Single-molecule vibrational Raman spectroscopy of malachite green adsorbed on planar metal surfaces is achieved by means of optical local-field enhancement provided by a scanning nanoscopic metallic tip. The single-molecule signature is evident from spectral diffusion and a discretization of Raman peak intensities. The optical tip-sample coupling gives rise to a localization of the response down to a sub-10 nm length scale and a Raman enhancement up to ϳ5 ϫ 10 9 . This combines vibrational spectroscopy with high resolution scanningprobe microscopy for ultrasensitive in situ analysis of individual molecules.

The optical near-field distribution and enhancement near the apex of model scanning probe tips are calculated within the quasistatic approximation. The optical tip-sample coupling sensitively depends on both the tip and sample material. This, in addition to the tip-sample distance and apex geometry, is found to affect the spatial resolution that can be obtained in scattering near-field microscopy (s-SNOM). A pronounced structural plasmon resonant behavior is found for gold tips, which redshifts upon tip-sample approach on the length scale given by the tip radius. This near-field tip-sample coupling also allows for surface plasmon excitation in the sample. With the critical dimensions of the tip apex in the range of 10 to several 10s of nanometers, the results are found to be in good agreement with experiment and more rigorous theoretical treatments.

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Sesqui-pushout (SqPO) rewriting is a variant of transformations of graph-like and other types of structures that fit into the framework of adhesive categories where deletion in unknown context may be implemented. We provide the first account of a concurrency theorem for this important type of rewriting, and we demonstrate the additional mathematical property of a form of associativity for these theories. Associativity may then be exploited to construct so-called rule algebras (of SqPO type), based upon which in particular a universal framework of continuous-time Markov chains for stochastic SqPO rewriting systems may be realized. * This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 753750.1 While non-linear rules in SqPO rewriting have interesting applications in their own right (permitting e.g. the cloning and fusing of vertices in graphs), this most general case is left for future work.SqPO Rewriting: Concurrency, Associativity and Rule Algebra Framework adhesive categories (see Assumption 1) in order to ensure certain technical properties necessary for our concurrency and associativity theorems to hold. To the best of our knowledge, apart from some partial results in the direction of developing a concurrency theorem for SqPO-type rewriting in [16,36,15], prior to this work neither of the aforementioned theorems had been available in the SqPO framework.Associativity of SqPO rewriting theories plays a pivotal role in our development of a novel form of concurrent semantics for these theories, the so-called SqPO-type rule algebras. Previous work on associative DPO-type rewriting theories [3, 5, 7] (see also [8]) has led to a category-theoretical understanding of associativity that may be suitably extended to the SqPO setting. In contrast to the traditional and well-established formalisms of concurrency theory for rewriting systems (see e.g. [42,25,23,15] for DPO-type semantics and [16,15] for a notion of parallel independence and a Local Church-Rosser theorem for SqPO-rewriting of graphs), wherein the focus of the analysis is mostly on derivation traces and their sequential independence and parallelism properties, the focus of our rule-algebraic approach differs significantly: we propose instead to put sequential compositions of linear rules at the center of the analysis (rather than the derivation traces), and moreover to employ a vector-space based semantics in order to encode the non-determinism of such rule compositions. It is for this reason that the concurrency theorem plays a quintessential role in our rule algebra framework, in that it encodes the relationship between sequential compositions of linear rules and derivation traces, which in turn gives rise to the socalled canonical representations of the rule algebras (see Section 4). This approach in particular permits to uncover certain combinatorial properties of rewriting systems that would otherwise not be accessible. While undoubtedly not a stand...