Continuous Map Research Articles

Final Topology for Preference Spaces

Abstract We say a model is continuous in utilities (resp., preferences) if small perturbations of utility functions (resp., preferences) generate small changes in the model’s outcomes. While similar, these two questions are different. They are only equivalent when the following two sets are isomorphic: the set of continuous mappings from preferences to the model’s outcomes, and the set of continuous mappings from utilities to the model’s outcomes. In this paper, we study the topology for preference spaces defined by such an isomorphism. This study is practically significant, as continuity analysis is predominantly conducted through utility functions, rather than the underlying preference space. Our findings enable researchers to extrapolate continuity in utility as indicative of continuity in underlying preferences.

Novel Categories of Spaces in the Frame of Generalized Fuzzy Topologies via Fuzzy $g\mu$-closed sets

One of the known approaches to studying topological concepts is to utilize subclasses of topology, such as clopen sets and generalized closed sets. In this study, we apply the notion of fuzzy generalized $\mu$-closed sets ($Fg\mu$-closed sets) to establish and analyze novel categories of spaces, namely $Fg\mu $-regular, $Fg\mu $-normal, and $F\mu $-symmetric spaces in the frame of generalized fuzzy topology($GFT$). We investigate the fundamental properties of these classes, exploring their unique characteristics and preservation theorems under $Fg\mu$-continuous maps. We establish the interrelationships between these classes and the other separation axioms in this setting, and we demonstrated that $F\mu $-regular, $F\mu $-normal, and $F\mu $-symmetric spaces are special cases of $Fg\mu $-regular, $Fg\mu $-normal, and $F\mu$-$T_{1}$ spaces respectively. Additionally, we show that the equivalence for these cases hold when the $GFT$ is $F\mu$-$T_{\frac{1}{2}}$. The connections between these classes and their counterparts in the crisp $GT$ are studied. Finally, we discuss these classes' hereditary and topological properties, further enhancing our comprehension of their behavior and implications.

Online lane mapping based on multi-sensor SLAM and Catmull–Rom splines

Abstract Traditional offline high-definition maps have significant limitations owing to their high cost and difficulty in updating. Online vector maps represent a new trend in the development of autonomous driving. In this paper, we propose an online lane mapping system, which leverages both multi-sensor simultaneous localization and mapping (SLAM) and Catmull–Rom splines. The map is constructed in real-time through the localization provided by SLAM and the fitting of splines. Existing SLAM algorithms perform poorly in complex urban environments, a deep learning network is incorporated into the visual front-end of our system to improve stability. Deep learning can extract and study deeper and less observable features, so that the system enhances the accuracy of visual tracking in scenarios characterized by low texture, lighting variations, and rapid motion. The system state is optimized by a factor graph framework to achieve precise localization. To construct continuous and consistent online lane maps, our system combines SLAM localization results with spline curves to achieve lane connection. Additionally, we develop a convenient method for initializing, extending, and optimizing the control points of splines to obtain more precise results. Our method combines the geometric characteristics with historical observations of lanes, enhancing the association and consistency of maps. Experiments conducted on the mainstream datasets demonstrate that our system exhibits greater robustness and improves the accuracy of localization and mapping across diverse environments.

Parametric Topological Entropy of Possibly Discontinuous Maps in Compact Hausdorff Spaces and Hyperspaces

Abstract It is well known that the topological entropy of the induced hyperspace map is positive, provided so is the topological entropy of the original continuous single-valued map. In this paper we ask whether this is also true for the sequences of multivalued maps. In particular, we consider various definitions (some of them for the first time) of topological entropy for multivalued nonautonomous dynamical systems in compact Hausdorff spaces and investigate their mutual relations. We explain that some of them can deal with arbitrary multivalued maps, i.e. without regularity restrictions. For upper semicontinuous multivalued maps, we study the parametric topological entropy of the induced possibly discontinuous dynamical systems in hyperspaces. We show, among other things, that a comparison of different types of entropy notions allows us to make the entropy estimates without direct calculations. Several illustrative examples are supplied.

On Generic Topological Embeddings

We show that an embedding of a fixed 0-dimensional compact space K into the Čech–Stone remainder ω∗ as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from K to compact metric spaces. Using categorical Fraïssé theory, we obtain some well-known theorems about Čech–Stone remainders of infinite discrete spaces. Furthermore, we prove several results concerning universality and homogeneity of the space κκ, with κ a regular uncountable cardinal, using generalized ultrametrics.

Maps on matrices preserving idempotency of a triadic relation

Let Mn be the algebra of all n×n real or complex matrices. In this paper, we give a full description of continuous maps on Mn such that Φ(A)(Φ(B)− Φ(C)) is idempotent if and only if A(B − C) is idempotent for all A, B, C ∈ Mn.

A comparative study of interpolation methods for the development of ore distribution maps

Mining projects require precise knowledge about tonnage and quality of ore reserves for planning and decision-making. This is hard to establish as exploration operations, which are costly, time-consuming and require an accurate description of the deposit. Geologists use deterministic and geostatistical methods as interpolation tools to generate a continuous (or prediction) ore distribution map from known sampled data. A comparative study between four different geostatistical interpolation techniques: Ordinary Kriging (OK), Simple Kriging (SK), Universal Kriging (UK) and Empirical Bayesian Kriging (EBK); and four deterministic techniques: Inverse Distance Weighting (IDW), Global Polynomial Interpolation (GPI), Radial Basis Function (RBF), and Local Polynomial Interpolation (LPI) were applied. The 95% confidence level was measured and the effectiveness of each interpolation method was assessed through cross-validation for the construction of the corresponding maps displaying the distribution of ore on the surface with the use of GIS. The output methods are ranked to perform a comprehensive analysis using the error statistics methods: Mean Error (ME), Root Mean Square Error (RMSE), Mean Standardized Error (MSE) and Root Mean Square Standardized Error (RMSSE). The results show that GPI, EBK and Kriging methods perform the best results while IDW has good statistical analysis factors but it came in the tail of the rank. Therefore, there is no appropriate interpolation method accurate for all cases, each method must be statistically evaluated before each application and essentially based on real data.

A note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds

In this note, Gromov’s reduction [No metrics with positive scalar curvatures on aspherical 5-manifolds, https://arxiv.org/abs/2009.05332, 2020], from the aspherical conjecture to the generalized filling radius conjecture, is generalized to the smooth Q \mathbb Q -homology vanishing conjecture in the case of hypersurface. In particular, we can show that any continuous map from a closed 4 4 -manifold admitting positive scalar curvature to an aspherical 5 5 -manifold induces zero map between H 4 ( ⋅ , Q ) H_4(\cdot ,\mathbb Q) . As a corollary, we obtain the following aspherical splitting theorem: if a complete orientable aspherical Riemannian 5 5 -manifold has non-negative scalar curvature and two ends, then it splits into the Riemannian product of a closed flat manifold and the real line.

Self-calibrating multiplexed microneedle electrode array for continuous mapping of subcutaneous multi-analytes in diabetes

Self-calibrating multiplexed microneedle electrode array for continuous mapping of subcutaneous multi-analytes in diabetes

Bivariate density estimation under marginal unimodality constraints

Empirical analyses of observed pairs of variables often suggest that the underlying joint distribution of these variables is highly dependent and each one of them is marginally unimodal. Exploiting marginal shape information can improve the bivariate density estimation. We develop a nonparametric density estimation method where the joint density of the bivariate outcome variables is estimated using a flexible class of mixtures of scaled Beta distributions and the mixing weights are suitably constrained to ensure marginal unimodality. Large sample consistency of the proposed density estimator is established using the method of sieves and Argmax Continuous Mapping Theorem. The proposed method enlarges the scope of previous studies in two important aspects: (i) joint density provides a simultaneous estimation of higher-order moments beyond mean and variance functions, and (ii) marginal shape constraints provide more efficient estimates. The proposed procedures are illustrated using simulated and real case study datasets on gestational age and infant birth weight.

Fixed point results on locally (δ,ϕ)-dominated and (δ,ϕ)-continuous mappings in dislocated quasi metric spaces

In this article, some concepts of locally (δ,ϕ)-dominated mappings, locally (δ,ϕ)-continuous mappings and LR−(δ,ϕ) complete dislocated quasi-metric spaces have been introduced. Fixed point results for such mappings under new contractive conditions in these spaces have been established. As an application, fixed point results endowed with order and graph in such spaces have been examined. Furthermore, the results are substantiated with specific examples. Our empirically validated results represent a novel contribution to the existing body of literature.

Topological Degree for Operators of Class (S)+ with Set-Valued Perturbations and Its New Applications

We investigate the topological degree for generalized monotone operators of class (S)+ with compact set-valued perturbations. It is assumed that perturbations can be represented as the composition of a continuous single-valued mapping and an upper semicontinuous set-valued mapping with aspheric values. This allows us to extend the standard degree theory for convex-valued operators to set-valued mappings whose values can have complex geometry. Several theoretical aspects concerning the definition and main properties of the topological degree for such set-valued mappings are discussed. In particular, it is shown that the introduced degree has the homotopy invariance property and can be used as a convenient tool in checking the existence of solutions to corresponding operator inclusions. To illustrate the applicability of our approach to studying models of real processes, we consider an optimal feedback control problem for the steady-state internal flow of a generalized Newtonian fluid in a 3D (or 2D) bounded domain with a Lipschitz boundary. By using the proposed topological degree method, we prove the solvability of this problem in the weak formulation.

A Novel Workflow for Mapping Forest Canopy Height by Synergizing ICESat-2 and Multi-Sensor Data

Precise information on forest canopy height (FCH) is critical for forest carbon stocks estimation and management, but mapping continuous FCH with satellite data at regional scale is still a challenge. By fusing ICESat-2, Sentinel-1/2 images and ancillary data, this study aimed to develop a workflow to obtain an FCH map using a machine learning algorithm over large areas. The vegetation-type map was initially produced by a phenology-based spectral feature selection method. A forest characteristic-based model was then proposed to map spatially continuous FCH after a multivariate quality control. Our results show that the overall accuracy (OA) and average F1 Score (F1) for eight main vegetation types were more than 90% and 89%, respectively, and the vegetation-type map agreed well with the census areas. The forest characteristic-based model demonstrated a greater potential in FCH prediction, with an R-value 60.47% greater than the traditional single model, suggesting that the addition of the multivariate quality control and forest structure characteristics could positively contribute to the prediction of FCH. We generated a 30 m continuous FCH map by the forest characteristic-based model and evaluated the product with about 35 km2 of airborne laser scanning (ALS) validation data (R = 0.73, RMSE = 2.99 m), which were 45.34% more precise than the China FCH, 2019. These findings demonstrate the potential of our proposed workflow for monitoring regional continuous FCH, and will greatly benefit accurate forest resources assessment.

Photon-counting in dual-contrast-enhanced computed tomography: a proof-of-concept quantitative biomechanical assessment of articular cartilage

This proof-of-concept study explores quantitative imaging of articular cartilage using photon-counting detector computed tomography (PCD-CT) with a dual-contrast agent approach, comparing it to clinical dual-energy CT (DECT). The diffusion of cationic iodinated CA4 + and non-ionic gadolinium-based gadoteridol contrast agents into ex vivo bovine medial tibial plateau cartilage was tracked over 72 h. Continuous maps of the contrast agents’ diffusion were created, and correlations with biomechanical indentation parameters (equilibrium and instantaneous moduli, and relaxation time constants) were examined at 28 specific locations. Cartilage at each location was analyzed as full-thickness to ensure a fair comparison, and calibration-based material decomposition was employed for concentration estimation. Both DECT and PCD-CT exhibit strong correlations between CA4 + content and biomechanical parameters, with PCD-CT showing superior significance, especially at later time points. DECT lacks significant correlations with gadoteridol-related parameters, while PCD-CT identifies noteworthy correlations between gadoteridol diffusion and biomechanical parameters. In summary, the experimental PCD-CT setup demonstrates superior accuracy and sensitivity in concentration estimation, suggesting its potential as a more effective tool for quantitatively assessing articular cartilage condition compared to a conventional clinical DECT scanner.

Generalized synchronization in the presence of dynamical noise and its detection via recurrent neural networks.

Given two unidirectionally coupled nonlinear systems, we speak of generalized synchronization when the responder "follows" the driver. Mathematically, this situation is implemented by a map from the driver state space to the responder state space termed the synchronization map. In nonlinear times series analysis, the framework of the present work, the existence of the synchronization map amounts to the invertibility of the so-called cross map, which is a continuous map that exists in the reconstructed state spaces for typical time-delay embeddings. The cross map plays a central role in some techniques to detect functional dependencies between time series. In this paper, we study the changes in the "noiseless scenario" just described when noise is present in the driver, a more realistic situation that we call the "noisy scenario." Noise will be modeled using a family of driving dynamics indexed by a finite number of parameters, which is sufficiently general for practical purposes. In this approach, it turns out that the cross and synchronization maps can be extended to the noisy scenario as families of maps that depend on the noise parameters, and only for "generic" driver states in the case of the cross map. To reveal generalized synchronization in both the noiseless and noisy scenarios, we check the existence of synchronization maps of higher periods (introduced in this paper) using recurrent neural networks and predictability. The results obtained with synthetic and real-world data demonstrate the capability of our method.

Approaching the Global Nash Equilibrium of Non-Convex Multi-Player Games.

Many machine learning problems can be formulated as non-convex multi-player games. Due to non-convexity, it is challenging to obtain the existence condition of the global Nash equilibrium (NE) and design theoretically guaranteed algorithms. This paper studies a class of non-convex multi-player games, where players' payoff functions consist of canonical functions and quadratic operators. We leverage conjugate properties to transform the complementary problem into a variational inequality (VI) problem using a continuous pseudo-gradient mapping. We prove the existence condition of the global NE as the solution to the VI problem satisfies a duality relation. We then design an ordinary differential equation to approach the global NE with an exponential convergence rate. For practical implementation, we derive a discretized algorithm and apply it to two scenarios: multi-player games with generalized monotonicity and multi-player potential games. In the two settings, step sizes are required to be O(1/k) and O(1/√k) to yield the convergence rates of O(1/ k) and O(1/√k), respectively. Extensive experiments on robust neural network training and sensor network localization validate our theory. Our code is available at https://github.com/GuanpuChen/Global-NE.

Mapping the exposure to outdoor radon in the German population

IntroductionData on outdoor radon are generally scarce compared to indoor radon. However, knowledge of the spatial distribution of outdoor radon is necessary to estimate the overall exposure of the population to radon, it supports the prediction of indoor radon and characterizes the natural radon background. Germany has a comprehensive dataset on long-term outdoor radon concentration and the equilibrium factor at national level, which allowed to produce what is probably the only spatially continuous outdoor radon map at national level so far. DataIn this study, outdoor radon concentration measurement data (n = 172) and equilibrium factors (n = 25) from a national survey from 2003 to 2006 were reanalyzed using state-of-the-art machine learning routines. Spatially comprehensive maps of distance to the sea, radon concentration in soil, sand content in topsoil and a terrain-based wind exposure index are used as predictors. MethodsQuantile regression forest was used to map the conditional distribution of outdoor radon concentration at 500 m grid resolution. The equilibrium factor was mapped using a linear regression model. Both maps were combined to derive the equivalent outdoor radon equilibrium concentration. Population weighting of the results was achieved by explicitly accounting for the population distribution using a probabilistic sampling procedure from the estimated conditional distributions. ResultsThe arithmetic mean and the interquartile range (25th to 75th percentile) for the population-weighted outdoor radon concentration for Germany are 9.3 Bq/m³ and 5.8 Bq/m³ to 11.2 Bq/m³, respectively. The mean equilibrium factor is 0.49. The arithmetic mean and the interquartile range (25th to 75th percentile) for the population-weighted outdoor radon equilibrium equivalent concentration are 4.7 Bq/m³ and 2.7 Bq/m³ to 5.9 Bq/m³ respectively. The estimated inhalation dose due to outdoor exposure to radon is 0.056 mSv/a (arithmetic mean), with less than 10 % of the population exceeding a value of 0.1 mSv/a. The unavoidable inhalation dose due to radon exposure (outdoors plus indoors) in Germany is estimated at an arithmetic mean of 0.37 mSv/a. The spatial distribution of radon outdoors is mainly determined by the distance to the sea. The predictors radon concentration in soil, sand in topsoil and wind exposure still have a significant influence, especially at local to regional level. ConclusionKnowledge about the spatial distribution of outdoor radon and its local variability for Germany was improved using a modern regression technique and relevant predictive information. The results confirm a low outdoor radon concentration with a small contribution to the effective dose received by the population from outdoor radon exposure.

A numerical solution of Schrödinger equation for the dynamics of early universe

Artificial neural networks (ANNs) have attained widespread success across varied disciplines. This study is designated for looking into an application of an integrated intelligent computing paradigm concerning dynamics in the early Universe through numerical solutions to the Schrödinger equation. To arrive at this we leverage the Levenberg–Marquardt backpropagation networks (LMBNs) to probe cosmic evolution in the early Universe with the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric for a flat minisuperspace model of the Universe in the background. This leads to bridging quantum mechanics and inflationary Universe dynamics conducing to quantum cosmology within the standard model. Wheeler–DeWitt equation corresponds to the time-independent Schrödinger equation obtained from the equations of motion for a single scalar field in flat spacetime with FLRW metric. Utilizing the ntstool the whole computing process is operated for simulation. To evaluate the accuracy and efficiency of the proposed scheme a comparative analysis is carried out. To construct continuous neural network mappings we employ the explicit Runge–Kutta method as the target parameter for generating datasets. To determine the solution datasets of different scenarios, the training, testing, and validation processes are employed to take advantage of these in the learning of neural network models established upon the backpropagation technique of Levenberg–Marquardt. By varying related parameters we develop three scenarios that produce nine cases, three for each. The data plots of performance, training state, error histogram, regression, time-series response, and error autocorrelation represent the visualization of the results. These plots show a complete case description by displaying all the necessary data values. The analysis of these plots is presented to validate all the cases. Performing the analysis by mean square error (MSE) validates the achieved accuracy of the results by validating and verifying neural networks. This work is motivated by the compelling need to develop innovative computational methods for solving complex cosmological questions to untangle the conundrums of the early universe. The attractive numerical solutions of the Schrödinger equation for the early Universe heralds a step towards quantum cosmology based on the interplay of the Wheeler–DeWitt equation and time-independent Schrödinger equation. There is an increasing trend to use computational methods to solve ordinary and partial differential equations with the help of code development in Matlab. For this purpose feed-forward artificial neural network is used for investigating the Schrödinger equation.

Putative past, present, and future spatial distributions of deep-sea coral and sponge microbiomes revealed by predictive models

Abstract Knowledge of spatial distribution patterns of biodiversity is key to evaluate and ensure ocean integrity and resilience. Especially for the deep ocean, where in situ monitoring requires sophisticated instruments and considerable financial investments, modelling approaches are crucial to move from scattered data points to predictive continuous maps. Those modelling approaches are commonly run on the macrobial level, but spatio-temporal predictions of host-associated microbiomes are not being targeted. This is especially problematic as previous research has highlighted that host-associated microbes may display distribution patterns that are not perfectly correlated with host biogeographies, but also with other factors such as prevailing environmental conditions. We here establish a new simulation approach and present predicted spatio-temporal distribution patterns of deep-sea sponge and coral microbiomes, making use of a combination of environmental data, host data, and microbiome data. This approach allows predictions of microbiome spatio-temporal distribution patterns on scales that are currently not covered by classical sampling approaches at sea. In summary, our presented predictions allow (i) identification of microbial biodiversity hotspots in the past, present, and future, (ii) trait-based predictions to link microbial with macrobial biodiversity, and (iii) identification of shifts in microbial community composition (key taxa) across environmental gradients and shifting environmental conditions.

S STAR p STAR Continuous Maps in Topological Spaces

Goal of this present study seeks to discover and to study the different kind of continuous map and irresolute map is known as semi star pre star continuous map (briefly s*p*continuous) and semi star pre star irresolute map (briefly s*p* irresolute) in topological spaces. As well as discuss a few fundamental characteristics of this continuous map. Further investigate the relationship between the newly defined map and the existing continuous map and irresolute map with suitable examples.