Complexity growth in integrable and chaotic models

We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O($$ \sqrt{N} $$ N ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

Characterisation of hardwood fibres used for wood fibre insulation boards (WFIB)

Wood fibre insulation boards (WFIB) are typically made from softwood fibres. However, due to the rapid decrease in softwood stands in Germany, the industry will be forced to adapt to the wood market. Therefore, alternative approaches for the substitution of softwood with hardwood will be needed in the fibre industry. The objective of this paper is to address the characterisation of hardwood fibres regarding their availability for the WFIB industry. The physico-mechanical properties of WFIB are significantly determined by the length of the fibres. Longer softwood fibres usually generate higher strength properties and a lower thermal conductivity than shorter hardwood fibres. In this paper, the potential application of hardwood fibres (up to 20,500 μm long) produced in a refiner by thermo-mechanical pulping (TMP) to WFIB production was examined. The scanner-based system FibreShape was used for the automatic optical analysis of the geodesic length distribution of fibres. The analysed hardwood fibres contained significantly more dust and were shorter than respectively produced softwood fibres, limiting their applicability for WFIB production. Thus, two analytical approaches were chosen to receive longer fibres and less dust: (1) blending hardwood fibres with supporting softwood fibres (20%, 50 and 80% proportion of softwood), and (2) mathematical fractionation of hardwood fibres based on the fibre length to remove all particles smaller than 500 μm. It was concluded that the practical fractionation seems to be economically and ecologically challenging and that blending hardwood fibres with at least 50% softwood fibres offers a promising approach, which should be further studied.

Holographic renormalization group flow effect on quantum correlations

We holographically study the finite-size scaling effects on macroscopic and microscopic quantum correlations deformed by excitation and condensation. The excitation (condensation) increases (decreases) the entanglement entropy of the system. We also investigate the two-point correlation function of local operators by calculating the geodesic length connecting two local operators. As opposed to the entanglement entropy case, the excitation (condensation) decreases (increases) the two-point function. This is because the screening effect becomes strong in the background with the large entanglement entropy. We further show that the holographic renormalization leads to the qualitatively same two-point function as the one obtained from the geodesic length.

Geodesic Length Measurement in Medical Images: Effect of the Discretization by the Camera Chip and Quantitative Assessment of Error Reduction Methods

After interventions such as bypass surgeries the vascular function is checked qualitatively and remotely by observing the blood dynamics inside the vessel via Fluorescence Angiography. This state-of-the-art method has to be improved by introducing a quantitatively measured blood flow. Previous approaches show that the measured blood flow cannot be easily calibrated against a gold standard reference. In order to systematically address the possible sources of error, we investigated the error in geodesic length measurement caused by spatial discretization on the camera chip. We used an in-silico vessel segmentation model based on mathematical functions as a ground truth for the length of vessel-like anatomical structures in the continuous space. Discretization errors for the chosen models were determined in a typical magnitude of 6%. Since this length error would propagate to an unacceptable error in blood flow measurement, counteractions need to be developed. Therefore, different methods for the centerline extraction and spatial interpolation have been tested and compared against their performance in reducing the discretization error in length measurement by re-continualization. In conclusion, the discretization error is reduced by the re-continualization of the centerline to an acceptable range. The discretization error is dependent on the complexity of the centerline and this dependency is also reduced. Thereby the centerline extraction by erosion in combination with the piecewise Bézier curve fitting performs best by reducing the error to 2.7% with an acceptable computational time.

Semiclassical torus blocks in the t-channel

We explicitly demonstrate the relation between the 2-point t-channel torus block in the large-c regime and the geodesic length of a specific geodesic diagram stretched in the thermal AdS3 spacetime.

Holographic Formulation of 3D Metric Gravity with Finite Boundaries

In this work we construct holographic boundary theories for linearized 3D gravity, for a general family of finite or quasi-local boundaries. These boundary theories are directly derived from the dynamics of 3D gravity by computing the effective action for a geometric boundary observable, which measures the geodesic length from a given boundary point to some center in the bulk manifold. We identify the general form for these boundary theories and find that these are Liouville-like with a coupling to the boundary Ricci scalar. This is illustrated with various examples, which each offer interesting insights into the structure of holographic boundary theories.

Optimal Surveillance of Covert Networks by Minimizing Inverse Geodesic Length

The inverse geodesic length (IGL) is a well-known and widely used measure of network performance. It equals the sum of the inverse distances of all pairs of vertices. In network analysis, IGL of a network is often used to assess and evaluate how well heuristics perform in strengthening or weakening a network. We consider the edge-deletion problem MINIGLED. Formally, given a graph G, a budget k, and a target inverse geodesic length T, the question is whether there exists a subset of edges X with |X| ≤ ck, such that the inverse geodesic length of G − X is at most T.In this paper, we design algorithms and study the complexity of MINIGL-ED. We show that it is NP-complete and cannot be solved in subexponential time even when restricted to bipartite or split graphs assuming the Exponential Time Hypothesis. In terms of parameterized complexity, we consider the problem with respect to various parameters. We show that MINIGL-ED is fixed-parameter tractable for parameter T and vertex cover by modeling the problem as an integer quadratic program. We also provide FPT algorithms parameterized by twin cover and neighborhood diversity combined with the deletion budget k. On the negative side we show that MINIGL-ED is W[1]-hard for parameter tree-width.

Geometric Operators in the Einstein–Hilbert Truncation

We review the study of the scaling properties of geometric operators, such as the geodesic length and the volume of hypersurfaces, in the context of the Asymptotic Safety scenario for quantum gravity. We discuss the use of such operators and how they can be embedded in the effective average action formalism. We report the anomalous dimension of the geometric operators in the Einstein–Hilbert truncation via different approximations by considering simple extensions of previous studies.