The Stern-Brocot tree (or rather half of it) can be defined as follows. Start with two fractions 0/1 and 1/1, forming an ordered set S0. (Throughout this paper, “fraction” means “fraction in lowest terms”.) At stage k, (k = 1, 2, . . .), form a new set Sk by inserting between each pair of adjacent fractions in Sk−1, say p/q and r/s, the fraction (p+r)/(q+s). Name the (ordered) set of fractions ...

1. Introducing the birank. A partition is defined as being a nonincreasing sequence of positive integers, λ = (λ1,λ2, . . . ,λr ). The set of all partitions, which includes the empty partition ∅, is denoted by . The sum of the parts of a given partition is called the weight of the partition, wt(λ) = λ1+λ2+···+λr . It is standard notation to write (z;q)∞ := ∏ t≥0(1−zq) and p−k(n) for the coeffic...

In this paper we will present a new construction of any real geometric (Clifford) algebra $${\mathbb {G}}^{(p,q)}$$ with signature (p, q) where $$p+q=n$$ by defining product on the vector space {R}}^{(2^n)}$$ in manner similar to Gauss’ ordered pair complex numbers ( {C}}$$ ) and Hamilton’s quadruple quaternions {H}}$$ ). We motivate definition generalizing tuple multiplication each . Similar w...

An ordered partition of [n] := {1, 2, . . . , n} is a sequence of its disjoint subsets whose union is [n]. The number of ordered partitions of [n] with k blocks is k!S(n, k), where S(n, k) is the Stirling number of second kind. In this paper we prove some refinements of this formula by showing that the generating function of some statistics on the set of ordered partitions of [n] with k blocks ...

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following ...

We prove the following theorem. Suppose that M is a trim DFA on the Boolean alphabet 0, 1. The language L(M) is well-ordered by the lex-icographic order < ℓ iff whenever the non sink states q, q.0 are in the same strong component, then q.1 is a sink. It is easy to see that this property is sufficient. In order to show the necessity, we analyze the behavior of a < ℓ-descending sequence of words....

It is shown through detailed inelastic neutron scattering experiments that the gapped collective magnetic excitation of the unconventional superconductor CeCoIn5, the spin resonance mode, is incommensurate and that the corresponding fluctuations are of an Ising nature. The incommensurate peak position of these fluctuations corresponds to the propagation vector of the adjacent field induced stat...

The BFKL and the unified angular-ordered equations are solved to determine the gluon distribution at small x. The impact of kinematic constraints is investigated. Predictions are made for observables sensitive to the gluon at small x. In particular comparison is made with measurements at the HERA electron-proton collider of the proton structure function F 2 (x, Q 2) as a function of ln Q 2 , th...

Using the concept of a mixed g-monotone mapping, we prove some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a Q-function q. The presented theorems are generalizations of the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham 2006 , Lakshmikantham and Ćirić 2009 and many ot...

In this paper, with a new idea, we redefine generalized fuzzy ideal and investigate some of its properties. We generalize fuzzy prime ideals and introduce the notions of (∈,∈ ∨q)−fuzzy prime ideal of ordered semigroups. We characterize (∈,∈ ∨q)−fuzzy prime ideal by its image and level ideals. We establish necessary and sufficient condition for fuzzy set to be a (∈,∈ ∨q)−fuzzy prime ideal. 2010 ...