Higher-Order Spreadsheets with Spilled Arrays

We develop a theory for two recently-proposed spreadsheet mechanisms: gridlets allow for abstraction and reuse in spreadsheets, and build on spilled arrays, where an array value spills out of one cell into nearby cells. We present the first formal calculus of spreadsheets with spilled arrays. Since spilled arrays may collide, the semantics of spilling is an iterative process to determine which arrays spill successfully and which do not. Our first theorem is that this process converges deterministically. To model gridlets, we propose the grid calculus, a higher-order extension of our calculus of spilled arrays with primitives to treat spreadsheets as values. We define a semantics of gridlets as formulas in the grid calculus. Our second theorem shows the correctness of a remarkably direct encoding of the Abadi and Cardelli object calculus into the grid calculus. This result is the first rigorous analogy between spreadsheets and objects; it substantiates the intuition that gridlets are an object-oriented counterpart to functional programming extensions to spreadsheets, such as sheet-defined functions.

Tilting theory via stable homotopy theory

We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillie’s strongly (co)cartesian n-cubes. As applications we construct abstract Auslander–Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi–Yau property. This is used to offer a new perspective on May’s axioms for monoidal, triangulated categories.

’Dette er ikke et hospital’: Rum, følelser og professionel ekspertise/funktion i dansk kræftrehabilitering

Artiklen ser på fremvæksten i det danske sundhedsvæsen af specialdesignede organisationer, hvis primære formål er at rehabilitere mennesker med eller efter kræftsygdom. Baseret på etnografisk-inspirerede feltstudier vises, at når professionelle rehabiliteringshjælpere er sat til at arbejde i potentielt mere humane og helende bygninger, i hvilke funktioner og adfærd ikke altid er fastlagt i overensstemmelse med gennemsigtige, formelle kalkuler eller professionelle rationaler, men i stedet er dannet efter nogle moralske antagelser om menneskelige behov og følelser i forsøg på at trække en klar demarkationslinje til den traditionelle behandlingsindustri – her tænkes først og fremmest på offentlige hospitaler og kræftafdelinger – har det en række konsekvenser for de professionelle rehabiliteringshjælpere; for måden, hvorpå disse tilgår deres funktioner samt indgår i relationer med hinanden og borgere, der kommer for at få hjælp. ‘This is not a hospital’: Space, emotion and professional expertise/function in Danish cancer rehabilitation. The article looks into the growth of special designed organizations in the Danish health care sector, whose primary purpose is to rehabilitate people with or after cancer. Based on ethnographic-inspired field studies it is shown, that when professional rehabilitation workers are put to work in potentially more humane and healing buildings, in which functions and behavior are not always determined in accordance with transparent, formal calculus, or pro-fessional rationales, but instead formed after some moral assumptions about human needs and

Programming chemistry in DNA-addressable bioreactors

We present a formal calculus, termed the chemtainer calculus , able to capture the complexity of compartmentalized reaction systems such as populations of possibly nested vesicular compartments. Compartments contain molecular cargo as well as surface markers in the form of DNA single strands. These markers serve as compartment addresses and allow for their targeted transport and fusion, thereby enabling reactions of previously separated chemicals. The overall system organization allows for the set-up of programmable chemistry in microfluidic or other automated environments. We introduce a simple sequential programming language whose instructions are motivated by state-of-the-art microfluidic technology. Our approach integrates electronic control, chemical computing and material production in a unified formal framework that is able to mimic the integrated computational and constructive capabilities of the subcellular matrix. We provide a non-deterministic semantics of our programming language that enables us to analytically derive the computational and constructive power of our machinery. This semantics is used to derive the sets of all constructable chemicals and supermolecular structures that emerge from different underlying instruction sets. Because our proofs are constructive, they can be used to automatically infer control programs for the construction of target structures from a limited set of resource molecules. Finally, we present an example of our framework from the area of oligosaccharide synthesis.

Formal Calculus and Umbral Calculus

We use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a very short proof which naturally arose as a motivating computation related to a certain crucial "associativity" property of an important class of vertex operator algebras. Very similar (somewhat forgotten) proofs had appeared by the 19-th century, of course without any motivation related to vertex operator algebras. Using this formula, we derive certain results, including especially the calculation of certain adjoint operators, of the classical umbral calculus. This is, roughly speaking, a reversal of the logical development of some standard treatments, which have obtained formulas for the higher derivatives of a composite function, most notably Faà di Bruno's formula, as a consequence of umbral calculus. We also show a connection between the Virasoro algebra and the classical umbral shifts. This leads naturally to a more general class of operators, which we introduce, and which include the classical umbral shifts as a special case. We prove a few basic facts about these operators.

Flag algebras

Asymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen sub-model of N with ∣M∣ elements is isomorphic to M. Which asymptotic relations exist between the quantities p(M1,N),…, p(Mh,N), where M1,…, M1, are fixed “template” models and ∣N∣ grows to infinity?In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density.