Transseries as germs of surreal functions

We show that Écalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper

A note on the Cops & Robber game on graphs embedded in non-orientable surfaces

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.Comment: 5 pages, 1 figur

Orthogonal Decomposition of Definable Groups

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. As an application, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures

Exponential fields and Conway's omega-map

Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields

Graph easy sets of mute lambda terms

Among the unsolvable terms of the lambda calculus, the mute ones are those having the highest degree of undefinedness. In this paper, we define for each natural number n, an infinite and recursive set M-n of mute terms, and show that it is graph-easy: for any closed term t of the lambda calculus there exists a graph model equating all the terms of M-n to t. Alongside, we provide a brief survey of the notion of undefinedness in the lambda calculus. (C) 2015 Elsevier B.V. All rights reserved

Untyped Recursion Schemes and Infinite Intersection Types

Abstract. A new framework for higher-order program verification has been recently proposed, in which higher-order functional programs are modelled as higher-order recursion schemes and then model-checked. As recursion schemes are essentially terms of the simply-typed lambda-calculus with recursion and tree constructors, however, it was not clear how the new framework applies to programs written in languages with more advanced type systems. To circumvent the limitation, this paper introduces an untyped version of recursion schemes and develops an in-finite intersection type system that is equivalent to the model checking of untyped recursion schemes, so that the model checking can be re-duced to type checking as in recent work by Kobayashi and Ong for typed recursion schemes. The type system is undecidable but we can obtain decidable subsets of the type system by restricting the shapes of intersection types, yielding a sound (but incomplete in general) model checking algorithm.

Insulin Degrading Enzyme Induces a Conformational Change in Varicella-Zoster Virus gE, and Enhances Virus Infectivity and Stability

Varicella-zoster virus (VZV) glycoprotein E (gE) is essential for virus infectivity and binds to a cellular receptor, insulin-degrading enzyme (IDE), through its unique amino terminal extracellular domain. Previous work has shown IDE plays an important role in VZV infection and virus cell-to-cell spread, which is the sole route for VZV spread in vitro. Here we report that a recombinant soluble IDE (rIDE) enhances VZV infectivity at an early step of infection associated with an increase in virus internalization, and increases cell-to-cell spread. VZV mutants lacking the IDE binding domain of gE were impaired for syncytia formation and membrane fusion. Pre-treatment of cell-free VZV with rIDE markedly enhanced the stability of the virus over a range of conditions. rIDE interacted with gE to elicit a conformational change in gE and rendered it more susceptible to proteolysis. Co-incubation of rIDE with gE modified the size of gE. We propose that the conformational change in gE elicited by IDE enhances infectivity and stability of the virus and leads to increased fusogenicity during VZV infection. The ability of rIDE to enhance infectivity of cell-free VZV over a wide range of incubation times and temperatures suggests that rIDE may be useful for increasing the stability of varicella or zoster vaccines

Disruption of PML Nuclear Bodies Is Mediated by ORF61 SUMO-Interacting Motifs and Required for Varicella-Zoster Virus Pathogenesis in Skin

Promyelocytic leukemia protein (PML) has antiviral functions and many viruses encode gene products that disrupt PML nuclear bodies (PML NBs). However, evidence of the relevance of PML NB modification for viral pathogenesis is limited and little is known about viral gene functions required for PML NB disruption in infected cells in vivo. Varicella-zoster virus (VZV) is a human alphaherpesvirus that causes cutaneous lesions during primary and recurrent infection. Here we show that VZV disrupts PML NBs in infected cells in human skin xenografts in SCID mice and that the disruption is achieved by open reading frame 61 (ORF61) protein via its SUMO-interacting motifs (SIMs). Three conserved SIMs mediated ORF61 binding to SUMO1 and were required for ORF61 association with and disruption of PML NBs. Mutation of the ORF61 SIMs in the VZV genome showed that these motifs were necessary for PML NB dispersal in VZV-infected cells in vitro. In vivo, PML NBs were highly abundant, especially in basal layer cells of uninfected skin, whereas their frequency was significantly decreased in VZV-infected cells. In contrast, mutation of the ORF61 SIMs reduced ORF61 association with PML NBs, most PML NBs remained intact and importantly, viral replication in skin was severely impaired. The ORF61 SIM mutant virus failed to cause the typical VZV lesions that penetrate across the basement membrane into the dermis and viral spread in the epidermis was limited. These experiments indicate that VZV pathogenesis in skin depends upon the ORF61-mediated disruption of PML NBs and that the ORF61 SUMO-binding function is necessary for this effect. More broadly, our study elucidates the importance of PML NBs for the innate control of a viral pathogen during infection of differentiated cells within their tissue microenvironment in vivo and the requirement for a viral protein with SUMO-binding capacity to counteract this intrinsic barrier