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Boaz Tsaban

Publications and source records attributed to Boaz Tsaban.

At least 91 records · Page 5Linked to original sources
arXiv2010

Permutation graphs, fast forward permutations, and sampling the cycle structure of a permutation

A permutation P on {1,..,N} is a_fast_forward_permutation_ if for each m the computational complexity of evaluating P^m(x)$ is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions. By studying the evolution of permutation graphs, we prove that the number of queries needed to distinguish a random cyclus from a random permutation on {1,..,N} is Theta(N) if one does not use queries of the form P^m(x), but is only Theta(1) if one is allowed to make such queries. We construct fast forward permutations which are indistinguishable from random permutations even when queries of the form P^m(x) are allowed. This is done by introducing an efficient method to sample the cycle structure of a random permutation, which in turn solves an open problem of Naor and Reingold.

cs.CR↗
arXiv2010

Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures

We consider the question, which of the major classes defined by topological diagonalizations of open or Borel covers is hereditary. Many of the classes in the open case are not hereditary already in ZFC, and none of them is provably hereditary. This is contrasted with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes d$ and b, respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.

math.LO↗
arXiv2010

o-bounded groups and other topological groups with strong combinatorial properties

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of the real line R (thus strictly o-bounded) which have the Hurewicz property but are not sigma-compact, and show that the product of two o-bounded subgroups of R^N may fail to be o-bounded, even when they satisfy the stronger property S1(Borel_Omega,Borel_Omega). This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups G of size continuum such that every countable Borel omega-cover of G contains a gamma-cover of G.

math.GN↗
arXiv2010

Several comments about the combinatorics of tau-covers

In a previous work with Mildenberger and Shelah, we showed that the combinatorics of the selection hypotheses involving tau-covers is sensitive to the selection operator used. We introduce a natural generalization of Scheepers' selection operators, and show that: (1) A slight change in the selection operator, which in classical cases makes no difference, leads to different properties when tau-covers are involved. (2) One of the newly introduced properties sheds some light on a problem of Scheepers concerning tau-covers. Improving an earlier result, we also show that no generalized Luzin set satisfies U_fin(Gamma,Tau).

math.GN↗
arXiv2010

Critical cardinalities and additivity properties of combinatorial notions of smallness

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (tau-covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivity numbers of the combinatorial notions corresponding to the topological diagonalization notions. This gives new insights on the structure of the eventual dominance ordering on the Baire space, the almost inclusion ordering on the Rothberger space, and the interactions between them.

math.LO↗
arXiv2010

Additivity properties of topological diagonalizations

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions. This is a very concise paper. For a self-contained, complete, and extended treatment of the topic see math.GN/0604451

math.LO↗
arXiv2010

Menger's covering property and groupwise density

We establish a surprising connection between Menger's classical covering property and Blass's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.

math.GN↗
arXiv2010

Combinatorial images of sets of reals and semifilter trichotomy

Using a dictionary translating a variety of classical and modern covering properties into combinatorial properties of continuous images, we get a simple way to understand the interrelations between these properties in ZFC and in the realm of the trichotomy axiom for upward closed families of sets of natural numbers. While it is now known that the answer to the Hurewicz 1927 problem is positive, it is shown here that semifilter trichotomy implies a negative answer to a slightly weaker form of this problem.

math.LO↗
arXiv2010

The Hurewicz covering property and slaloms in the Baire space

According to a result of Kocinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals $X$ satisfies the Hurewicz property if, and only if, each large open cover of $X$ contains a groupable subcover. This solves in the affirmative a problem of Scheepers. The proof uses a rigorously justified abuse of notation and a "structure" counterpart of a combinatorial characterization, in terms of slaloms, of the minimal cardinality b of an unbounded family of functions in the Baire space. In particular, we obtain a new characterization of $\b$.

math.GN↗
arXiv2010

Scales, fields, and a problem of Hurewicz

Menger's basis property is a generalization of $σ$-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non $σ$-compact sets of reals with Menger's property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis. The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results can be stated in a Ramsey-theoretic manner. Since we believe that this paper is of interest to a diverse mathematical audience, we have made a special effort to make it self-contained and accessible.

math.GN↗
arXiv2010

Solving random equations in Garside groups using length functions

We give a systematic exposition of memory-length algorithms for solving equations in noncommutative groups. This exposition clarifies some points untouched in earlier expositions. We then focus on the main ingredient in these attacks: Length functions. After a self-contained introduction to Garside groups, we describe length functions induced by the greedy normal form and by the rational normal form in these groups, and compare their worst-case performances. Our main concern is Artin's braid groups, with their two known Garside presentations, due to Artin and due to Birman-Ko-Lee (BKL). We show that in $B_3$ equipped with the BKL presentation, the (efficiently computable) rational normal form of each element is a geodesic, i.e., is a representative of minimal length for that element. (For Artin's presentation of $B_3$, Berger supplied in 1994 a method to obtain geodesic representatives in $B_3$.) For an arbitrary number of strands, finding the geodesic length of an element is NP-hard, by a 1991 result of by Paterson and Razborov. We show that a good estimation of the geodesic length of a braid in Artin's presentation is measuring the length of its rational form in the \emph{BKL} presentation. This is proved theoretically for the worst case, and experimental evidence is provided for the generic case.

math.GR↗
arXiv2010

SPM Bulletin 30

This is the 30th issue of this bulletin, dedicated to mathematical selection principles and related areas. Now in a concise format.

math.GN↗
arXiv2010

Sequential properties of function spaces with the compact-open topology

Let M be the countably infinite metric fan. We show that C_k(M,2) is sequential and contains a closed copy of Arens space S_2. It follows that if X is metrizable but not locally compact, then C_k(X) contains a closed copy of S_2, and hence does not have the property AP. We also show that, for any zero-dimensional Polish space X, C_k(X,2) is sequential if and only if X is either locally compact or the derived set X' is compact. In the case that X is a non-locally compact Polish space whose derived set is compact, we show that all spaces C_k(X, 2) are homeomorphic, having the topology determined by an increasing sequence of Cantor subspaces, the n-th one nowhere dense in the (n+1)-st.

math.GN↗
arXiv2009

SPM Bulletin 29

In addition to 29 announcements in related areas, this issue contains several contributions to "core" SPM: Measurable cardinals and the cardinality of Lindelof spaces; Topological games and covering dimension; Menger's and Hurewicz's Problems: Solutions from "The Book" and refinements; Point-cofinite covers in the Laver model; Projective versions of selection principles

math.GN↗
arXiv2009

SPM Bulletin 28

Contents: 1. Editor's note; 2. gamma-sets from a weak hypothesis; 3. Research announcements; 3.1. Ultrafilters with property (s); 3.2. On a converse to Banach's Fixed Point Theorem; 3.3. Analytic groups and pushing small sets apart 3.4. Club-guessing, stationary reflection, and coloring theorems; 3.5. More on the pressing down game; 3.6. A note on discrete sets; 3.7. Antidiamond principles and topological applications; 3.8. Partitions and indivisibility properties of countable dimensional vector spaces; 3.9. Group-valued continuous functions with the topology of pointwise convergence; 3.10. Stationary and convergent strategies in Choquet games; 3.11. Linear sigma-additivity and some applications; 4. Unsolved problems from earlier issues.

math.GN↗
arXiv2009

SPM Bulletin 27

Contents: 1. Editor's note 2. Research announcements 2.1. Pseudocompact group topologies with no infinite compact subsets 2.2. Selective coideals on (FIN[1]k) 2.3. Entire functions mapping uncountable dense sets of reals onto each other monotonically 2.4. Effective refining of Borel coverings 2.5. Symmetry and colorings: Some results and open problems 2.6. Many partition relations below density 2.7. Lindelof indestructibility, topological games and selection principles 2.8. Locally precompact groups: (Local) realcompactness and connectedness 2.9. The group Aut is Roelcke precompact 2.10. An infinite combinatorial statement with a poset parameter 2.11. The Schur `1 Theorem for fifilters 2.12. On uniform asymptotic upper density in locally compact abelian groups 2.13. A c0-saturated Banach space with no long unconditional basic sequences 2.14. Zero subspaces of polynomials on `1(2.15. MAD Families and SANE Player 2.16. On the consistency of d_k > cov_k(M) 2.17. o-Boundedness of free topological groups 2.18. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. V 2.19. Menger subsets of the Sorgenfrey line 2.20. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. VI 3. Unsolved problems from earlier issues

math.GN↗
arXiv2008

SPM Bulletin 26

This festive issue concludes the civilian year 2008 with details on a special issue of Topology and its Applications dedicated to SPM, and with a quite large list of research announcements.

math.GN↗