of CM-diffeomorphisms is a CM-Lie group but not better.     

Ball and spindle convexity with respect to a convex body

Let ${C \subset \mathbb{R}^n}$ be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ${\emptyset}$ , or ${\mathbb{R}^n}$ . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.     

Monomorphisms and epimorphisms in pro-categories

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. We give characterizations of monomorphisms (respectively, epimorphisms) in pro-category pro-C, provided C has direct sums (respectively, pushouts).Let E(C) (respectively, M(C)) be the subcategory of C whose morphisms are epimorphisms (respectively, monomorphisms) of C. We give conditions in some categories C for an object X of pro-C to be isomorphic to an object of pro-E(C) (respectively, pro-M(C)).A related class of objects of pro-C consists of X such that there is an epimorphism X→P∈Ob(C) (respectively, a monomorphism P∈Ob(C)→X). Characterizing those objects involves conditions analogous (respectively, dual) to the Mittag-Leffler property. One should expect that the object belonging to both classes ought to be stable. It is so in the case of pro-groups. The natural environment to discuss those questions are balanced categories with epimorphic images. The last part of the paper deals with that question in pro-homotopy.     

Means of unitaries, conjugations, and the Friedrichs operator

If C is a conjugation (an isometric, conjugate-linear involution) on a separable complex Hilbert space H, then T∈B(H) is called C-symmetric if T=CT^∗C. In this note we prove that each C-symmetric contraction T is the mean of two C-symmetric unitary operators. We discuss several corollaries and an application to the Friedrichs operator of a planar domain.     

Groups of homeomorphisms on C(X) and pointwise limits

This study looks at some subgroups of the group H(C(X)) of homeomorphisms on the space C(X) of continuous real-valued functions on a topological space X, where C(X) has the compact-open topology. The main result shows that, for certain spaces X, the subgroup of H(C(X)) generated by the algebraic and vertical homeomorphisms on C(X) is dense in H(C(X)) with the pointwise topology. Also, for X equal to the unit interval, a subgroup of H(C(X)) is developed using integration of the members of C(X), and this subgroup is used as an example and to illustrate certain properties that subgroups of H(C(X)) can have.     

An example of the Weierstrass semigroup of a pointed curve on K3 surfaces

Let X be a K3 surface with Picard number one which is given by a double cover π:X→?². Let C be a smooth curve on X with π ^?1 π(C)=C which is not the ramification divisor of π, and let P be a ramification point of π| _C :C→π(C). In this paper, in the case where the intersection multiplicity at π(P) of the curve π(C) and the tangent line at π(P) on π(C) is equal to deg(π(C)) or deg(π(C))?1, we investigate the Weierstrass semigroup of the pointed curve (C,P).     

A geometric Gordan-Stiemke theorem

Let C and K be closed cones in Rⁿ. Denote by φ (K∩C) the face of C generated by K ∩ C, by φ(K ∩ D)^D the dual face of φ(K ∩ C) in C¹, and by φ(-K¹ ∩ C¹) the face of C¹ generated by -K¹ ∩ C¹. It is proved that φ(K ∩ C¹) if and only if -C¹ ∩ [span(K ∩ C)] ⊥ ? C¹ + K¹. In particular, the closedness of C¹ + K¹ is a sufficient condition. Our result contains a generalization of the Gordon-Stiemke theorem which appeared in a recent paper of Saunders and Schneider.     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.     

The Cn WP-Bailey chain

The purpose of this paper is to introduce the concept of C_n WP-Bailey pairs. The C_n WP-Bailey transform is obtained by applying the Cn 6φ5 summation formula. From this result, the Cn WP-Bailey lemma is deduced by making use of the Cn q-Dougall summation formula. Some applications are investigated. Finally, the case of elliptic Cn WP-Bailey pairs is discussed.     

Internal coverings of sets by asymmetric relations

A non-empty finite collection $\text{C}$ of non-empty finite sets is internally coverable iff there exists an asymmetric binary relation P on ∪{C; C ∈ $\text{C}$} such that C ? {x: yPx for some y ∈ C} for all C ? $\text{C}$. Necessary and sufficient conditions for internal coverability in terms of “bad sets” are specified for all $\text{C}$ with |$\text{C}$| ≤ 5. Relationships between internal coverability and aspects of tournaments are discussed.     

Concerning some variants of C-embedding in pointfree topology

We study three types of quotient maps of frames which are closely related to C- and C^?-quotient maps. We call them C₁-, strong C₁-, and uplifting quotient maps. C₁-quotient maps are precisely those whose induced ring homomorphisms contract maximal ideals to maximal ideals. We show that every homomorphism onto a frame is a C₁-, a strong C₁-, or an uplifting quotient map iff the frame is pseudocompact, compact, or almost compact and normal, respectively. These quotient maps are used to characterize normality and also certain weaker forms of normality in a manner akin to the characterization of normal frames as those for which every closed quotient map is a C-quotient map. Under certain conditions, we show that the Stone extension of a quotient map is C₁-, strongly C₁- or uplifting if the map has the corresponding property.     

Some isomorphisms between pairs of latin squares

Let A, B, C, D be latin squares with A orthogonal to B and C orthogonal to D. The pair A, B is isomorphic with the pair C, D if the graph of A, B is graph-isomorphic with the graph of C, D. A characterization is given for determining when a pair A, B of latin squares is isomorphic with a self-orthogonal square C and its transpose. Self-orthogonal squares are important because they are both abundant and easy to store. An algorithm either displays a self-orthogonal square C and an isomorphism from A, B to C, C^T or, if none exists, gives a small set of blocks to the existence of such a square isomorphism.     

On the structure of some group codes

In this paper, we study the following problem: Which characteristics does a codeC possess when the syntactic monoidsyn(C ^*) of the star closureC ^* ofC is a group? For a codeC, if the syntactic monoidsyn(C ^*) is a group, then we callC a group code. This definition of a group code is different from the one in [1] (see [1], 46–47). Schützenberger had characterized the structure of finite group codes and had proved thatC is a finite group code if and only ifC is a full uniform code (see [5], [8]). Fork-prefix andk-suffix codes withk≥2,k-infix,k-outfix,p-infix,s-infix, right semaphore codes and left semaphore codes, etc., we obtain similar results. It is proved that the above mentioned codes are group codes if and only if they are uniform codes.     

Strongly rational comodules and semiperfect Hopf algebras over QF rings

Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right C-comodule. Using this notion a number of characterizations of right semiperfect coalgebras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show that a Hopf algebra H over a QF ring R is right semiperfect if and only if it is left semiperfect or — equivalently — the (left) integrals form a free R-module of rank 1.     

On the automorphism group of a cone

Let V be a real finite dimensional vector space, and let C be a full cone in C. In Sec. 3 we show that the group of automorphisms of a compact convex subset of V is compact in the uniform topology, and relate the group of automorphisms of C to the group of automorphisms of a compact convex cross-section of C. This section concludes with an application which generalizes the result that a proper Lorentz transformation has an eigenvector in the light cone. In Sec. 4 we relate the automorphism group of C to that of its irreducible components. In Sec. 5 we show that every compact group of automorphisms of C leaves a compact convex cross-section invariant. This result is applied to show that if C is a full polyhedral cone, then the automorphism group of C is the semidirect product of the (finite) automorphism group of a polytopal cross-section and a vector group whose dimension is equal to the number of irreducible components of C. An example shows that no such result holds for more general cones.     

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S,S]^C. An idempotent e of this ring will split the homotopy category: [X,Y]^C≅e[X,Y]^C⊕(1−e)[X,Y]^C. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L_eSC×L_(1−e)SC and [X,Y]^L_eSC≅e[X,Y]^C. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.