Logarithmic Mapping Research Articles

Three-dimensional Phan-Thien-Tanner viscoelastic flows around a sphere

This study investigates the steady and unsteady flow of a viscoelastic fluid around a sphere in three-dimensional space. Numerical simulations using the finite volume method incorporated a logarithmic mapping technique to overcome divergence issues at high elastic numbers. The fluid's behavior was characterized by the Phan-Thien-Tanner model, renowned for its accuracy and parametric simplicity. Key findings reveal that viscosity variations and stress relaxation times are pivotal in shaping the fluid's viscoelastic properties, surpassing the influence of other factors. Notably, the drag coefficient exhibited diverse behaviors—in some cases increasing, in others decreasing, and at times remaining constant—across different elasticity numbers. The study also explored the impact of fluid dilution, elasticity, and viscosity ratio on shear stress and drag coefficient variations, highlighting the profound role of elasticity in modulating the drag coefficient. A rise in velocity, elasticity number, viscosity ratio, and slip parameter was found to correspond with an increase in the drag coefficient, whereas an enhancement of the first model parameter reduced velocity, allowing viscous losses to dominate flow dynamics. The critical Reynolds number for Newtonian fluids was determined to be 300, with a noticeable decline as elasticity increased. The viscosity ratio demonstrated a strong influence on the critical Reynolds number, while the model parameters had minimal impact. Additionally, it was observed that as the vortex separation ratio increased, vortex length extended, and the separation angle decreased.

Hermite interpolation with retractions on manifolds

Interpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. A novel procedure relying on the general concept of retractions is proposed to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. The well-posedness of the method is analyzed by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. A classical result on the asymptotic interpolation error of Hermite interpolation is extended to the manifold setting. Finally numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns illustrate these results and the effectiveness of the method.

Questioning weber-fechner law in young children's numerical estimation strategies

The subjective number-space mapping and, especially, its evolution in young children has been the subject of intense controversy among different competing models. Many studies point out that: (i) young children's innate estimates follow a logarithmic mapping (Weber-Fechner law) and (ii) driven by education, children evolve into a linear mapping. In this paper we show, in consonance with other investigations, that innate numerical intuitions of young children are in fact a linear mapping, and not a logarithmic one. We found that young children extrapolate linearly from a reference point. The apparent logarithmic mapping is an artifact produced by placing a huge range of numbers within a small number line. We show how the appearance of the mapping can be modulated by changing the size and contour conditions of the number lines used in experiments.

Rolling bearing fault diagnosis method based on wavelet packet logarithmic energy map

Aiming at the fault diagnosis problem of rolling bearings, a diagnosis method based on wavelet packet logarithmic energy map is proposed. Firstly, a new wavelet packet node logarithmic energy formula is improved and proposed to overcome the shortcomings of cumbersome parameter determination and strong subjectivity in the traditional wavelet packet energy formula, improve the recognition of high-frequency faults and the distinction of low-frequency fault categories, so as to fully extract the initial time-frequency domain features; secondly, the Gram angle and field idea is used to realize the conversion from one-dimensional features to two-dimensional image features, so as to construct the logarithmic energy map feature based on wavelet packet, which further considers the spatial information between adjacent features, thereby optimizing the initial time-frequency domain features and improving the significance of the obtained features. On this basis, the residual network is used to improve the accuracy of fault diagnosis classification results. Finally, through the simulation verification of the standard rolling bearing data set of Case Western Reserve University, it can be seen that the fault diagnosis model constructed by the proposed method has high diagnostic accuracy and strong generalization ability.

On the v-Picard Group of Stein Spaces

Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $\mathbf{Z}_{p}$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $1$.

Universality for tropical and logarithmic maps

We prove that every toric monoid appears in a space of maps from tropical curves to an orthant. It follows that spaces of logarithmic maps to Artin fans exhibit arbitrary toric singularities: a virtual universality theorem for logarithmic maps to pairs. The target rank depends on the chosen singularity: we show that the cone over the 7-gon never appears in a space of maps to a rank 1 target. We obtain similar results for tropical maps to affine space.Comment: 22 pages. Comments welcome. v4: Published version in journal style

Wasserstein principal component analysis for circular measures

We consider the 2-Wasserstein space of probability measures supported on the unit-circle, and propose a framework for Principal Component Analysis (PCA) for data living in such a space. We build on a detailed investigation of the optimal transportation problem for measures on the unit-circle which might be of independent interest. In particular, building on previously obtained results, we derive an expression for optimal transport maps in (almost) closed form and propose an alternative definition of the tangent space at an absolutely continuous probability measure, together with fundamental characterizations of the associated exponential and logarithmic maps. PCA is performed by mapping data on the tangent space at the Wasserstein barycentre, which we approximate via an iterative scheme, and for which we establish a sufficient a posteriori condition to assess its convergence. Our methodology is illustrated on several simulated scenarios and a real data analysis of measurements of optical nerve thickness.

Optimisation of Convolution-Based Image Lightness Processing.

In the convolutional retinex approach to image lightness processing, an image is filtered by a centre/surround operator that is designed to mitigate the effects of shading (illumination gradients), which in turn compresses the dynamic range. Typically, the parameters that define the shape and extent of the filter are tuned to provide visually pleasing results, and a mapping function such as a logarithm is included for further image enhancement. In contrast, a statistical approach to convolutional retinex has recently been introduced, which is based upon known or estimated autocorrelation statistics of the image albedo and shading components. By introducing models for the autocorrelation matrices and solving a linear regression, the optimal filter is obtained in closed form. Unlike existing methods, the aim is simply to objectively mitigate shading, and so image enhancement components such as a logarithmic mapping function are not included. Here, the full mathematical details of the method are provided, along with implementation details. Significantly, it is shown that the shapes of the autocorrelation matrices directly impact the shape of the optimal filter. To investigate the performance of the method, we address the problem of shading removal from text documents. Further experiments on a challenging image dataset validate the method.

Divisors and curves on logarithmic mapping spaces

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

Cubic Hermite interpolators on the space of probability measures

In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures . First, we develop new approaches to capture the Riemannian geometric structure of when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on , namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail.

Genus 0 logarithmic and tropical fixed‐domain counts for Hirzebruch surfaces

AbstractFor a non‐singular projective toric variety , the virtual logarithmic Tevelev degrees are defined as the virtual degree of the morphism from the moduli stack of logarithmic stable maps to the product . In this paper, after proving that Mikhalkin's correspondence theorem holds in genus 0 for logarithmic virtual Tevelev degrees, we use tropical methods to provide closed formulas for the case in which is a Hirzebruch surface. In order to do so, we explicitly list all the tropical curves contributing to the count.

FEA-Based Stress-Strain Barometers as Forecasters for Corneal Refractive Power Change in Orthokeratology.

To improve the effectivity of patient-specific finite element analysis (FEA) to predict refractive power change (RPC) in rigid Ortho-K contact lens fitting. Novel eyelid boundary detection is introduced to the FEA model to better model the effects of the lid on lens performance, and stress and strain outcomes are investigated to identify the most effective FEA components to use in modelling. The current study utilises fully anonymised records of 249 eyes, 132 right eyes, and 117 left eyes from subjects aged 14.1 ± 4.0 years on average (range 9 to 38 years), which were selected for secondary analysis processing. A set of custom-built MATLAB codes was built to automate the process from reading Medmont E300 height and distance files to processing and displaying FEA stress and strain outcomes. Measurements from before and after contact lens wear were handled to obtain the corneal surface change in shape and power. Tangential refractive power maps were constructed from which changes in refractive power pre- and post-Ortho-K wear were determined as the refractive power change (RPC). A total of 249 patient-specific FEA with innovative eyelid boundary detection and 3D construction analyses were automatically built and run for every anterior eye and lens combination while the lens was located in its clinically detected position. Maps of four stress components: contact pressure, Mises stress, pressure, and maximum principal stress were created in addition to maximum principal logarithmic strain maps. Stress and strain components were compared to the clinical RPC maps using the two-dimensional (2D) normalised cross-correlation and structural similarity (SSIM) index measure. On the one hand, the maximum principal logarithmic strain recorded the highest moderate 2D cross-correlation area of 8.6 ± 10.3%, and contact pressure recorded the lowest area of 6.6 ± 9%. Mises stress recorded the second highest moderate 2D cross-correlation area with 8.3 ± 10.4%. On the other hand, when the SSIM index was used to compare the areas that were most similar to the clinical RPC, maximum principal stress was the most similar, with an average strong similarity percentage area of 26.5 ± 3.3%, and contact pressure was the least strong similarity area of 10.3 ± 7.3%. Regarding the moderate similarity areas, all components were recorded at around 34.4% similarity area except the contact pressure, which was down to 32.7 ± 5.8%. FEA is an increasingly effective tool in being able to predict the refractive outcome of Ortho-K treatment. Its accuracy depends on identifying which clinical and modelling metrics contribute to the most accurate prediction of RPC with minimal ocular complications. In terms of clinical metrics, age, Intra-ocular pressure (IOP), central corneal thickness (CCT), surface topography, lens decentration and the 3D eyelid effect are all important for effective modelling. In terms of FEA components, maximum principal stress was found to be the best FEA barometer that can be used to predict the performance of Ortho-K lenses. In contrast, contact pressure provided the worst stress performance. In terms of strain, the maximum principal logarithmic strain was an effective strain barometer.

Manifold fitting with CycleGAN

Manifold fitting, which offers substantial potential for efficient and accurate modeling, poses a critical challenge in nonlinear data analysis. This study presents an approach that employs neural networks to fit the latent manifold. Leveraging the generative adversarial framework, this method learns smooth mappings between low-dimensional latent space and high-dimensional ambient space, echoing the Riemannian exponential and logarithmic maps. The well-trained neural networks provide estimations for the latent manifold, facilitate data projection onto the manifold, and even generate data points that reside directly within the manifold. Through an extensive series of simulation studies and real data experiments, we demonstrate the effectiveness and accuracy of our approach in capturing the inherent structure of the underlying manifold within the ambient space data. Notably, our method exceeds the computational efficiency limitations of previous approaches and offers control over the dimensionality and smoothness of the resulting manifold. This advancement holds significant potential in the fields of statistics and computer science. The seamless integration of powerful neural network architectures with generative adversarial techniques unlocks possibilities for manifold fitting, thereby enhancing data analysis. The implications of our findings span diverse applications, from dimensionality reduction and data visualization to generating authentic data. Collectively, our research paves the way for future advancements in nonlinear data analysis and offers a beacon for subsequent scholarly pursuits.

A Grassmann manifold handbook: basic geometry and computational aspects

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.

BCLM: a novel chaotic map for designing cryptography-based security mechanism for IEEE C37.118.2 PMU communication in smart grid

Abstract Today, creating a smart grid that is resistant to cyberattacks is a subject of utmost significance. One of the components of the smart grid that is most susceptible to a cyber-attack is the phasor measuring unit (PMU). The reason is that PMU employs IEEE C37.118.2 communication standards, which specify the structure and sequencing of data packets but offer no security measures. Users must implement the security techniques to ensure the protection of PMU data. Additionally, PMU communicates via a public wide-area network, raising the risk to security. In addition, PMU is a crucial component of the smart grid, enabling different crucial choices for the reliable functioning of the smart grid to be made using its data. This research suggests a chaos-based data encryption solution to close the knowledge gap and reduce the confidentiality assault on PMU data. For this, a brand-new boost converter logarithmic map (BCLM), a one-dimensional (1D) chaotic map, has been presented. The research demonstrates how the suggested chaotic map has better chaotic qualities than conventional chaotic maps. The pseudorandom generator is the chaotic BCLM system. The PMU data are encrypted using the random sequence produced by the BCLM chaotic system. The suggested chaotic map is not computationally demanding, making it simple to implement in a PMU device with limited resources.

Low-illumination image enhancement with logarithmic tone mapping

Abstract For low-illumination video sequences, some existing enhancement algorithms have some problems, such as image over-enhancement, color distortion, and inadequate detail processing. Based on luminance detection, we add logarithmic tone mapping to optimize the existing algorithms. The color space of low-illumination video image is converted from the red, green, blue mode to the hue-saturation-intensity mode, and then, logarithmic tone enhancement is applied to the image. Algorithm in this study has an obvious effect on image luminance enhancement and details processing, which makes the low-illumination video show a clear image with more natural visual effect, thus improving the quality of low-illumination video. This algorithm can avoid the problems of overexposure, color distortion, and blurring of detail processing under low illumination. The infrared spectrum of the object can be captured by infrared detection equipment, and the purpose of image enhancement can be achieved by applying the infrared spectrum of the object.

Robot Learning by Demonstration with Dynamic Parameterization of the Orientation: An Application to Agricultural Activities

This work proposes a Learning by Demonstration framework based on Dynamic Movement Primitives (DMPs) that could be effectively adopted to plan complex activities in robotics such as the ones to be performed in agricultural domains and avoid orientation discontinuity during motion learning. The approach resorts to Lie theory and integrates into the DMP equations the exponential and logarithmic map, which converts any element of the Lie group SO(3) into an element of the tangent space so(3) and vice versa. Moreover, it includes a dynamic parameterization for the tangent space elements to manage the discontinuity of the logarithmic map. The proposed approach was tested on the Tiago robot during the fulfillment of four agricultural activities, such as digging, seeding, irrigation and harvesting. The obtained results were compared to the one achieved by using the original formulation of the DMPs and demonstrated the high capability of the proposed method to manage orientation discontinuity (the success rate was 100 % for all the tested poses).

Adaptive high-gray image enhancement algorithm based on logarithmic mapping and simulated exposure

In the fields of industry, security, military and other areas, high gray-level low contrast images have been widely used. However, due to the large dynamic range of gray level, high noise, blurry detail information and low contrast, it brings difficulties to subsequent image enhancement, target detection and other processes. Ordinary image enhancement algorithms are only suitable for conventional 8-bit images, which are not effective to deal with high-gray and low-contrast images. To address these issues, this paper proposed an adaptive high-gray image enhancement algorithm based on logarithmic mapping and simulated exposure, which was used for processing X-ray film images, infrared images, SAR images and other high gray-level low contrast images. Firstly, a detailed algorithm model construction method was provided for high gray-level low contrast images. Secondly, in order to verify the universality of the algorithm proposed in this paper, qualitative and quantitative experimental analyses were carried out using X-ray film images, infrared images and SAR images respectively. Finally, taking the most complex infrared image in the application environment as an example, the images, enhanced by the algorithm proposed in this paper, were sent to the YOLOv8 neural network for target detection, which improved the detection accuracy. The experiment shows that the algorithm, proposed in this paper, can significantly improve the image quality of high gray-level low contrast images, the detail information is obvious and the contrast is significantly improved. Various indicators are far higher than those of traditional algorithms. At the same time, this algorithm can provide reliable technical support for applications, such as target detection and target tracking in this field.

A different kind of meantone temperament and a novel mapping from the harmonic series to Pythagorean pitch classes

The logarithm mapping of natural numbers is a sum of products of coefficients. If these coefficients are arbitrary parameters, a new mapping of natural numbers to some subset of real numbers appears. This mapping preserves some crucial logarithm properties and constructs a new musical sound with a spectrum of inharmonic overtones. The simplest one-parameter mapping to a subset of polynomials with integer coefficients is constructed. The parameter defines a new “perfect fifth” similarly to the meantone temperament. It is an interesting case when the mapping parameter is defined from the linear relation between the new “major third” and the new “perfect fifth.” The most interesting case of a linear relation is the condition of zeroing the syntonic comma. Here, the new meantone temperament, based on the new “perfect fifth,” simultaneously coincides with Pythagorean- and five-limit-like tunings.

Rendezvous Control Design for the Generalized Cucker–Smale Model on Riemannian Manifolds

In this paper, we consider a rendezvous problem for the generalized Cucker-Smale model, which is a double-integrator multi-agent system, on complete Riemannian manifolds. With the help of the covariant derivative, parallel transport and logarithm map on the Riemannian manifold, we design a rendezvous feedback law that enables all agents to converge at a given target in the Riemannian manifold, under some a priori conditions. Furthermore, we consider three concrete complete Riemannian manifolds, such as the unit circle, unit sphere, and hyperboloid, and present the explicit feedback laws for rendezvous on them by calculating the corresponding covariant derivatives, parallel transports, and logarithm maps. Meanwhile, numerical examples are given for the manifolds as mentioned above to verify and illustrate the theoretical results.