Landry, Minsky and Taylor (LMT) introduced two polynomial invariants of veering triangulations—the taut the polynomial. Here, we consider a pair polynomials associated to one triangulation, upper lower one, analogously polynomials. We prove that are equal. In contrast, same triangulation may differ by more than unit. give algorithms compute all these invariants. LMT related Teichmüller fibered ...

Given two observers, we define the “relative velocity” of one observer with respect to the other in four different ways. All four definitions are given intrinsically, i.e. independently of any reference system. Two of them are given in the framework of spacelike simultaneity and, analogously, the other two are given in the framework of observed (lightlike) simultaneity. Properties and physical ...

We use unified algebraic methods to investigate the properties of polynomial Bezoutians with respect to a general basis. Not only can three known results be easily verified, but also some new properties of polynomial Bezoutians are obtained. Nonsymmetric Lyapunov-type equations of polynomial Bezoutians are also discussed. It turns out that most properties of classical Bezoutians can be analogou...

The idea of expected utility, to transform payments into their utilities before calculating expectation, traces back at least to Bernoulli (1738). It is a very natural idea to transform, analogously, probabilities. This paper gives heuristic visual arguments to showthat the, at first sight, natural way to do this, at second thought seems questionable. At second thought a sound and natural way i...

In developing electronic gadgets, meters and oscilloscopes are used to probe into their workings, so that engineers can understand how they are behaving and how they can be improved. Analogously, this paper suggests a variety of conceptual probes that can be used to explore the process of design. The probes focus designers’ attention onto successfully managing complexity and in coping with soci...

This paper establishes a link between Bayesian inference (learning) and predicate and state transformer operations from programming semantics and logic. Specifically, a very general definition of backward inference is given via first applying a predicate transformer and then conditioning. Analogously, forward inference involves first conditioning and then applying a state transformer. These def...

We are interested in subsets S of Z which admit simultaneous orderings, that is, sequences {an}n≥0 such that Qn−1 k=0 (an−ak) divides Qn−1 k=0 (x− ak) for every x ∈ S (analogously to n! which divides (m+n)!/m! for every m). In particular, we characterize the polynomials f of degree 2 such that f(N) admits a simultaneous ordering.

In this paper definitions of many sorted closure system and many sorted closure operator are introduced. These notations are also introduced in [9], but in another meaning. In this article closure system is absolutely multiplicative subset family of many sorted sets and in [9] is many sorted absolutely multiplicative subset family of many sorted sets. Analogously, closure operator is function b...

Metallic nanoparticles co-functionalised with monolayers of UV- and CO2-sensitive ligands were prepared and shown to respond to these two types of stimuli reversibly and in an orthogonal fashion. The composition of the coating could be tailored to yield nanoparticles capable of aggregating exclusively when both UV and CO2 were applied at the same time, analogously to the behaviour of an AND log...