Search results for "Combinatorics"
showing 10 items of 1770 documents
Optimal selection of thek best of a sequence withk stops
1997
We first consider the situation in which the decision-maker is allowed to have five choices with purpose to choose exactly the five absolute best candidates fromN applicants. The optimal stopping rule and the maximum probability of making the right five-choice are given for largeN eN, the maximum asymptotic value of the probability of the best choice being limN→∝P (win) ≈ 0.104305. Then, we study the general problem of selecting thek best of a sequence withk stops, constructing first a rough solution for this problem. Using this suboptimal solution, we find an approximation for the optimal probability valuesPk of the form $$P_k \approx \frac{1}{{(e - 1)k + 1}}$$ for any k eN.
Derived length and character degrees of solvable groups
2003
We prove that the derived length of a solvable group is bounded in terms of certain invariants associated to the set of character degrees and improve some of the known bounds. We also bound the derived length of a Sylow p-subgroup of a solvable group by the number of different p-parts of the character degrees of the whole group.
A characterization of the line set of an odd-dimensional Baer subspace
1990
Generalizing a theorem of Beutelspacher and Seeger, we consider line sets\(\mathcal{L}\) inP=PG(2t + 1,q),t ∈ IN, with the following properties: (1) any (t + 1)-dimensional subspace ofP contains at least one line of\(\mathcal{L}\), (2) if a pointx ofP is incident with at least two lines of\(\mathcal{L}\) then the points in the factor geometryP/x which are induced by the lines of\(\mathcal{L}\) throughx form a blocking set of type (t, 1) inP/x, (3) any line of\(\mathcal{L}\) is coplanar with at least one further line of\(\mathcal{L}\). We will show that the examples of minimal cardinality are exactly the line sets of Baer subspaces ofP.
Degrees of characters in the principal block
2021
Abstract Let G be a finite group. We prove that if the set of degrees of characters in the principal p-block of G has size at most 2 then G is p-solvable, and G / O p ′ ( G ) has a metabelian normal Sylow p-subgroup. The general question of proving that if an arbitrary p-block has two degrees then their defect groups are metabelian remains open.
On finite products of soluble groups
1998
Let the finite groupG =AB be the product of two soluble subgroupsA andB, and letπ be a set of primes. We investigate under which conditions for the maximal normalπ-subgroups ofA, B andG the following holds:Oπ(G) ∩Oπ(G) ⊆Oπ(G).
Forbidden Factors and Fragment Assembly
2002
In this paper we approach the fragment assembly problem by using the notion of minimal forbidden factors introduced in previous paper. Denoting by M(w) the set of minimal forbidden factors of a word w, we first focus on the evaluation of the size of elements in M(w) and on designing of an algorithm to recover the word w from M(w). Actually we prove that for a word w randomly generated by a memoryless source with identical symbol probabilities, the maximal length m(w) of words in M(w) is logarithmic and that the reconstruction algorithm runs in linear time. These results have an interesting application to the fragment assembly problem, i.e. reconstruct a word w from a given set I of substrin…
Kontsevich–Zagier Periods
2017
We compare the set of Kontsevich–Zagier periods defined by integrals over semi-algebraic subsets of \(\mathbb {R}^n\) with cohomological periods.
On the packing sums of pairs
1993
Abstract This paper is concerned with the determination of the length of the largest interval of consecutive integers of the set hA k , where A k is a sequence of integers which is a B h -sequence.
Some Generalizations of a Simion Schmidt Bijection
2007
In 1985, Simion and Schmidt gave a constructive bijection φ from the set of all length (n-1) binary strings having no two consecutive 1s to the set of all length n permutations avoiding all patterns in {123,132,213}. In this paper, we generalize φ to an injective function from {0,1}n-1 to the set Sn of all length n permutations and derive from it four bijections φ : P →Q where P⊆{0,1}n-1 and Q ⊂ Sn. The domains are sets of restricted binary strings and the codomains are sets of pattern-avoiding permutations. As a particular case we retrieve the original Simion–Schmidt bijection. We also show that the bijections obtained are actually combinatorial isomorphisms, i.e. closeness-preserving bije…
On the number of factors of Sturmian words
1991
Abstract We prove that for m ⩾1, card( A m ) = 1+∑ m i =1 ( m − i +1) ϕ ( i ) where A m is the set of factors of length m of all the Sturmian words and ϕ is the Euler function. This result was conjectured by Dulucq and Gouyou-Beauchamps (1987) who proved that this result implies that the language (∪ m ⩾0 A m ) c is inherently ambiguous. We also give a combinatorial version of the Riemann hypothesis.