A class of bilinear differential operators is introduced through assigning appropriate signs and used to create bilinear differential equations which generalize Hirota bilinear equations. The resulting bilinear differential equations are characterized by a special kind of Bell polynomials and the linear superposition principle is applied to the construction of their linear subspaces of solution...

Currently there are two main working hypotheses that purport to answer the first of these questions: what do neural populations represent? The first (standard model) claims that populations encode the value of a stimulus. Whilst the second, more recent perspective, claims they encode a probability distribution over the possible values of a stimulus. The standard model can be caricatured in the ...

The generalized Stirling numbers Ss;h(n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k;α, β, r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n, k) and the associated Bell numbers Bs;h(n) and Bell polynomials Bs;h|n(x) are derived. The particular case s = 2 and h = −1 corres...

The paper presents an algorithm for automatic transcription of recordings of bell-playing clocks. Bell-playing clocks are clocks containing a hidden bell-playing mechanism that is periodically activated to play a melody. Clocks from the eighteenth century give us unique insight into the musical taste of their owners, so we are interested in studying their repertoire and performances thus the ne...

Given a prime number p, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition β(n) = kβ(0) in Fpp in order to resolve the congruence B(n) ≡ k (mod p), where B(n) is the n-th Bell number, and k is any fixed integer. Several applications of the formula and of the condition are inclu...

Babson and Steingŕımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is given by the Bell numbers. For the rem...

We prove that the ordinary generating function of Bell numbers satisfies no algebraic differential equation over C(x) (in fact, over a larger field). We investigate related numbers counting various set partitions (the Uppuluri–Carpenter numbers, the numbers of partitions with j mod i blocks, the Bessel numbers, the numbers of connected partitions, and the numbers of crossing partitions) and pro...

A class of trilinear differential operators is introduced through a technique of assigning signs to derivatives and used to create trilinear differential equations. The resulting trilinear differential operators and equations are characterized by the Bell polynomials, and the superposition principle is applied to the construction of resonant solutions of exponential waves. Two illustrative exam...

Applying the enumeration of sparse set partitions, we show that the number of set systems H ⊂ exp({1, 2, . . . , n}) such that ∅ 6∈ H, ∑E∈H |E| = n and ⋃ E∈H E = {1, 2, . . . ,m}, m ≤ n, equals (1/ log(2)+o(1))bn where bn is the n-th Bell number. The same asymptotics holds ifH may be a multiset. If vertex degrees in H are restricted to be at most k, the asymptotics is (1/αk + o(1)) bn where αk ...

The 2-adic valuations of Bell and complementary Bell numbers is determined. The complementary Bell numbers are known to be zero at n = 2 and H. S. Wilf conjectured that this is the only case where vanishing occurs. N. C. Alexander and J. An prove (independently) that there are at most two indices where this happens. This paper presents yet an alternative proof of the latter.