Position Uncertainty Research Articles

Particle in a Box with Generalized Uncertainty Principle

in ordinary quantum mechanics for the position uncertainty since the minimum value for the uncertainty can be tend to zero. Different theories of quantum gravity, such as string theory, loop quantum gravity, and black hole physics, confirm a minimal observable length. Kempf et al have formulated The minimal length is a fundamental length for position measurement, although there is no lower bound a generalized uncertainty principle (GUP) by modifying the Heisenberg uncertainty principle to include an appearance of the length in quantum mechanics. In this study, we have found out the influence of the GUP on the particle for its confinement in a 1D potential box and calculated energy eigenvalues. The modified Schrödinger equation for the particle is factorized to be of second order to obtain the eigenvalues in a handy way. The obtained energy levels are modified from the usual result with GUP parameter dependency. The modification in the energy spectrum due to the GUP characterized by the presence of the minimal length has been also explored graphically.

A random error estimation method for satellites being launched into orbit based on ellipsoid theory

Accurate and fast random error estimation is essential to improve satellite control. In this paper, the random error of the satellite being launched into orbit is explored. According to the uncertainty matrix of six satellite orbit parameters associated with the project, the uncertainty of the orbit position in the three axes of the geocentric inertial frame of reference is calculated via the propagation rate of the covariance matrix. Then, the random error probability density of the satellite position is constructed for normal distribution. The surface of the error with the same probability density is approximated as an ellipsoid, the size is deduced along with the ellipsoid's shape and direction, and the satellite's spatial position probability is obtained. Monte Carlo method and STK software were employed to simulate low-earth orbit micro-nano satellite launching into the same orbital position. It is confirmed that the results obtained via the theoretical model are in line with practice. Therefore, the ellipsoid theory proposed in this paper can be used to describe the satellite random position error.

Phase-Dependent Asymmetry of Pattern Masking in Natural Images Explained by Intrinsic Position Uncertainty

Phase-Dependent Asymmetry of Pattern Masking in Natural Images Explained by Intrinsic Position Uncertainty

Validation of robust radiobiological optimization algorithms based on the mixed beam model for intensity-modulated carbon-ion therapy.

Currently, treatment planning systems (TPSs) that can compute the intensities of intensity-modulated carbon-ion therapy (IMCT) using scanned carbon-ion beams are limited. In the present study, the computational efficacy of the newly designed IMCT algorithms was analyzed for the first time based on the mixed beam model with respect to the physical and biological doses; moreover, the validity and effectiveness of the robust radiobiological optimization were verified. A dose calculation engine was independently generated to validate a clinical dose determined in the TPS. A biological assay was performed using the HSGc-C5 cell line to validate the calculated surviving fraction (SF). Both spot control (SC) and voxel-wise worst-case scenario (WC) algorithms were employed for robust radiobiological optimization followed by their application in a Radiation Therapy Oncology Group benchmark phantom under homogeneous and heterogeneous conditions and a clinical case for range and position errors. Importantly, for the first time, both SC and WC algorithms were implemented in the integrated TPS platform that can compute the intensities of IMCT using scanned carbon-ion beams for robust radiobiological optimization. For assessing the robustness, the difference between the maximum and minimum values of a dose-volume histogram index in the examined error scenarios was considered as a robustness index. The relative biological effectiveness (RBE) determined by the independent dose calculation engine exhibited a -0.6% difference compared with the RBE defined by the TPS at the isocenter, whereas the measured and the calculated SF were similar. Regardless of the objects, compared with the conventional IMCT, the robust radiobiological optimization enhanced the sensitivity of the examined error scenarios by up to 19% for the robustness index. The computational efficacy of the novel IMCT algorithms was verified according to the mixed beam model with respect to the physical and biological doses. The robust radiobiological optimizations lowered the impact of range and position uncertainties considerably in the examined scenarios. The robustness of the WC algorithm was more enhanced compared with that of the SC algorithm. Nevertheless, the SC algorithm can be used as an alternative to the WC IMCT algorithm with respect to the computational cost.

Enhancing the alignment of the optically brightGaiareference frame with respect to the International Celestial Reference System

Context.The link of theGaiaframe in terms of non-rotation with respect to the International Celestial Reference System (ICRS), which is realized via very long baseline interferometry (VLBI) at radio wavelengths, has to be conducted for the wide range of optical magnitudes in which the spacecraft observes. There is a sufficient number of suitable counterparts between the two measurement systems for optically faint objects. However, the number of common optically bright (G≤ 13 mag) objects is sparse as most are faint at radio frequencies, and only a few objects suitable for astrometry have been observed by VLBI in the past. As a result, rotation parameters for the optically brightGaiareference frame are not yet determined with sufficient accuracy.Aims.The verification of theGaiabright frame of DR2 and EDR3 is enhanced by the reevaluation of existing VLBI observations and the addition of newly acquired data for a sample of optically bright radio stars.Methods.Historical data from the literature were reevaluated, ensuring that the calibrator positions and uncertainties (used for the determination of the absolute star positions in the phase-referencing analysis) were updated and homogeneously referred to the ICRF3, the third realization of the International Celestial Reference Frame. We selected 46 suitable optically bright radio stars from the literature for new radio observations, out of which 32 were detected with the VLBA in continuum mode in the X or C band, along with radio-bright calibrators in the ICRF3. ImprovedGaia-VLBI rotation parameters were obtained by adding new observations and utilizing more realistic estimates of the absolute position uncertainties for all phase-referenced radio observations.Results.The homogenization greatly improved the steadiness of the results when the most discrepant stars were rejected one after another through a dedicated iterative process. ForGaiaDR2, this homogenization reduced the magnitude of the orientation parameters to less than 0.5 mas but increased that of the spin parameters, with the largest component being the rotation around the Y axis. An adjustment of the position uncertainties improved the reliability of the orientation parameters and the goodness of fit for the iterative solutions. Introducing the new single-epoch positions to the analysis reduced the correlations between the rotation parameters. The final spin forGaiaDR2 as determined by VLBI observations of radio stars is (−0.056, −0.113, +0.033) ± (0.046, 0.058, 0.053) mas yr−1. A comparison of the new results with external, independently derived spin parameters forGaiaDR2 reveals smaller differences than when using the historical data from the literature. Applying the VLBI data toGaiaEDR3, which was already corrected for spin duringGaiaprocessing, the derived residual spin is (+0.022, +0.065, −0.016) ± (0.024, 0.026, 0.024)mas yr−1, showing that the component in Y is significant at the 2.4σlevel.Conclusions.Even though our analysis provides a more accurate frame tie, more VLBI data are needed to refine the results and reduce the scatter between iterative solutions.

Contour integration deficits at high spatial frequencies in children treated for anisometropic amblyopia.

This study was conducted to reexamine the question of whether children treated for anisometropic amblyopia have contour integration deficits. To do so, we used psychophysical methods that require global contour processing while minimizing the influence of low-level deficits: visibility, shape perception, and positional uncertainty. Thirteen children with anisometropic amblyopia (age: 10.1 ± 1.8 years) and thirteen visually normal children (age: 10.8 ± 2.0 years) participated in this study. The stimuli were closed figures made up of Gabor patches either in noise or on a blank field. The contrast thresholds to detect a circular contour on a blank field, as well as the thresholds of aspect ratio and contour element number to discriminate a circular or elliptical contour in noise, were measured at Gabor spatial frequencies of 1.5, 3, and 6 cpd for amblyopic eyes (AEs), fellow eyes (FEs), and normal control eyes. Visual acuities and contrast sensitivity functions for AEs and FEs and the Randot stereoacuity were measured before testing. The AEs showed contrast deficits and degraded shape perception compared to the FEs at higher spatial frequencies (6 cpd). When the influence of abnormal contrast sensitivity and shape perception were minimized, the AEs showed contour integration deficits at spatial frequencies 3 and 6 cpd. These deficits were not related to basic losses in contrast sensitivity and acuity, stereoacuity, and visual crowding. Besides, no significant difference was found between the fellow eyes of the amblyopic children and the normal control eyes in the performance of contour integration. After eliminating or compensating for the low-level deficits, children treated for anisometropic amblyopia still show contour integration deficits, primarily at higher spatial frequencies, which might reflect the deficits in global processing caused by amblyopia. Contour integration deficits are likely independent of spatial vision deficits. Refractive correction and/or occlusion therapies may not be sufficient to fully restore contour integration deficits, which indicates the need for the development of clinical treatments to recover these deficits.

Performance of inhomogeneous Poisson point process models under different scenarios of uncertainty in species presence-only data

Haphazard and opportunistic species occurrence (PO) data are widely used in species distribution models (SDMs) instead of high-quality species data gathered using appropriate and structured sampling methods, which is expensive and often spatially limited. Despite their widespread use in ecology, PO data are prone to errors and uncertainties, such as imperfect detectability, positional imprecision, and spatial niche truncation, which make their use analytically challenging for effective and adaptive biodiversity management and conservation. Using simulated data, this study investigates the effects of these uncertainties on the performance of spatial point process based presence-only and integrated SDMs. We investigated three SDMs in this study, one that ignores imperfect detectability: the presence-only model (PO model), and two that account for it: the thinned presence-only model (THINPO model) and the integrated model (PBPC model). The ability of these SDMs to produce accurate maximum likelihood estimates of intensity model coefficients and reliable predictions of species distributions under different data quality scenarios was investigated. The results show that SDMs that account for imperfect detectability (THINPO or PBPC models) are not applicable in situations of high detectability. In this situation, the PO model produces the most accurate maximum likelihood estimates of the models’ coefficients ({hat{beta }}_k), and consequently the most accurate predictions of species distributions ({hat{lambda }}(s)). The effects of positional uncertainty and spatial niche truncation on this SDM output are minimal. However, in situations of low detectability, it is preferable to use the PBPC model. Positional uncertainty and spatial niche truncation have negligible effects on the output of this SDM, except when positionally uncertain PO data are analyzed along with truncated PC data. These minimal effects of spatial niche truncation on SDM outputs demonstrate the transferability of SDMs. However, the effects of all these uncertainties may depend on the characteristics of the species. Prior to modeling species distributions, a multivariate environmental similarity surface analysis should be performed to test the similarity between data from the restricted region to be used for model calibration and data from the entire range. If this analysis reveals dissimilarities, larger spatial and ecological scales should be considered to address the issue of spatial niche truncation. Further efforts could address the effects of species characteristics on SDMs performance and assess the effects of species-specific uncertainties.

Source Localization Using RSS Measurements with Sensor Position Uncertainty

Received signal strength- (RSS-) based localization has attracted considerable attention for its low cost and easy implementation. In plenty of existing work, sensor positions, which play an important role in source localization, are usually assumed perfectly known. Unfortunately, they are often subject to uncertainties, which directly leads to effect on localization result. To tackle this problem, we study the RSS-based source localization with sensor position uncertainty. Sensor position uncertainty will be modeled as two types: Gaussian random variable and unknown nonrandom variable. For either of the models, two semidefinite programming (SDP) methods are proposed, i.e., SDP-1 and SDP-2. The SDP-1 method proceeds from the nonconvex problem with respect to the maximum likelihood (ML) estimation and then obtains an SDP problem using proper approximation and relaxation. The SDP-2 method first transfers the sensor position uncertainties to the source position and then obtains an SDP formulation following a similar idea as in SDP-1 method. Numerical examples demonstrate the performance superiority of the proposed methods, compared to some existing methods assuming perfect sensor position information.

Numerical Analysis of GNSS Signal Outage Effect on EOPs Solutions Using Tightly Coupled GNSS/IMU Integration: A Simulated Case Study in Sweden.

The absence of a reliable Global Navigation Satellite System (GNSS) signal leads to degraded position robustness in standalone receivers. To address this issue, integrating GNSS with inertial measurement units (IMUs) can improve positioning accuracy. This article analyzes the performance of tightly coupled GNSS/IMU integration, specifically the forward Kalman filter and smoothing algorithm, using both single and network GNSS stations and the post-processed kinematic (PPK) method. Additionally, the impact of simulated GNSS signal outage on exterior orientation parameters (EOPs) solutions is investigated. Results demonstrate that the smoothing algorithm enhances positioning uncertainty (RMSE) for north, east, and heading by approximately 17-43% (e.g., it improves north RMSE from 51 mm to a range of 42 mm, representing a 17% improvement). Orientation uncertainty is reduced by about 60% for roll, pitch, and heading. Moreover, the algorithm mitigates the effects of GNSS signal outage, improving position uncertainty by up to 95% and orientation uncertainty by up to 60% using the smoothing algorithm instead of the forward Kalman filter for signal outages up to 180 s.

Remote Sensing of Riverbank Migration Using Particle Image Velocimetry

AbstractMobile river channels endanger human life and property and over centuries shape ecosystems, landscapes, and stratigraphy. Quantifying channel movements from remote sensing is difficult, in part due to the diversity of river mobility processes (e.g., channel migration, cutoffs, avulsion) and planform morphologies (e.g., meandering, braided). Here, we present a framework for quantifying riverbank migration from remote sensing that upscales recent methodological advances from laboratory flume studies utilizing particle image velocimetry (PIV). We apply PIV to image time series of 21 rivers worldwide, showing PIV ignores cutoff and avulsion processes by design and is well suited for tracking riverbank migration regardless of planform morphology. We show that PIV‐derived results for riverbank migration are consistent with published results from centerline‐ and bank‐based Lagrangian methods. Unlike existing methods, PIV offers a grid‐based Eulerian framework where defining channel centerlines is unnecessary and quantified uncertainty in riverbank positions is propagated into uncertainty in migration rates. PIV offers means to efficiently extract global patterns in riverbank migration from decades of satellite data, as well as investigate river response to climate change and human activities in our rapidly changing world.

A deep neural network architecture for reliable 3D position and size determination for Lagrangian particle tracking using a single camera

Microfluidic flows feature typically fully three-dimensional velocity fields. However, often the optical access for measurements is limited. Astigmatism or defocus particle tracking velocimetry is a technique that enables the 3D position determination of individual particles by the analysis of astigmatic/defocused particle images. The classification and position determination of particles is a task well suited to deep neural networks (DNNs). In this work, two DNNs are used to extract the class and in-plane position (object detection) as well as the depth position (regression). The performance of both DNNs is assessed by the position uncertainties as well as the precision of the size classes and the amount of recalled particles. The DNNs are evaluated on a synthetic dataset and establish a new benchmark of DNNs in defocus tracking applications. The recall is higher than compared to classic methods and the in-plane errors are always subpixel accurate. The relative uncertainty in the depth position is below 1% for all examined particle seeding concentrations. Additionally, the performance on experimental images, using four different particle sizes, ranging from 1.14 to 5.03 is analyzed. The particle images are systematically rearranged to produce comprehensive datasets of varying particle seeding concentrations. The distinction between particles of similar size is more challenging but the DNNs still show very good results. A precision above 96% is reached with a high recall above 95%. The error in the depth position remains below 1% and the in-plane errors are subpixel accurate with respect to the labels. The work shows that first, DNNs can be trained with artificially rearranged data sets based on individual experimental images and are therefore easily adaptable to various experimental setups and applicable by non-experts. Second, the DNNs can be successfully adapted to determine additional variables as in this case the size of the suspended particles.

3D Point Cloud Object Detection Algorithm Based on Temporal Information Fusion and Uncertainty Estimation

In autonomous driving, LiDAR (light detection and ranging) data are acquired over time. Most existing 3D object detection algorithms propose the object bounding box by processing each frame of data independently, which ignores the temporal sequence information. However, the temporal sequence information is usually helpful to detect the object with missing shape information due to long distance or occlusion. To address this problem, we propose a temporal sequence information fusion 3D point cloud object detection algorithm based on the Ada-GRU (adaptive gated recurrent unit). In this method, the feature of each frame for the LiDAR point cloud is extracted through the backbone network and is fed to the Ada-GRU together with the hidden features of the previous frames. Compared to the traditional GRU, the Ada-GRU can adjust the gating mechanism adaptively during the training process by introducing the adaptive activation function. The Ada-GRU outputs the temporal sequence fusion features to predict the 3D object in the current frame and transmits the hidden features of the current frame to the next frame. At the same time, the label uncertainty of the distant and occluded objects affects the training effect of the model. For this problem, this paper proposes a probability distribution model of 3D bounding box coordinates based on the Gaussian distribution function and designs the corresponding bounding box loss function to enable the model to learn and estimate the uncertainty of the positioning of the bounding box coordinates, so as to remove the bounding box with large positioning uncertainty in the post-processing stage to reduce the false positive rate. Finally, the experiments show that the methods proposed in this paper improve the accuracy of the object detection without significantly increasing the complexity of the algorithm.

Multirobot Navigation Using Partially Observable Markov Decision Processes with Belief-Based Rewards

For the safe deployment of multirobot systems, accurate position estimation is necessary. We present a multirobot navigation algorithm that accomplishes the dual objective of reaching designated goal positions and maintaining low position uncertainty. We pose the problem as a -partially observable Markov decision process (POMDP) to capture position uncertainty through a belief-based reward. The multirobot system employs a centralized extended Kalman filter (EKF) for state estimation, whose output is used as the observation as well as the belief in the POMDP. Using the EKF output as the observation and belief allows for the integration of the EKF in the action planning module of the system. To solve the POMDP, we adapt the online information particle filter tree algorithm to be compatible with the EKF closed-form output being used as both the observation and belief. We present results for the proposed navigation algorithm that show emergent behaviors for a multirobot system, where the robots move to provide inter-ranging measurements that minimize position uncertainty for all robots in the system. We also demonstrate that the proposed method outperforms a state-of-the-art solver (partially observable Monte Carlo planning with observation widening) and a direct-to-goal planner in terms of minimizing position uncertainty.

Long‐term drivers of shoreline change over two centuries on a headland‐embayment beach

AbstractShoreline evolution over the last two centuries was analysed for Dundrum Bay, Co. Down, Northern Ireland, using historical and recent shoreline datasets from 1833 to 2020. The area of interest comprises two sandy beaches and vegetated coastal dune fields, Newcastle–Murlough and Ballykinler, separated by an inlet channel which connects the inner with the outer bay. Twenty‐four temporal shorelines were extracted, and a quantitative assessment of their positional uncertainty was performed. This was combined with analysis of foredune volume variations by applying the Structure‐from‐Motion‐Multi‐View‐Stereo technique to 1963 aerial photography and comparing it with a 2014 Light Detection And Ranging (LiDAR) dataset to better inform on links between the sediment dynamics and the observed shoreline changes. Storm events were identified using recorded extreme water levels (EWLs) (1901–2020) and hindcasted wave data (1948–2020). In the first 87 years, the shoreline was largely stable, and change was focused at the inlet area. In the 20th century, localised retreat characterised the western (Newcastle–Murlough) sector, whereas the eastern sector (Ballykinler) experienced general shoreline advance. The rate and extent of shoreline retreat in the western sector increased post‐1997 in concert with accelerated accretion at Ballykinler. The strongest erosional episodes were recorded in 1920–1951, 1997–2005, and 2012–2014. Although no direct link was established between single storms and shoreline retreat rates, the three major retreat periods coincided with several consecutive EWLs or EWLs with a return period greater than 100 years. These were generated by storm directions ranging from south to southwest. The long‐term pattern of shoreline change points to a complex coastal system that is still evolving.

СОЦИОКУЛЬТУРНАЯ МАРГИНАЛЬНОСТЬ КАК ПРОБЛЕМНОЕ СВОЙСТВО СТУДЕНЧЕСКОЙ МОЛОДЕЖИ В СОВРЕМЕННОМ ОБЩЕСТВЕ РИСКА

The authors of this article consider the sociocultural marginality of student youth in the modern risk society as a problem property. Properties inherent in student youth: the passage of the process of secondary socialization and the assimilation of various social functions; the problem of choosing social values and guidelines; struggle between responsibility and irresponsibility; rationality and irrationality; the impact of the information environment on consciousness; being at the intersection of the real and virtual worlds, balancing between them. The main criteria for the marginal position of student youth are: the state of uncertainty and transitivity of status positions; exclusion or partial and incomplete inclusion of student youth in functioning social and professional structures; the destructive state of previously developed social norms, the inability to follow them; anomie as a social disorganization of the risk society, which determines the behavioral strategies of young people.

EAAINet: An Element-Wise Attention Network With Global Affinity Information for Accurate Indoor Visual Localization

Visual localization, a vital component of many visual applications, has been tackled by scene coordinates regression (SCoRe) methods that leverage neural networks to predict scene coordinates, followed by a PnP algorithm to recover camera pose. However, these methods do not consider the relationship between image patches, known as relative features or affinity information, which is instrumental for network to perform complete scene parsing. Besides, owing to the visual similarity between image patches, these methods are weak in extracting reliable absolute features that represent the context information of the image patches, resulting in inferior localization performance. In response, we propose EAAINet that is based on classical SCoRe approaches and consists of two novel modules: the Global Affinity Aggregation Module (GAAM) and the Element-wise Attention Module (EAM). Specifically, GAAM employs an interval sampling strategy to sample image patches to construct sparse graph neural networks (GNNs), from which global affinity information between image patches is retrieved, hence ensuring precise scene parsing. EAM integrates multi-level features to generate reliable absolute features to regress accurate scene coordinates, with the key insight that the structure information is essential to differentiate similar image patches and the semantic information assists in modeling regression problems. Technically, EAM predicts element-wise soft attention masks to reconcile multi-level feature maps, enabling efficient feature fusion. Positional encoding and uncertainty modeling are also employed to enhance visual localization performance. Experimental results show that EAAINet significantly outperforms the state-of-the-arts on multiple benchmarks with faster speed and less model parameters. Source code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/DK-HU/EAAINet</uri> .

Relativistic probability densities for location

Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ( x , t) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes yield normalized functions ψ( x , t) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ μ ( x , t) = (ℏ/2i)[ψ +( x , t)∂ μ ψ( x , t) − ∂ μ ψ +( x , t) · ψ( x , t)]. However, in the case of bosonic fields, the wave packets ψ( x , t) are nonlocally related to the corresponding relativistic quantum fields ϕ( x , t), and therefore the canonical local energy-momentum densities and differ from ℘ μ ( x , t) and appear nonlocal in terms of the wave packets ψ( x , t). We examine the relation between the canonical energy density , the canonical charge density ϱ( x , t), the energy pseudo-density , and the Born density ∣ψ( x , t)∣2 for the massless free Klein–Gordon field. We find that those four proxies for particle location are tantalizingly close even in this extremely relativistic case: in spite of their nonlocal mathematical relations, they are mutually local in the sense that their maxima do not deviate beyond a common position uncertainty Δx. Indeed, they are practically indistinguishable in cases where we would expect a normalized quantum state to produce particle-like position signals, viz. if we are observing quanta with momenta p ≫ Δp ≥ ℏ/2Δx. We also translate our results to massless Dirac fields. Our results confirm and illustrate that the normalized energy density provides a suitable measure for positions of bosons, whereas normalized charge density ϱ( x , t)/q provides a suitable measure for fermions.

A Two-Stage Aerial Target Localization Method Using Time-Difference-of-Arrival Measurements with the Minimum Number of Radars

Distributed radar systems promise to significantly enhance target localization by virtue of the superiority of multi-view observations from widely separated radars, compared to their monostatic counterparts. Nevertheless, when the radar number is limited, performing target localization bears the brunt of the parameter identifiability requirement that the parameter number must be no less than the number of independent measurements. In this way, the canonical two-stage target localization method, as well as its developments, is no longer appropriate for direct application. Hence, in this paper, we propose a novel target localization method using time-difference-of-arrival (TDOA) measurements with the minimum number of radars under platform position uncertainties. The referred distributed system is a bistatic multi-receiver system, where the primary signal is transmitted by a geostationary Earth orbit (GEO) satellite while receivers are equipped on several unmanned aerial vehicles (UAVs). In the first stage, the reference range from the reference radar to the target is estimated by a quadratic function, and then the weighted least squares (WLS) solution of the target location is updated by substituting the range estimate back into it. In the second stage, we invoke the Taylor series approximation to further refine the target localization obtained by the first stage. It can be foreseen that the developed method is beneficial for scenarios with a limited number of radars, including engineering projects such as fire control, surveillance, and guidance, to support high-accuracy target localization. The simulation results show the superiority of the localization performance of the proposed method over other existing methods.

Determination of the reference point of a radio telescope using a multilateration-based coordinate measurement prototype

A measurement campaign was carried out at the Geodetic Observatory Wettzell in order to determine the reference point of a radio telescope used for Very Long Baseline Interferometry (VLBI) at better than 1 mm. This point has been determined by measuring the trajectories of targets installed on the telescope, using two commercial total stations, but also a prototype of multilateration system composed of one absolute distance meter and of four measurement heads. The use of prototypes is always subject to risk, so the produced data must be carefully checked.This paper evaluates independently the data from the multilateration system to ensure that they are compatible with the intended purpose. Despite the unstable position of one of the measurement heads, position uncertainties between 46 μm and 304 μm were assessed for the targets installed on the radio telescope, depending on uncertainties of the distance measurements, target visibility, and the positions of the targets relatively to the heads. These results validate the use of the multilateration system, even if the position measurements of some survey pillars of the local site network have shown larger uncertainties, of several hundred of micrometres.At the end, the measurements of the multilateration system have been combined with those of the total stations, a global network adjustment was performed, and the coordinates of the VLBI reference point were determined with related standard deviations of 0.06 mm, 0.04 mm, and 0.10 mm for the x, y, and z axes, respectively.

Extended GUP formulation and the role of momentum cut-off

We analyze the extension of the GUP theory deriving from the modified uncertainty principle in agreement with the string low energy limit, which represents one of the most general formulations satisfying the Jacobi identity, in the context of the associative algebras. After providing some physical insights on the nature of the considered approaches exploiting the cosmological arena, first, we show how a natural formulation of the theory in an infinite momentum space does not lead to the emergence of a nonzero minimal uncertainty in position, then we construct a truncated formulation of the theory in momentum space, proving that only in this case we can recover the desired feature of the presence of a nonzero minimal uncertainty in position, which – as usual in these theories – can be interpreted as a phenomenological manifestation of cut-off physics effects. Both quantization schemes are completely characterized and finally applied to study wave packets’ behavior and their evolution in time. The obtained results can shed light on which generalizations of the GUP theory are more coherent with the string low energy limit, in view of the existence of a minimum length in the form of a minimal uncertainty in position.