Expert and novice facilitated modelling: A case of a Viable System Model workshop in a local food network

This paper provides an empirical study based on action research in which expert and novice facilitators in facilitated modelling workshops are compared. There is limited empirical research analysing the differences between expert and novice facilitators. Aiming to address this gap we study the behaviour of one expert and two novice facilitators during a Viable System Model workshop. The findings suggest common facilitation patterns in the behaviour of experts and novices. This contrasts literature claiming that experts and novices behave and use their available knowledge differently, and empirically supports the claim that facilitation skills can be taught to participants to enable them to self-facilitate workshops. Differences were also found, which led to the introduction of a new dimension—‘internal versus external’ facilitation. The implications of our findings for effective training and facilitation strategies in contexts in which external, expert facilitation is not always possible are also discussed, and limitations of this study are provided.     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

Cartan Subalgebras in C*-Algebras of Haus dorff étale Groupoids

The reduced C*-algebra of the interior of the isotropy in any Hausdorff étale groupoid G embeds as a C*-subalgebra M of the reduced C*-algebra of G. We prove that the set of pure states of M with unique extension is dense, and deduce that any representation of the reduced C*-algebra of G that is injective on M is faithful. We prove that there is a conditional expectation from the reduced C*-algebra of G onto M if and only if the interior of the isotropy in G is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, M is a Cartan subalgebra. We prove that for a large class of groupoids G with abelian isotropy—including all Deaconu–Renault groupoids associated to discrete abelian groups—M is a maximal abelian subalgebra. In the specific case of k-graph groupoids, we deduce that M is always maximal abelian, but show by example that it is not always Cartan.     

An axiomatic look at the Erdős-Trost problem

The Erd?s-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erd?s-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erd?s-Trost statement.     

On the computation of <Emphasis Type="Italic">C</Emphasis>* certificates

The cone of completely positive matrices C* is the convex hull of all symmetric rank-1-matrices xx ^T with nonnegative entries. While there exist simple certificates proving that a given matrix \({B\in C^*}\) is completely positive it is a rather difficult problem to find such a certificate. We examine a simple algorithm which—for a given input B—either determines a certificate proving that \({B\in C^*}\) or converges to a matrix \({\bar S}\) in C* which in some sense is “close” to B. Numerical experiments on matrices B of dimension up to 200 conclude the presentation.     

On classifying unitary modules by their Dirac cohomology

Let G be a connected real reductive group with maximal compact subgroup K of the same rank as G. Dirac cohomology of an A_q(λ) module can be identified with a geometric object—the k-dominant part of a face of the convex hull of the Weyl group orbit of the parameter λ + ρ. We show how Dirac cohomology can be used as a parameter to classify the A_q(λ) modules.     

Improving integrability via absolute summability: a general version of Diestel’s Theorem

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the \((p,\sigma )\)-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces—including C(K), \(L^p\) and Hilbert spaces—and operators—p-summing, (q, p)-summing and p-approximable operators.     

Neumann heat flow and gradient flow for the entropy on non-convex domains

For large classes of non-convex subsets Y in \(\mathbb R^n\) or in Riemannian manifolds (M, g) or in RCD-spaces (X, d, m) we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space \((Y,d_Y,m_Y)\) exists—despite the fact that the entropy is not semiconvex—and coincides with the heat flow on Y with Neumann boundary conditions.     

An axiomatization of the Choquet integral in the context of multiple criteria decision making without any commensurability assumption

An axiomatization of the Choquet integral is proposed in the context of multiple criteria decision making without any commensurability assumption. The most essential axiom—named Commensurability Through Interaction—states that the importance of an attribute i takes only one or two values when a second attribute k varies. When the importance takes two values, the point of discontinuity is exactly the value on the attribute k that is commensurate to the fixed value on attribute i. If the weight of criterion i does not depend on criterion k, for any value of the other criteria than i and k, then criteria i and k are independent. Applying this construction to any pair i, k of criteria, one obtains a partition of the set of criteria. In each block, the criteria interact one with another, and it is thus possible to construct vectors of values on the attributes that are commensurate. There is complete independence between the criteria of any two blocks in this partition. Hence one cannot ensure commensurability between two blocks in the partition. But this is not a problem since the Choquet integral is additive between subsets of criteria that are independent.     

Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

Abstract—We study analytical and arithmetical properties of the complexity function for infinite families of circulant C _n (s₁, s_2,…, s _k ) C_2n(s₁, s_2,…, s _k , n). Exact analytical formulas for the complexity functions of these families are derived, and their asymptotics are found. As a consequence, we show that the thermodynamic limit of these families of graphs coincides with the small Mahler measure of the accompanying Laurent polynomials.     

The characterization problem for designs with the parameters of AG<Subscript>d</Subscript>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We start a new characterization of the geometric 2-design AG _d (n,q) among all simple 2-designs with the same parameters by handling the cases d ∈ {1,2,3,n — 2}. For d ≠ 1, our characterization is in terms of line sizes, and for d = 1 in terms of the number of affine hyperplanes. We also show that the number of non-isomorphic resolvable designs with the parameters of AG₁(n,q) grows exponentially with linear growth of n.     

Translation Invariant Asymptotic Homomorphisms and Extensions of <Emphasis Type="Italic">C</Emphasis>*-Algebras

Let A and B be C*-algebras, let A be separable, and let B be σ-unital and stable. We introduce the notion of translation invariance for asymptotic homomorphisms from S A = C₀(?) ? A to B and show that the Connes—Higson construction applied to any extension of A by B is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of A by B from a translation invariant asymptotic homomorphism. This leads to our main result that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.     

Mortality Risk and Life Expectancy

Changing the mortality risks we face would change human life expectancy. As a special case, one could imagine adding a fixed increment R to all the age-specific mortality rates from age zero upwards. For this case we seek a constant K(A) such that K(A) x R approximates the resulting change in life expectancy remaining at age A, at least for small values of R. The formula for K(A) derived here corrects a heuristic argument that appeared in JORS earlier. An estimate of K(0) suggests that the permanent addition of a one-in-a-million risk at each year of life would reduce life expectancy at birth by about 1 day—a useful fact for risk communication.     

Möbius disjointness for models of an ergodic system and beyond

Given a topological dynamical system (X, T) and an arithmetic function u: ? → ?, we study the strong MOMO property (relatively to u) which is a strong version of u-disjointness with all observable sequences in (X, T). It is proved that, given an ergodic measure-preserving system (Z, \(\mathcal{D}\), к, R),the strong MOMO propertly (relately to u) of a uniquely ergodic midel (X, T)of R yields all other uniquely ergodic midel of R to be u-disjiont. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue—Viorse and Rudin—Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak5s conjecture implies the strong MOMO property relatively to μ in all zero entropy systems; in particular, it makes μ-disjointness uniform. The absence of the strong MOMO property in positive entropy systems is discussed and it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.     

Stabilization by output of continuous and pulse-modulated uncertain systems

The system ? _i = ? _i (?) + x _i+2, \(i \in \overline {1,n - 2} \), ? _n?1 = ? _n?1(?) + u ₁, ? _n = ? _n (?) + u ₂,where ? _i (·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u ₁ and u ₂ are the controls is considered. It is assumed that only the outputs x ₁ and x ₂ are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method.     

Characteristic modules and tensor products over quasi-hereditary algebras

Let A be a monomial quasi-hereditary algebra with a pure strong exact Borel subalgebra B.It is proved that the category of induced good modules over B is contained in the category of good modules over A;that the characteristic module of A is an induced module of that of B via the exact functor-(?)_B A if and only if the induced A-module of an injective B-module remains injective as a B-module.Moreover,it is shown that an exact Borel subalgebra of a basic quasi-hereditary serial algebra is right serial and that the characteristic module of a basic quasi-hereditary serial algebra is exactly the induced module of that of its exact Borel subalgebra.     

Optimal linear estimator of origin-destination flows with redundant data

Suppose given a network endowed with a multiflow. We want to estimate some quantities connected with this multiflow, for instance the value of an s–t flow for one of the sources–sinks pairs s–t, but only measures on some arcs are available, at least on one s–t cocycle (set of arcs having exactly one endpoint in a subset X of vertices with s∈X and t?X). These measures, supposed to be unbiased, are random variables whose variances are known. How can we combine them optimally in order to get the best estimator of the value of the s–t flow?This question arises in practical situations when the OD matrix of a transportation network must be estimated. We will give a complete answer for the case when we deal with linear combinations, not only for the value of an s–t flow but also for any quantity depending linearly from the multiflow. Interestingly, we will see that the Laplacian matrix of the network plays a central role.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.