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In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craigs interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maeharas lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and (partly) overcome the failure of interpolation for the implication-free fragment.

To address the issues of large anomalous triangulations, invalid interpolations, and uneven boundary interpolations in kriging interpolation, we propose research on a coal seam modeling construction method based on improved kriging interpolation. The work methodology assumes that by introducing kriging interpolation and analyzing its problems, we improve the interpolation method via a local interpolation algorithm for large anomalous triangulations, an optimization algorithm for locally redundant interpolation points, and a nonuniform boundary adaptive local interpolation algorithm. These improvements allow the interpolation method to better reflect the variability and realistic nature of coal seams. The research results indicate that applying this method to the construction of the Dananhu No. 2 open-pit mine coal seam model has improved the issue of coal seam transition stiffness, such as abnormal large-area triangulation in areas with significant elevation differences. This approach appropriately reduces the memory space usage without altering the coal seam morphology (which saves approximately 27,000 KB of memory, equivalent to the space occupied by 4 out of 21 coal seams). It has also prevented inaccuracies in boundary line positioning and transitions caused by too low a density of points on the coal seam reserve boundary line, resulting in smoother model transitions at the boundaries that better align with the actual coal seam change trends, the error rate in coal quality estimation decreased by 62.69%. This study provides data support for mining planning and reduces costs. This method can be extended to the construction of all mine models.

Time-domain backprojection algorithms are widely used in state-of-the-art synthetic aperture radar (SAR) imaging systems that are designed for applications where motion error compensation is required. These algorithms include an interpolation procedure, under which an unknown SAR range-compressed data parameter is estimated based on complex-valued SAR data samples and backprojected into a defined image plane. However, the phase of complex-valued SAR parameters estimated based on existing interpolators does not contain correct information about the range distance between the SAR imaging system and the given point of space in a defined image plane, which affects the quality of reconstructed SAR scenes. Thus, a phase-control procedure is required. This paper introduces extensions of existing linear, cubic, and sinc interpolation algorithms to interpolate complex-valued SAR data, where the phase of the interpolated SAR data value is controlled through the assigned a priori known range time that is needed for a signal to reach the given point of the defined image plane and return back. The efficiency of the extended algorithms is tested at the Nyquist rate on simulated and real data at THz frequencies and compared with existing algorithms. In comparison to the widely used nearest-neighbor interpolation algorithm, the proposed extended algorithms are beneficial from the lower computational complexity perspective, which is directly related to the offering of smaller memory requirements for SAR image reconstruction at THz frequencies.

For many decades, kriging and deterministic interpolation techniques, such as inverse distance weighting and nearest neighbour interpolation, have been the most popular spatial interpolation techniques. Kriging with external drift and regression kriging have become basic techniques that benefit both from spatial autocorrelation and covariate information. More recently, machine learning techniques, such as random forest and gradient boosting, have become increasingly popular and are now often used for spatial interpolation. Some attempts have been made to explicitly take the spatial component into account in machine learning, but so far, none of these approaches have taken the natural route of incorporating the nearest observations and their distances to the prediction location as covariates. In this research, we explored the value of including observations at the nearest locations and their distances from the prediction location by introducing Random Forest Spatial Interpolation (RFSI). We compared RFSI with deterministic interpolation methods, ordinary kriging, regression kriging, Random Forest and Random Forest for spatial prediction (RFsp) in three case studies. The first case study made use of synthetic data, i.e., simulations from normally distributed stationary random fields with a known semivariogram, for which ordinary kriging is known to be optimal. The second and third case studies evaluated the performance of the various interpolation methods using daily precipitation data for the 2016–2018 period in Catalonia, Spain, and mean daily temperature for the year 2008 in Croatia. Results of the synthetic case study showed that RFSI outperformed most simple deterministic interpolation techniques and had similar performance as inverse distance weighting and RFsp. As expected, kriging was the most accurate technique in the synthetic case study. In the precipitation and temperature case studies, RFSI mostly outperformed regression kriging, inverse distance weighting, random forest, and RFsp. Moreover, RFSI was substantially faster than RFsp, particularly when the training dataset was large and high-resolution prediction maps were made.

Monge–Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities—it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for interpolation of distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of recent developments in physics, probability theory, image processing, time-series analysis, and several other fields. In spite of extensive work and theoretical developments, the computation of OMT for large-scale problems has remained a challenging task. An alternative framework for interpolating distributions, rooted in statistical mechanics and large deviations, is that of the Schrödinger bridge problem (SBP), which leads to entropic interpolation. SBP may be seen as a stochastic regularization of OMT, and can be cast as the stochastic control problem of steering the probability density of the state-vector of a dynamical system between two marginals. The actual computation of entropic flows, however, has received hardly any attention. In our recent work on Schrödinger bridges for Markov chains and quantum channels, we showed that the solution can be efficiently obtained from the fixed point of a map which is contractive in the Hilbert metric. Thus, the purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which (i) leads to a new proof of a classical result on SBP and (ii) provides an efficient computational scheme for both SBP and OMT. We illustrate this new computational approach by obtaining interpolation of densities in representative examples such as interpolation of images.

In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. We then extend the

The widely scattered pattern of meteorological stations in large watersheds and remote locations, along with a need to estimate meteorological data for point sites or areas where little or no data have been recorded, has encouraged the development and implementation of spatial interpolation techniques. The various interpolation techniques featured in GIS software allow for the extraction of this new information from spatially distinct point data. Since no one interpolation method can be accurate in all regions, each method must be evaluated prior to each geographically distinct application. Many methods have been used for interpolating minimum temperature (Tmin), maximum temperature (Tmax) and precipitation data, and few have been used in the Zayandeh-Rud River basin, Iran, and no comparison of methods has ever been carried out in the area. The accuracies of six spatial interpolation methods [Inverse Distance Weighting, Natural Neighbor (NN), Regularized Spline, Tension Spline, Ordinary Kriging, Universal Kriging] were compared in this study simultaneously, and the best method for mapping monthly precipitation and temperature extremes was determined in a large semi-arid watershed with high temperature and rainfall variation. A cross-validation technique and long-term (1970–2014) average monthly Tmin, Tmax and precipitation data from meteorological stations within the basin were used to identify the best interpolation method for each variable dataset. For Tmin, Kriging (Gaussian) proved to be the most accurate interpolation method (MAE = 1.827 °C), whereas, for Tmax and precipitation the NN method performed best (MAE = 1.178 °C and 0.5241 mm, respectively). Accordingly, these variable-optimized interpolation methods were used to define spatial patterns of newly generated climatic maps.

To address the issue of invalid values in the measured wind speed data of the wind lidar, a data interpolation experiment was conducted by using the wind speed data outputted by the wind lidar as the base data source. Based on data preprocessing, the interpolation model was constructed by integrating MLP and the mean wind speed shear index. The accuracy of the interpolated wind speed data was evaluated. The experimental results show that the proposed interpolation method for wind lidar data achieved high accuracy, with a root mean square error of less than 0.75 m/s and an average relative error of less than 3.5%. This approach enhances the reliability and integrity of wind lidar data, providing a reference for the use of wind lidar data in wind resource assessment.

The escalation in the risk of extreme weather events due to climate system warming has caused a distinct alteration in the frequency and intensity of cold waves in China. Moreover, these changes exhibit distinct regional variations in their altering characteristics. Using daily temperature data from 70 weather stations in Fujian Province of China from 1961 to 2018, this study looks at how cold waves move through the province and how they change over time. It does this by combining linear regression trend, spatial interpolation, and spatial autocorrelation analyses. The findings indicate that the occurrence frequency of cold waves in the designated study area is notably higher in the western and northern regions, while it is comparatively lower in the eastern coastal cities. Furthermore, there is an overall downward trend in the frequency of cold waves, with a declining frequency from 1961 to 2000, followed by a significant increase in frequency after 2000. Furthermore, the number of cold waves in the study area increased significantly after 2010.