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...

We complete the reformulation of the holographic correspondence as a highly efficient RG flow that can also determine the UV data in the field theory in the strong coupling and large N limit. We introduce a special way to define operators at any given scale in terms of appropriate coarse-grained collective variables, without requiring the use of the elementary fields. The Wilsonian construction is generalised by promoting the cut-off to a functional of these collective variables. We impose three criteria to determine the coarse-graining. The first criterion is that the effective Ward identities for local conservation of energy, momentum, etc. should preserve their standard forms, but in new scale-dependent background metric and sources which are functionals of the effective single trace operators. The second criterion is that the scale-evolution equations of the operators in the actual background metric should be state-independent, implying that the collective variables should not explicitly appear in them. The final required criterion is that the endpoint of the scale-evolution of the RG flow can be transformed to a fixed point corresponding to familiar non-relativistic equations with a finite number of parameters, such as incompressible non-relativistic Navier-Stokes, under a certain universal rescaling of the scale and of the time coordinate.Using previous work, we explicitly show that in the hydrodynamic limit each such highly efficient RG flow reproduces a unique classical gravity theory with precise UV data that satisfy our IR criterion and also lead to regular horizons in the dual geometries. We obtain the explicit coarse-graining which reproduces Einstein's equations. In a simple example, we are also able to construct a low energy effective action and compute the beta function. Finally, we show how our construction can be interpolated with the traditional Wilsonian RG flow at a suitable scale, and can be used to develop new non-perturbative frameworks for QCD-like theories.

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We investigate how the holographic correspondence can be reformulated as a generalisation of Wilsonian RG flow in a strongly interacting large N quantum field theory. We firstly define a highly efficient RG flow as one in which the Ward identities related to local conservation of energy, momentum and charges preserve the same form at each scale -to achieve this it is necessary to redefine the background metric and external sources at each scale as functionals of the effective single trace operators. These redefinitions also absorb the contributions of the multi-trace operators to these effective Ward identities. Thus the background metric and external sources become effectively dynamical reproducing the dual classical gravity equations in one higher dimension. Here, we focus on reconstructing the pure gravity sector as a highly efficient RG flow of the energy-momentum tensor operator, leaving the explicit constructive field theory approach for generating such RG flows to the second part of the work. We show that special symmetries of the highly efficient RG flows carry information through which we can decode the gauge fixing of bulk diffeomorphisms in the corresponding gravity equations. We also show that the highly efficient RG flow which reproduces a given classical gravity theory in a given gauge is unique provided the endpoint can be transformed to a non-relativistic fixed point with a finite number of parameters under a universal rescaling.The results obtained here are used in the second part of this work, where we do an explicit field-theoretic construction of the RG flow, and obtain the dual classical gravity theory.