On the effective reconstruction of expectation values from ab initio quantum embedding

Quantum embedding is an appealing route to fragment a large interacting quantum system into several smaller auxiliary `cluster' problems to exploit the locality of the correlated physics. In this work we critically review approaches to recombine these fragmented solutions in order to compute non-local expectation values, including the total energy. Starting from the democratic partitioning of expectation values used in density matrix embedding theory, we motivate and develop a number of alternative approaches, numerically demonstrating their efficiency and improved accuracy as a function of increasing cluster size for both energetics and non-local two-body observables in molecular and solid state systems. These approaches consider the $N$-representability of the resulting expectation values via an implicit global wave~function across the clusters, as well as the importance of including contributions to expectation values spanning multiple fragments simultaneously, thereby alleviating the fundamental locality approximation of the embedding. We clearly demonstrate the value of these introduced functionals for reliable extraction of observables and robust and systematic convergence as the cluster size increases, allowing for significantly smaller clusters to be used for a desired accuracy compared to traditional approaches in ab initio wave~function quantum embedding.Comment: 20 page

Neural Dynamics of Motion Processing and Speed Discrimination

A neural network model of visual motion perception and speed discrimination is presented. The model shows how a distributed population code of speed tuning, that realizes a size-speed correlation, can be derived from the simplest mechanisms whereby activations of multiple spatially short-range filters of different size are transformed into speed-tuned cell responses. These mechanisms use transient cell responses to moving stimuli, output thresholds that covary with filter size, and competition. These mechanisms are proposed to occur in the Vl→7 MT cortical processing stream. The model reproduces empirically derived speed discrimination curves and simulates data showing how visual speed perception and discrimination can be affected by stimulus contrast, duration, dot density and spatial frequency. Model motion mechanisms are analogous to mechanisms that have been used to model 3-D form and figure-ground perception. The model forms the front end of a larger motion processing system that has been used to simulate how global motion capture occurs, and how spatial attention is drawn to moving forms. It provides a computational foundation for an emerging neural theory of 3-D form and motion perception.Office of Naval Research (N00014-92-J-4015, N00014-91-J-4100, N00014-95-1-0657, N00014-95-1-0409, N00014-94-1-0597, N00014-95-1-0409); Air Force Office of Scientific Research (F49620-92-J-0499); National Science Foundation (IRI-90-00530

Twisted Blanchfield pairings, twisted signatures and Casson-Gordon invariants

This paper decomposes into two main parts. In the algebraic part, we prove an isometry classification of linking forms over $\mathbb{R}[t^{\pm 1}]$ and $\mathbb{C}[t^{\pm 1}]$. Using this result, we associate signature functions to any such linking form and thoroughly investigate their properties. The topological part of the paper applies this machinery to twisted Blanchfield pairings of knots. We obtain twisted generalizations of the Levine-Tristram signature function which share several of its properties. We study the behavior of these twisted signatures under satellite operations. In the case of metabelian representations, we relate our invariants to the Casson-Gordon invariants and obtain a concrete formula for the metabelian Blanchfield pairings of satellites. Finally, we perform explicit computations on certain linear combinations of algebraic knots, recovering a non-slice result of Hedden, Kirk and Livingston.Comment: 81 pages, 1 figur

Generalised Weber Functions

A generalised Weber function is given by \w_N(z) = \eta(z/N)/\eta(z), where $\eta(z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power \w_N^e evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating \w_N(z) and $j(z)$. Our ultimate goal is the use of these invariants in constructing reductions of elliptic curves over finite fields suitable for cryptographic use

Comparison of numerical wind modelling on hectometric and km scales at Kvitfjell wind power plant

Evaluating the performance of numerical weather models (NWMs) is crucial to identify the best choice of model for wind resource assessments. This thesis has investigated how well NWMs with different horizontal resolutions reproduced the wind directions, the wind speeds and the associated power production at a wind power plant located in complex terrain. The evaluated models included NORA3, WRF1km, and AROME Troms and Finnmark (ATF300m) throughout the 9 months from January to September 2022, as well as WRF111m for some case studies of shorter duration. The models were evaluated by comparing the simulations’ results to hub-height wind measurements. ATF300m outperformed the other models in terms of lower errors when considering the time period as a whole, and also for a case with wind directions similar to the main wind direction registered for the park. This could be as a result of the better topographic representation by the model, thereby including terrain induced effects on the wind more accurately such as orographic blocking. However, NORA3 outperformed the other models in terms of lower errors for two cases with wind directions coming from the NW sector. These results may suggest that 3 km grid spacing gives a sufficient representation of the wind fields when the wind is coming from the open ocean and is less influenced by the terrain before entering the park. Another suggestion may be that the coarser spatial resolution of NORA3 provides lower simulated wind speeds which counteracts the overestimation related to the absence of a wind farm parameterization

Relational Graph Models at Work

We study the relational graph models that constitute a natural subclass of relational models of lambda-calculus. We prove that among the lambda-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between lambda-terms. We focus on the two main observational equivalences in the lambda-calculus, the theory H+ generated by taking as observables the beta-normal forms, and H* generated by considering as observables the head normal forms. On the one hand we introduce a notion of lambda-K\"onig model and prove that a relational graph model is fully abstract for H+ if and only if it is extensional and lambda-K\"onig. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H*

Positive scalar curvature on manifolds with odd order abelian fundamental groups

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimension at least five which have odd order abelian fundamental groups, are nonspin and atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fundamental groups.Comment: 31 pages; 2 figures; minor edits; published versio