Search results for "cardinality"
showing 10 items of 42 documents
Ranking and unrankingk-ary trees with a 4k –4 letter alphabet
1997
Abstract The problem of the direct generation in A-order of binary trees was stated by Zaks in 1980. In 1988 Roelants van Baronaigien and Ruskey gave a solution for k-ary trees with n internal nodes using an encoding sequence of kn+1 integers between 1 and n. Vajnovszki and Pallo improved this result for binary trees in 1994 using words of length n–1 on a four letter alphabet. Recently Korsh generalized the Vajnovszki and Pallo’s generating algorithm to k-ary trees using an alphabet whose cardinality depends on k but not on n. We give in this paper ranking and unranking algorithms for k-ary trees using the Korsh’s encoding scheme.
A note on rank 2 diagonals
2020
<p>We solve two questions regarding spaces with a (G<sub>δ</sub>)-diagonal of rank 2. One is a question of Basile, Bella and Ridderbos about weakly Lindelöf spaces with a G<sub>δ</sub>-diagonal of rank 2 and the other is a question of Arhangel’skii and Bella asking whether every space with a diagonal of rank 2 and cellularity continuum has cardinality at most continuum.</p>
Point counting on Picard curves in large characteristic
2005
We present an algorithm for computing the cardinality of the Jacobian of a random Picard curve over a finite field. If the underlying field is a prime field Fp, the algorithm has complexity O(p).
A smallest irregular oriented graph containing a given diregular one
2004
AbstractA digraph is called irregular if its vertices have mutually distinct ordered pairs of semi-degrees. Let D be any diregular oriented graph (without loops or 2-dicycles). A smallest irregular oriented graph F, F=F(D), is constructed such that F includes D as an induced subdigraph, the smallest digraph being one with smallest possible order and with smallest possible size. If the digraph D is arcless then V(D) is an independent set of F(D) comprising almost all vertices of F(D) as |V(D)|→∞. The number of irregular oriented graphs is proved to be superexponential in their order. We could not show that almost all oriented graphs are/are not irregular.
On the cardinality of almost discretely Lindelof spaces
2016
A space is said to be almost discretely Lindelof if every discrete subset can be covered by a Lindelof subspace. Juhasz et al. (Weakly linearly Lindelof monotonically normal spaces are Lindelof, preprint, arXiv:1610.04506 ) asked whether every almost discretely Lindelof first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhasz et al. (First-countable and almost discretely Lindelof $$T_3$$ spaces have cardinality at most continuum, preprint, arXiv:1612.06651 ). We conclude with a few rel…
Combinatorial aspects of L-convex polyominoes
2007
We consider the class of L-convex polyominoes, i.e. those polyominoes in which any two cells can be connected with an ''L'' shaped path in one of its four cyclic orientations. The paper proves bijectively that the number f"n of L-convex polyominoes with perimeter 2(n+2) satisfies the linear recurrence relation f"n"+"2=4f"n"+"1-2f"n, by first establishing a recurrence of the same form for the cardinality of the ''2-compositions'' of a natural number n, a simple generalization of the ordinary compositions of n. Then, such 2-compositions are studied and bijectively related to certain words of a regular language over four letters which is in turn bijectively related to L-convex polyominoes. In …
The article <i>a(n)</i> in English quantifying expressions: A default marker of cardinality
2020
Certain English quantificational expressions feature what appears to be an indefinite article, e.g. a bunch, a few, a hundred. These can be divided into three types of quantifying expressions: pseudopartitives (a lot, a bunch, a ton), article-requiring quantifiers (a few, a couple, a hundred), and article-free quantifiers (three, many, several); article-free quantifiers have an article under certain circumstances, e.g. modification by an adjective (a surprising 30 …). While standard analyses would take the article in these expressions to be a D head, it is argued here that the article is not in D, nor is it singular or count, as evidenced by its (lack of an) interaction with verbal agreemen…
Weak regularity and consecutive topologizations and regularizations of pretopologies
2009
Abstract L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ -compact pretopology. On the other hand, it is proved that for each n ω there is a (regular) pretopology ρ (on a set of cardinality c ) such that ( RT ) k ρ > ( RT ) n ρ for each k n and ( RT ) n ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT -order ⩾ ω 0 . Moreover, all these pretopologies have the property…
Countably compact weakly Whyburn spaces
2015
The weak Whyburn property is a generalization of the classical sequential property that was studied by many authors. A space X is weakly Whyburn if for every non-closed set \({A \subset X}\) there is a subset \({B \subset A}\) such that \({\overline{B} \setminus A}\) is a singleton. We prove that every countably compact Urysohn space of cardinality smaller than the continuum is weakly Whyburn and show that, consistently, the Urysohn assumption is essential. We also give conditions for a (countably compact) weakly Whyburn space to be pseudoradial and construct a countably compact weakly Whyburn non-pseudoradial regular space, which solves a question asked by Angelo Bella in private communica…
On approximation of a class of stochastic integrals and interpolation
2004
Given a diffusion Y = (Y_{t})_{t \in [0,T]} we give different equivalent conditions so that a stochastic integral has an L 2-approximation rate of n −η, {\rm \eta \in (0,1/2],} if one approximates by integrals over piece-wise constant integrands where equidistant time nets of cardinality n + 1 are used. In particular, we obtain assertions in terms of smoothness properties of g(Y T ) in the sense of Malliavin calculus. After optimizing over non-equidistant time-nets of cardinality n + 1 in case {\rm \eta > 0} , it turns out that one always obtains a rate of n^{ - 1/2}, which is optimal. This applies to all functions g obtained in an appropriate way by the real interpolation method between th…