Homotopy theory of monoids and derived localization

We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and new. This includes a recent generalization of Adams’s cobar-construction to the non-simply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an $$\infty $$ ∞ -category of discrete monoids.

The Robin function and conformal welding – A new proof of the existence

Green’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

‘That Miserable Party Spirit’

Chapter 4 considers Mary’s relationship with the Conservative Party in the late 1860s and early 1870s. In order to examine the influential role Mary played at the heart of the Conservative Party, this chapter considers three connected case studies. The first considers the disquiet over Disraeli’s leadership of the Conservative Party during the years 1868–1874 and examines the attempts by Mary and her political allies to oust him as leader. The second case study examines the intersections between Mary and the 1874 Conservative cabinet. It pays particular attention to her pivotal role in the formation of that cabinet. The third case study develops this narrative and explores her involvement in the processes and tensions of cabinet government during the years 1874–1876. In doing so, it considers the challenges and constraints offered up by the post-1867 landscape. Significantly, this chapter also casts new light on the fragility of Disraeli’s leadership of the Conservative Party.

Antipodes of monoidal decomposition spaces

We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as [Formula: see text], just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors [Formula: see text], and it is a refinement of the general Möbius inversion construction of Gálvez–Kock–Tonks, but exploiting the monoidal structure.

A Linear Multiport Network Approach for Elastically Coupled Underactuated Grippers

This paper presents a framework based on multiport network theory for modeling underactuated grippers where the actuators produce finger motion by deforming an elastic transmission mechanism. If the transmission is synthesized from compliant components joined together with series (equal force) or parallel (equal displacement) connections, the resulting multiport immittance (stiffness) matrix for the entire transmission can be used to deduce how the object will behave in the grasp. To illustrate this, a three-fingered gripper is presented in which each finger is driven by one of two linear two-port spring networks. The multiport approach predicts contact force distribution with good fidelity even with asymmetric objects. The parallel-connected configuration exhibited object rotation and was more prone to object ejection than the series-connected case, which balanced the contact forces evenly.

The Carathéodory Reflection Principle and Osgood–Carathéodory Theorem on Riemann Surfaces

The Osgood–Carathéodory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the Carathéodory reflection principle.

Formality and the Lefschetz property in symplectic and cosymplectic geometry

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b

Asymptotic Properties of Some Minor-Closed Classes of Graphs

Let${\cal A}$be a minor-closed class of labelled graphs, and let${\cal G}_{n}$be a random graph sampled uniformly from the set ofn-vertex graphs of${\cal A}$. Whennis large, what is the probability that${\cal G}_{n}$is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.Using exact enumeration, we study a collection of classes${\cal A}$excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating functionC(z) that counts connected graphs of${\cal A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-Gaussian limit laws (Beta and Gamma), and clearly merits a systematic investigation.