Homogeneous permutations.
Homological Dimension and Stationary Sets.
Homology theory in the alternative set theory I. Algebraic preliminaries
The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called commutative -group), is introduced. Commutative -groups are studied from the set-theoretical point of view and from the point of view of free groups. Commutativity of the operator which is a special kind of inverse limit and factorization, is proved. Tensor product is defined, commutativity of direct product (also a free group construction and tensor product) with the special...
Homology theory in the AST II. Basic concepts, Eilenberg-Steenrod's axioms
Homology functor in the spirit of the AST is defined, its basic properties are studied. Eilenberg-Steenrod axioms for this functor are formulated and established.
How complicated can be one-dimensional dynamical systems: descriptive estimates of sets
How large are left exact functors?
How many clouds cover the plane?
The plane can be covered by n + 2 clouds iff .
How many normal measures can carry?
We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for to be measurable and to carry exactly τ normal measures, where is any regular cardinal. This contrasts with the fact that assuming AD + DC, is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.
How to handle fuzzy-quantities?
How to recognize a true Σ^0_3 set
Let X be a Polish space, and let be a sequence of hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether is a true subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true .
Hurewicz properties, non distinguishing convergence properties and sequence selection properties
Hurewicz scheme
Hybrid Prikry forcing
We present a new forcing notion combining diagonal supercompact Prikry forcing with interleaved extender based forcing. We start with a supercompact cardinal κ. In the final model the cofinality of κ is ω, the singular cardinal hypothesis fails at κ, and GCH holds below κ. Moreover we define a scale at κ which has a stationary set of bad points in the ground model.
Hydrological applications of a model-based approach to fuzzy set membership functions
Since the common approach to defining membership functions of fuzzy numbers is rather subjective, another, more objective method is proposed. It is applicable in situations where two models, say and , share the same uncertain input parameter . Model is used to assess the fuzziness of , whereas the goal is to assess the fuzziness of the -dependent output of model . Simple examples are presented to illustrate the proposed approach.
Hyperplanes in matroids and the axiom of choice
We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC, the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?
Hyperstructures Associated with Ordered sets