Geometric Bounds Research Articles

Quantum geometric bound and ideal condition for Euler band topology

Quantum geometric bound and ideal condition for Euler band topology

Unifying speed limit, thermodynamic uncertainty relation and Heisenberg principle via bulk-boundary correspondence

The bulk-boundary correspondence provides a guiding principle for tackling strongly correlated and coupled systems. In the present work, we apply the concept of the bulk-boundary correspondence to thermodynamic bounds described by classical and quantum Markov processes. Using the continuous matrix product state, we convert a Markov process to a quantum field, such that jump events in the Markov process are represented by the creation of particles in the quantum field. Introducing the time evolution of the continuous matrix product state, we apply the geometric bound to its time evolution. We find that the geometric bound reduces to the speed limit relation when we represent the bound in terms of the system quantity, whereas the same bound reduces to the thermodynamic uncertainty relation when expressed based on quantities of the quantum field. Our results show that the speed limits and thermodynamic uncertainty relations are two aspects of the same geometric bound.

A stochastic exposure model for seismic risk assessment and pricing of catastrophe bonds

Risk-based catastrophe bonds require the estimation of losses from the convolution of hazard, exposure and vulnerability models. These models are affected by different uncertainties that arise from the definition of their input parameters. In this paper, we propose a stochastic approach to treat the uncertainty in the asset location and attributes of the exposure model. The proposed method uses the Monte Carlo sampling approach to generate a stochastic exposure database, where each asset location is generated randomly within the geometric bound of the administration, while the asset attributes (i.e. construction type and material, number of storey, building activity type and number of dwelling) are sampled from distributions built from census data. Finally, a sensitivity analysis is performed to investigate the influence of the spatial resolution of the exposure model on the average annual losses (AAL) and catastrophe bond prices. To this end, we implement four exposure models, with spatial resolution at the asset, municipality and province levels, on a study region comprising ten provinces in southern Italy. Compared to the proposed model, the exposure model where assets are relocated and aggregated at the geometric centroid of the municipality underestimates AAL by 8%, while a higher difference (up to 20%) is observed for the exposure model where assets are relocated and aggregated at the geometric centroid of the province. We also consider an exposure model whose asset locations are extracted from publicly available building footprints. Yet, this latter model was incomplete for some provinces resulting in underestimation of AAL up to 90%. Differences in catastrophe bond prices obtained from the four exposure models are less evident, with the exposure model built based on the building footprints showing a difference up to 9.5%.

Explainable human activity recognition based on probabilistic spatial partitions for symbiotic workplaces

ABSTRACT In recent years, smart workplaces that adapt to human activity and motion behavior have been proposed for cognitive production systems. In this respect, methods for identifying the feelings and activities of human workers are being investigated to improve the cognitive capability of smart machines such as robots in shared working spaces. Recognizing human activities and predicting the possible next sequence of operations may simplify robot programming and improve collaboration efficiency. However, human activity recognition still requires explainable models that are versatile, robust, and interpretative. Therefore, recognizing and analyzing human action details using continuous probability density estimates in different workplace layouts is essential. Three scenarios are considered: a standalone, a one-piece flow U-form, and a human-robot hybrid workplace. This work presents a novel approach to human activity recognition based on a probabilistic spatial partition (HAROPP). Its performance is compared to the geometric-bounded activity recognition method. Results show that spatial partitions based on probabilistic density contain 20% fewer data frames and 10% more spatial areas than the geometric bounding box. The approach, on average, detects human activities correctly for 81% of the cases for a pre-known workplace layout. HAROPP has scalability and applicability potential for cognitive workplaces with a digital twin in the loop for pushing the cognitive capabilities of machine systems and realizing human-centered environments.

Numerically probing the universal operator growth hypothesis.

Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, nonintegrable many-body systems, the so-called Lanczos coefficients associated with an autocorrelation function grow asymptotically linear, with a logarithmic correction in one-dimensional systems. In contrast, the growth is expected to be slower in integrable or free models. In this paper, we numerically test this hypothesis for a variety of exemplary systems, including one-dimensional and two-dimensional Ising models as well as one-dimensional Heisenberg models. While we find the hypothesis to be practically fulfilled for all considered Ising models, the onset of the hypothesized universal behavior could not be observed in the attainable numerical data for the Heisenberg model. The proposed linear bound on operator growth associated with the hypothesis eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We derive and investigate a related geometric bound, and we find that while the bound itself is not sharply achieved for any considered model, the hypothesis is nonetheless fulfilled in most cases.

Geometric bounds on the power of adiabatic thermal machines.

We analyze the performance of slowly driven meso- and microscale refrigerators and heat engines that operate between two thermal baths with a small temperature difference. Using a general scaling argument, we show that such devices can work arbitrarily close to their Carnot limit only if heat leaks between the baths are fully suppressed. Their power output is then subject to a universal geometric bound that decays quadratically to zero at the Carnot limit. This bound can be asymptotically saturated in the quasistatic limit if the driving protocols are suitably optimized and the temperature difference between the baths goes to zero with the driving frequency. These results hold under generic conditions for any thermodynamically consistent dynamics admitting a well-defined adiabatic-response regime and a generalized Onsager symmetry. For illustration, we work out models of a qubit-refrigerator and a coherent charge pump operating as a cooling device.

Geomagnetic spikes on the core-mantle boundary

Extreme variations of Earth’s magnetic field occurred in the Levant region around 1000 BC, when the field intensity rapidly rose and fell by a factor of 2. No coherent link currently exists between this intensity spike and the global field produced by the core geodynamo. Here we show that the Levantine spike must span >60° longitude at Earth’s surface if it originates from the core–mantle boundary (CMB). Several low intensity data are incompatible with this geometric bound, though age uncertainties suggest these data could have sampled the field before the spike emerged. Models that best satisfy energetic and geometric constraints produce CMB spikes 8–22° wide, peaking at O(100) mT. We suggest that the Levantine spike reflects an intense CMB flux patch that grew in place before migrating northwest, contributing to growth of the dipole field. Estimates of Ohmic heating suggest that diffusive processes likely govern the ultimate decay of geomagnetic spikes.

Geometric Bounds for Buckling-Induced Auxetic Metamaterials Undergoing Large Deformation

The performance of a metamaterial is dominated by the geometric features and deformation mechanisms of its microstructure. For a certain mechanism, the geometric features have bounds in which the performance of a metamaterial such as negative Poisson’s ratio (NPR) can be designed. Previous investigation on buckling-induced auxetic metamaterial revealed that there is a geometric limit for its microstructure to exhibit auxetic behaviour in infinitesimal deformation. However, the limit for auxetic metamaterials undergoing large deformation is different from that under small deformation and has not been reported yet. In this paper, the geometric limit was investigated in an elastic and infinitesimal deformation range using linear buckling analysis. Furthermore, experimentally validated finite element models were used to identify the geometric limits for auxetic metamaterials undergoing large deformation. Depending on the control parameters of the topology, the bounds were represented by a line strip for one control parameter, an area for two control parameters and spatial domain surrounded by a 3D surface for three parameters. The limit was determined by the shape and size of the void of the metamaterials and it was identified through the large deformation analysis as well as the linear buckling analysis. We found that there was a significant difference in the geometric bounds obtained through those two methods. The results from this study can be used to design an auxetic metamaterial for different applications and to control the auxetic performance.

Geometric bounding box interpolation: an alternative for efficient video annotation

In video annotation, instead of annotating every frame of a trajectory, usually only a sparse set of annotations is provided by the user: typically its endpoints plus some key intermediate frames, interpolating the remaining annotations between these key frames in order to reduce the cost of the video labeling. While a number of video annotation tools have been proposed, some of which are freely available, and bounding box interpolation is mainly based on image processing techniques whose performance is highly dependent on image quality, occlusions, etc. We propose an alternative method to interpolate bounding box annotations, based on cubic splines and the geometric properties of the elements involved, rather than image processing techniques. The algorithm proposed is compared with other bounding box interpolation methods described in the literature, using a set of selected videos modeling different types of object and camera motion. Experiments show that the accuracy of the interpolated bounding boxes is higher than the accuracy of the other evaluated methods, especially when considering rigid objects. The main goal of this paper is related with the bounding box interpolation step, and we believe that our design can be integrated seamlessly with any annotation tool already developed.

Asymptotic Bounds on the Power-Delay Tradeoff for Fading Point-to-Point Links From Geometric Bounds on the Stationary Distribution of the Queue Length

The optimal power-delay tradeoff is studied for a time-slotted independently and identically distributed fading point-to-point link, with perfect channel state information at both transmitter and receiver, and with random packet arrivals to the transmitter queue. It is assumed that the transmitter can control the number of packets served by controlling the transmit power in the slot. The optimal tradeoff between average power and average delay is analyzed for stationary and monotone transmitter policies. For such policies, an asymptotic lower bound on the minimum average delay of the packets is obtained, when average transmitter power approaches the minimum average power required for transmitter queue stability. The asymptotic lower bound on the minimum average delay is obtained from geometric upper bounds on the stationary distribution of the queue length. This approach, which uses geometric upper bounds, also leads to an intuitive explanation of the asymptotic behavior of average delay. The asymptotic lower bounds, along with previously known asymptotic upper bounds, are used to identify three new cases where the order of the asymptotic behavior differs from that obtained from a previously considered approximate model, in which the transmit power is a strictly convex function of real valued service batch size for every fade state.

Geometric reliability analysis applied to wave overtopping of sea defences

Surge levels and waves are mutually dependent random variables, and this is reflected in their joint confidence regions or probability density contours (PDCs). The PDC generalises the concept of confidence intervals of a single variable in order to deal with multiple quantiles, so that the contour implies a geometric bound of observations falling inside it. This study introduces an efficient numerical scheme for quantifying the reliability index of a sea defence using a distance ratio of two PDCs, i.e., a dispersed PDC that just reaches the limit state surface and a one-standard-deviation PDC. The joint PDCs are defined in the original space of random variables and represented via a series of discrete vertices, which do not necessarily need to be smooth or elliptical in shape in order to fit different scattering patterns of the observations. Two numerical examples involving coastal wave overtopping problems indicate that the proposed contour-based expanding method (CBEM) provides flexibility to adopt various parametric or non-parametric joint distributions. The numerical implementation of the proposed algorithm graphically demonstrates an intuitive interpretation of the reliability index, which makes the relation between the joint PDCs and the limit state function more explicit.

On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound

The second largest eigenvalue in absolute value determines the rate of convergence of the Markov chain Monte Carlo methods. In this paper we consider the Gibbs sampler for the 1-D Ising model. We apply the geometric bound by Diaconis and Stroock (1991) to calculate an upper bound of the second largest eigenvalue, which we show is also a bound of the second largest eigenvalue in absolute value. Based on this upper bound, we derive that the convergence time is O(n2), where n is the number of sites. Our result includes a constant of proportionality, which enables us to give a precise bound of the convergence time. The results presented in this paper provide the lowest bound compared to those with a constant of proportionality in the literature.

A Note on the Elastic and Geometric Bounds for Composite Laminates

In this paper it is shown that a laminate composed of identical anisotropic layers, like for instance composite plies, cannot reach all the possible elastic states that are allowed by the classical elastic bounds on anisotropic constants. The analysis is carried on and facilitated by the use of particular tensor invariants, the so-called polar parameters.

Geometric bounding techniques for underwater localization using range-only sensors

This paper describes the application of geometric bounding techniques to range-only navigation of an underwater vehicle. A geometric technique is defined to obtain a position fix of an underwater vehicle using a combination of dead-reckoning navigation and acoustic measurements of range between the underwater vehicle and a global positioning system (GPS)-equipped ship. An assessment is made of the accuracy to which navigational parameters can be estimated using these methods.

Geometric Bounding Techniques for Underwater Navigation

This paper describes the application of geometric bounding techniques to navigation of an underwater vehicle. A technique is described for obtaining a fix of submersible position based on measurements of range from a supporting ship and the propagation of dead-reckoning errors. Also described is the estimation of bounds on a constant drift rate and the use of these bounds in the determination of velocity sensor misalignment angle. An assessment is made of the accuracy to which navigational parameters can be estimated using these methods.

Geometric Bounds: A Noniterative Analysis Technique for Closed Queueing Networks

We propose the Geometric Bounds (GBs), a new family of fast and accurate noniterative bounds on closed queueing network performance metrics that can be used in the online optimization of distributed applications. Compared to state-of-the-art techniques such as the Balanced Job Bounds (BJBs), GB achieves higher accuracy at similar computational costs, limiting the worst- case bounding error typically within 5-13 percent when, for the BJB, it is usually in the range of 15-35 percent. Optimization problems that are solved with GBs return solutions that are much closer to the global optimum than with existing bounds. We also show that the GB technique generalizes as an accurate approximation to closed fork-join networks commonly used in disk, parallel, and database models, thus extending the applicability of the method beyond the optimization of basic product-form networks.

Cusps of Hilbert modular varieties

AbstractMotivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifoldMto be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped nonsingular Hilbert modular surface.

Tight geometric bound for Marcum Q-function

A new geometric bound is proposed for the Marcum Q (MQ)-function. The proposed bounds are derived by the geometric interpretation of the first-order MQ-function, Q(a,b). It has been shown by previous researchers that Q(a,b) can be viewed as the probability that a complex, Gaussian random variable Z with real, nonzero mean lies outside of a circular region. Based on the interpretation, the new multi-sectors bound proves to be very tight and outperforms all previously proposed bounds.

Bounds on coefficients of reciprocals of formal power series with rapidly decreasing coefficients

This paper studies reciprocals of formal power series whose coefficients are monotone and bounded by a geometrically decaying sequence. Explicit and applicable, optimal decay rates are provided for the coefficients of the reciprocal series in terms of the parameters of the geometric bound. The results imply a best possible lower bound on the zeros of the series being considered.

3D data driven prediction for active contour models based on geometric bounding volumes

Active contour models have proven to be a promising approach for data driven object tracking without knowledge about the problem domain and the object. Problems arise in case of fast moving objects and in natural scenes with heterogeneous background. In these cases, a prediction step is an essential part of the tracking mechanism. In this paper we describe a new approach for modelling the contour of moving objects in the 3D world. The key point is the description of moving objects by a simplified geometric model, the so-called bounding volume. The contour of moving objects is predicted by estimating the movement and the shape of the bounding volume in the 3D world and by projecting its contour to the image plane. Stochastic optimization algorithms are used to estimate shape parameters of the bounding volume. The 2D contour of the bounding volume is used to initialize the active contour, which then extracts the contour of the moving object. Thus, an update of the motion and model parameters of the bounding volume is possible. No task specific knowledge and no a priori knowledge about the object is necessary. We will show that in the case of convex polyhedral bounding volumes, this method can be applied to real-time closed-loop object tracking on standard Unix workstations. Furthermore, we present experiments which prove that the robustness for tracking moving objects in front of a heterogeneous background can be improved.