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Background: The Rotation Invariant Vision Transformer (RViT) is a novel deep learning model tailored for brain tumor classification using MRI scans. Methods: RViT incorporates rotated patch embeddings to enhance the accuracy of brain tumor identification. Results: Evaluation on the Brain Tumor MRI Dataset from Kaggle demonstrates RViT's superior performance with sensitivity (1.0), specificity (0.975), F1-score (0.984), Matthew's Correlation Coefficient (MCC) (0.972), and an overall accuracy of 0.986. Conclusion: RViT outperforms the standard Vision Transformer model and several existing techniques, highlighting its efficacy inmedical imaging. The study confirms that integrating rotational patch embeddings improves the model's capability to handle diverse orientations, a common challenge in tumor imaging. The specialized architecture and rotational invariance approach of RViT have the potential to enhance current methodologies for brain tumor detection and extend to other complex imaging tasks. [ABSTRACT FROM AUTHOR]

Deep learning has proven promising in biological and chemical applications, aiding in accurate predictions of properties such as atomic forces, energies, and material band gaps. Traditional methods with rotational invariance, one of the most crucial physical laws for predictions made by machine learning, have relied on Fourier transforms or specialized convolution filters, leading to complex model design and reduced accuracy and efficiency. However, models without rotational invariance exhibit poor generalization ability across datasets. Addressing this contradiction, this work proposes a rotationally invariant graph neural network, named RotNet, for accurate and accelerated quantum mechanical calculations that can overcome the generalization deficiency caused by rotations of molecules. RotNet ensures rotational invariance through an effective transformation and learns distance and angular information from atomic coordinates. Benchmark experiments on three datasets (protein fragments, electronic materials, and QM9) demonstrate that the proposed RotNet framework outperforms popular baselines and generalizes well to spatial data with varying rotations. The high accuracy, efficiency, and fast convergence of RotNet suggest that it has tremendous potential to significantly facilitate studies of protein dynamics simulation and materials engineering while maintaining physical plausibility. [ABSTRACT FROM AUTHOR]

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BackgroundThe Rotation Invariant Vision Transformer (RViT) is a novel deep learning model tailored for brain tumor classification using MRI scans.MethodsRViT incorporates rotated patch embeddings to enhance the accuracy of brain tumor identification.ResultsEvaluation on the Brain Tumor MRI Dataset from Kaggle demonstrates RViT's superior performance with sensitivity (1.0), specificity (0.975), F1-score (0.984), Matthew's Correlation Coefficient (MCC) (0.972), and an overall accuracy of 0.986.ConclusionRViT outperforms the standard Vision Transformer model and several existing techniques, highlighting its efficacy in medical imaging. The study confirms that integrating rotational patch embeddings improves the model's capability to handle diverse orientations, a common challenge in tumor imaging. The specialized architecture and rotational invariance approach of RViT have the potential to enhance current methodologies for brain tumor detection and extend to other complex imaging tasks.

Despite the popularity of factor models with simple loading matrices, little attention has been given to formally address the identifiability of these models beyond standard rotation-based identification such as the positive lower triangular (PLT) constraint. To fill this gap, we review the advantages of variance identification in simple factor analysis and introduce the generalized lower triangular (GLT) structures. We show that the GLT assumption is an improvement over PLT without compromise: GLT is also unique but, unlike PLT, a non-restrictive assumption. Furthermore, we provide a simple counting rule for variance identification under GLT structures, and we demonstrate that within this model class, the unknown number of common factors can be recovered in an exploratory factor analysis. Our methodology is illustrated for simulated data in the context of post-processing posterior draws in sparse Bayesian factor analysis. [ABSTRACT FROM AUTHOR]

A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization of traditional fuzzy sets and complex fuzzy sets based on quaternions. The novelty of QFSs is that the range of the membership function is the set of quaternions with modulus less than or equal to one, of which the real and quaternionic imaginary parts can be used for four different features. A discussion is made on the intuitive interpretation of quaternion-valued membership grades and the possible applications of QFSs. Several operations, including quaternionic fuzzy complement, union, intersection, and aggregation of QFSs, are presented. Quaternionic fuzzy relations and their composition are also investigated. QFS is designed to maintain the advantages of traditional FS and CFS, while benefiting from the properties of quaternions. Cuts of QFSs and rotational invariance of quaternionic fuzzy operations demonstrate the particularity of quaternion-valued grades of membership. [ABSTRACT FROM AUTHOR]

We present a method for determining which of a large set of pixels are inside or on the boundary of a polygon. The method works much quicker than the standard method used for points in general and also provides an ordering of the boundary pixels. Once the edge pixels have been chosen and ordered, we move on to the characterization of the edge. We present a method for characterizing the edge slope of digitized images, which is—to a desired tolerance—invariant to rotation. We demonstrate that this characterization is good enough for subsequent use in edge identification and matching algorithms. This allows us to continue, e.g., to juxtaposing corresponding pieces of a jigsaw puzzle, or "sewing" back together, ripped pieces of material, paper or glass. [ABSTRACT FROM AUTHOR]

Can we accurately model the spin state of a quantum particle? If so, we should be able to make identical copies of such a state and also obtain its mirror image. In quantum mechanics, many subatomic particles can form entangled pairs that are mirror images of each other, although the state of an individual particle cannot be duplicated or cloned as experimentally demonstrated by Aspect, Clauser and Zeilinger, the winners of the Nobel Prize in Physics 2022. We show that there is a higher-order symmetry associated with the SL(2,C) group that underlies the singlet state, which means that the singlet pairing preserves Lorentz transformations independently of the metric used. The Pauli exclusion principle can be derived from this symmetry. [ABSTRACT FROM AUTHOR]

Despite the popularity of factor models with simple loading matrices, little attention has been given to formally address the identifiability of these models beyond standard rotation-based identification such as the positive lower triangular (PLT) constraint. To fill this gap, we review the advantages of variance identification in simple factor analysis and introduce the generalized lower triangular (GLT) structures. We show that the GLT assumption is an improvement over PLT without compromise: GLT is also unique but, unlike PLT, a non-restrictive assumption. Furthermore, we provide a simple counting rule for variance identification under GLT structures, and we demonstrate that within this model class, the unknown number of common factors can be recovered in an exploratory factor analysis. Our methodology is illustrated for simulated data in the context of post-processing posterior draws in sparse Bayesian factor analysis.

The topic of joint angle and frequency estimation (JAFE) has aroused extensive interests in the past decades. Current estimation algorithms mainly rely on the Nyquist sampling criterion. In order not to cause ambiguity for parameter estimation, the space–time intervals must be smaller than given thresholds, which results in complicated hardware costs and a huge computational burden. This paper aims to reduce the complexity for JAFE, and a novel sparsity-aware framework is proposed. Unlike the current uniform sampling architectures, the incoming narrow-band singles are sampled by a series of space–time coprime samplers. An improved rotational invariance estimator is introduced, which offers closed-form solutions for both angle and frequency estimation. The mathematical treatments indicate that our methodology is inherent in larger spatial/temporal aperture than the uniform sampling architectures; hence, it provides more accurate JAFE compared to alternative approaches relying on uniform sampling. Additionally, it attains nearly the same complexity as the current rotational invariance approach. Numerical results agree with the theoretical advantages of our methodology. [ABSTRACT FROM AUTHOR]

A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization of traditional fuzzy sets and complex fuzzy sets based on quaternions. The novelty of QFSs is that the range of the membership function is the set of quaternions with modulus less than or equal to one, of which the real and quaternionic imaginary parts can be used for four different features. A discussion is made on the intuitive interpretation of quaternion-valued membership grades and the possible applications of QFSs. Several operations, including quaternionic fuzzy complement, union, intersection, and aggregation of QFSs, are presented. Quaternionic fuzzy relations and their composition are also investigated. QFS is designed to maintain the advantages of traditional FS and CFS, while benefiting from the properties of quaternions. Cuts of QFSs and rotational invariance of quaternionic fuzzy operations demonstrate the particularity of quaternion-valued grades of membership.

The rise of novel materials such as graphene compounds or carbon nanotubes recently revived the interest on the role of symmetries upon electrical properties. Among possible solids, symmetrical polytopes, widely realized in natural and artificial matter, deserve particular attention. When equal resistors are attached between neighbouring nodes of Perfect (Platonic) solids (PS) along the edges, the two-point resistance between any pair of nodes was established in the 1990s by van Steenwijk using an elegant and efficient method. Using van Steenwijk’s method, Moddy and Aravind subsequently derived the resistance values for two Archimedean and two Catalan solids. Here, with the same method, we derive the exact expression for the two-point resistance between any two nodes for resistor networks made of equal resistors placed along the edges of the thirteen Archimedean solids (AS). While the calculation remains elementary for the simplest AS, a dedicated method using computer assistance was developed for the Snub Cube, the Snub Dodecahedron and the three AS with three different rotation symmetries (the Great Rhombicuboctahedron, the Small Rhombicosidodecahedron and the Great Rhombicosidodecahedron). The largest resistance value (191/90) is obtained between the two opposite nodes of the Truncated Dodecahedron, and the second largest (42815/21114) is obtained between the two opposite nodes of the Great Rhombicosidodecahedron. The smallest resistance value (14137/38016) is obtained between some of the neighbouring nodes of the Snub Cube, whereas the resistance between neighbouring nodes of the Icosahedron (11/30) is slightly smaller. Some general symmetry relations between two-point resistances in AS networks are also derived, as well as some relations with two-point resistances in PS networks. The complete set of exact two-point resistance values for PS and AS networks can be used to check numerical codes or to evaluate the capacity of regular solids to provide appropriate models for given experimental situations.

Sun, P.; Zhou, J.; Liu, Y., and Wang, Y., 2022. Lattice Boltzmann method for simulations of nozzle flows in coastal environments. Journal of Coastal Research, 38(1), 204–217. Coconut Creek (Florida), ISSN 0749-0208. A better understanding of flow in pipes is considered to be critical for marine equipment because it helps to improve pipe performance related to vibration and structure problems. In this study, flow simulations were conducted using a large–eddy-simulation lattice Boltzmann method (LES-LBM) in a nozzle under various Reynolds numbers (Re) and the effect of the three-dimensional cubic lattice models D3Q15, D3Q19, and D3Q27 was examined. The results show that when Re < 200 (i.e. the asymmetry of the results calculated using lattice models D3Q15 and D3Q19 is below 5%) or when the lattice model D3Q27 is adopted, computational results can satisfy the requirement of rotational invariance. When 250 < Re < 300 or the flow suddenly traverses an abrupt expansion, the isotropy of lattice models D3Q15 and D3Q19 experience "collapse". Defective planes cannot be used to completely solve complex flows and accurately transfer flow information. The resultant problems in the computational results include preference in the circumferential distribution of axial velocity and contraction effects of the phase portrait of axial and radial velocities. More complex flows (e.g., for situations with large Re or fluid traversing abrupt expansion) universally cause more significant lattice model effects, where nonlinear flow phenomena become dominant. Therefore, it is necessary to ensure that solutions in the entire computational domain are independent of lattice orientation during flow simulation using LBM. Overall, the results in this paper have important potential applications in predicting flow characteristics for pipe flows in marine structures such as marine risers in coastal environments and zones farther out to sea. [ABSTRACT FROM AUTHOR]

Can we accurately model the spin state of a quantum particle? If so, we should be able to make identical copies of such a state and also obtain its mirror image. In quantum mechanics, many subatomic particles can form entangled pairs that are mirror images of each other, although the state of an individual particle cannot be duplicated or cloned as experimentally demonstrated by Aspect, Clauser and Zeilinger, the winners of the Nobel Prize in Physics 2022. We show that there is a higher-order symmetry associated with the SL(2,C) group that underlies the singlet state, which means that the singlet pairing preserves Lorentz transformations independently of the metric used. The Pauli exclusion principle can be derived from this symmetry.

The topic of joint angle and frequency estimation (JAFE) has aroused extensive interests in the past decades. Current estimation algorithms mainly rely on the Nyquist sampling criterion. In order not to cause ambiguity for parameter estimation, the space–time intervals must be smaller than given thresholds, which results in complicated hardware costs and a huge computational burden. This paper aims to reduce the complexity for JAFE, and a novel sparsity-aware framework is proposed. Unlike the current uniform sampling architectures, the incoming narrow-band singles are sampled by a series of space–time coprime samplers. An improved rotational invariance estimator is introduced, which offers closed-form solutions for both angle and frequency estimation. The mathematical treatments indicate that our methodology is inherent in larger spatial/temporal aperture than the uniform sampling architectures; hence, it provides more accurate JAFE compared to alternative approaches relying on uniform sampling. Additionally, it attains nearly the same complexity as the current rotational invariance approach. Numerical results agree with the theoretical advantages of our methodology.

Due to the lack of rotation invariance in traditional convolution operations, even acting a slight rotation on the input can severely degrade the performance of Convolutional Neural Networks (CNNs). To address this, we propose a Rotation-Invariant Coordinate Convolution (RIC-C), which achieves natural invariance to arbitrary rotations around the input center without additional trainable parameters or data augmentation. We first evaluate the rotational invariance of RIC-C using the MNIST dataset and compare its performance with most previous rotation-invariant CNN models. RIC-C achieves state-of-the-art classification on the MNIST-rot test set without data augmentation and with lower computational costs. Then, the interchangeability of RIC-C with traditional convolution operations is demonstrated by seamlessly integrating it into common CNN models like VGG, ResNet, and DenseNet. We conduct remote sensing image classification on the NWPU VHR-10, MTARSI and AID datasets and patch matching experiments on the UBC benchmark dataset, showing that RIC-C significantly enhances the performance of CNN models across different applications, especially when training data is limited. Our codes can be downloaded from https://github.com/HanlinMo/Rotation-Invariant-Coordinate-Convolutional-Neural-Network.git. • We propose RIC-C: a novel convolutional operation naturally invariant to all input center rotations, no extra parameters or data augmentation. • Without data augmentation, RIC-CNN shows superior performance on MNIST compared to previous rotation invariant CNNs. • RIC-C is successfully deployed to popular CNN models, and enhances their performance in various tasks, particularly with limited training data. [ABSTRACT FROM AUTHOR]

A computational technique called opposition-based learning (OBL) has gained a lot of attention in the field of soft computing. Among the different forms of OBL, the iBetaCOBL variant, which is a stochastic OBL, has shown the best results in improving the performance of differential evolution (DE). However, this approach may not be effective in handling complex problems as it is susceptible to rotations in the coordinate system. To address this issue, we present a new and improved version of iBetaCOBL called iBetaCOBL-eig. This new technique uses an eigenvector-based multiple exponential crossover operator in the partial dimensional change method, making it rotationally invariant. We conducted experiments to evaluate the performance enhancements of DE using iBetaCOBL-eig on 29 challenging benchmark problems. Our results showed that the new algorithm is able to outperform not only ten strong OBL variants but also its previous version iBetaCOBL. • A rotationally invariant stochastic OBL called iBetaCOBL-eig is introduced. • iBetaCOBL-eig can direct the distribution of solutions toward the global optimum. • iBetaCOBL-eig can speed up the convergence of differential evolution variants. [ABSTRACT FROM AUTHOR]

The HLL solver for the generalized Riemann problem (HLL-GRP solver) is attractive because it is easy to implement for 1D hyperbolic problems and performs well. Based on the rotational invariance of multidimensional magnetohydrodynamics (MHD) equations, this paper provides a means of implementing the 1D HLL-GRP solver into the multidimensional ideal MHD simulations within the framework of the finite volume method. Furthermore, the Godunov-Powell source term and the locally divergence-free WLS-ENO reconstruction are adopted to constrain the divergence of the magnetic field. To maintain positive density and pressure, a positivity-preserving (PP) limiter is performed on the reconstructed polynomials of density and pressure. Some comparisons with other methods are listed in 1D, 2D and 3D MHD benchmark cases. The numerical results demonstrate that the scheme is stable and can well control the divergence of the magnetic field. [ABSTRACT FROM AUTHOR]

We present a new approach for constructing data-driven subgrid stress models for large eddy simulation of turbulent flows. The key to our approach is representation of model input and output tensors in the filtered strain rate eigenframe. Provided inputs and outputs are selected and non-dimensionalized in a suitable manner, this yields a model form that is symmetric, Galilean invariant, rotationally invariant, reflectionally invariant, and unit invariant. We use this model form to train a simple and efficient neural network model using only one time step of filtered direct numerical simulation data from a forced homogeneous isotropic turbulence simulation. We demonstrate the accuracy of this model as well as the model's ability to generalize to previously unseen filter widths, Reynolds numbers, and flow physics using a priori and a posteriori tests. [ABSTRACT FROM AUTHOR]

The problem of wind estimation by multirotor drone is suitable for machine learning (ML), because of the unknown drag coefficient changing with orientation. In previous work, we studied this problem experimentally to train an ML model that estimated wind by drone state alone. The primary drawback of this work was a decrease in performance between randomly selected and new flight data. This work applies data rotation and reduction to overcome this. Rotation allows the model to generalize to arbitrary coordinates, and reduction addresses data imbalance present in the original data. Two models are trained on experimental flight data: a gated-recurrent-unit (GRU) and a long-short-term-memory (LSTM) neural-network. The better performing GRU achieved 0.48 m/s root-mean-square-error on new flight data, although the LSTM performs similarly. These models approach the experimental limits of performance, which is determined by the spatial variation of wind as measured by the error between two separate anemometer readings. • Multirotor drone estimates wind field using a machine learning disturbance observer. • Data rotation helps models learn rotational invariance properties. • Data reduction addresses flight state imbalance and reduces training time. • Rotation and reduction produce models with robust performance metrics. • Experimental limits of performance account for ∼ 80% of model error. [ABSTRACT FROM AUTHOR]

In the companion paper (Bellet et al., 2021), a spherical harmonic subspace associated to the Cubed Sphere has been introduced. This subspace is further analyzed here. In particular, it permits to define a new Cubed Sphere based quadrature. This quadrature inherits the rotational invariance properties of the spherical harmonic subspace. Contrary to Gaussian quadrature, where the set of nodes and weights is solution of a nonlinear system, only the weights are unknown here. Despite this conceptual simplicity, the new quadrature displays an accuracy comparable to optimal quadratures, such as the Lebedev rules. [ABSTRACT FROM AUTHOR]

• Novel explicit upwind approaches are proposed for 2D basal resistance discretization. • The classical explicit upwind discretization generates mesh-alignment in the flow. • The implicit centered method for quadratic basal resistance fails in steady states. • The proposed explicit upwind approaches ensures mesh independence in realistic flows. In the context of two-dimensional models for complex geophysical surface flows such as debris flows, muddy slurries, oil spills over land, hyperconcentrated floods, lava flows, etc, depth-averaged rheological models relate the shear stress state within the fluid column to the depth-averaged local flow features. Despite it is the most influencing term on the mobility of complex shallow flows, the numerical treatment of the resistance contribution to the flow momentum is still a challenging topic, especially when dealing with 2D large-scale applications. In this work, two novel strategies for the explicit upwind discretization of generalized non-Newtonian resistance terms in two-dimensional numerical models are proposed, called integral and differential approaches. These new strategies are applicable to generalized rheological formulations in any type of mesh topology. Results from benchmark tests running in orthogonal, triangle structured and triangle unstructured meshes demonstrate that both approaches represent an improvement for the explicit upwind integration of the 2D resistance force compared with previous procedures. It is shown that the alignment of the flow with the mesh main-axis, which has been previously attributed to faults of 2D FV numerical methods and insufficient mesh refinements, is directly related to the loss of the rotational invariance of the integrated resistance force. This is caused by the erroneous procedure for including the 2D resistance term into the local flux balance at the cell edges. Furthermore, a novel implicit centered method for the integration of the 2D resistance force has also been derived for the quadratic frictional non-linear resistance formulation. Despite the implicit procedure fails to converge to steady uniform flow states, the differential explicit upwind and the implicit centered methods show similar level of accuracy, robustness and computational efficiency for transient 2D frictional visco-plastic flows. [ABSTRACT FROM AUTHOR]

• Archimedean networks obtained when unity resistors on edges of Archimedean solids. • Expressions obtained for two-point resistances of the thirteen Archimedean networks. • van Steenwijk's method used, solved with computer assistance in complicated cases. • Largest resistance 191/90 between opposite nodes of the Truncated Dodecahedron. • Smallest resistance 14137/38016 between neighbour nodes of the Snub Cube. The rise of novel materials such as graphene compounds or carbon nanotubes recently revived the interest on the role of symmetries upon electrical properties. Among possible solids, symmetrical polytopes, widely realized in natural and artificial matter, deserve particular attention. When equal resistors are attached between neighbouring nodes of Perfect (Platonic) solids (PS) along the edges, the two-point resistance between any pair of nodes was established in the 1990s by van Steenwijk using an elegant and efficient method. Using van Steenwijk's method, Moddy and Aravind subsequently derived the resistance values for two Archimedean and two Catalan solids. Here, with the same method, we derive the exact expression for the two-point resistance between any two nodes for resistor networks made of equal resistors placed along the edges of the thirteen Archimedean solids (AS). While the calculation remains elementary for the simplest AS, a dedicated method using computer assistance was developed for the Snub Cube, the Snub Dodecahedron and the three AS with three different rotation symmetries (the Great Rhombicuboctahedron, the Small Rhombicosidodecahedron and the Great Rhombicosidodecahedron). The largest resistance value (191/90) is obtained between the two opposite nodes of the Truncated Dodecahedron, and the second largest (42815/21114) is obtained between the two opposite nodes of the Great Rhombicosidodecahedron. The smallest resistance value (14137/38016) is obtained between some of the neighbouring nodes of the Snub Cube, whereas the resistance between neighbouring nodes of the Icosahedron (11/30) is slightly smaller. Some general symmetry relations between two-point resistances in AS networks are also derived, as well as some relations with two-point resistances in PS networks. The complete set of exact two-point resistance values for PS and AS networks can be used to check numerical codes or to evaluate the capacity of regular solids to provide appropriate models for given experimental situations. [ABSTRACT FROM AUTHOR]

The Internet of things (IoT) enables mobile devices to connect and exchange information with others over the Internet with a lot of applications in consumer, commercial, and industrial products. With the rapid development of machine learning, IoT with image recognition capability is a new research area to assist mobile devices with processing image information. In this research, we propose the rotation-invariant multi-scale convolutional neural network (RIMS-CNN) to recognize rotated objects, which are commonly seen in real situations. Based on the dihedral group D4 transformations, the RIMS-CNN equips a CNN with multiple rotated tensors and its processing network. Furthermore, multi-scale features and shared weights are employed in the RIMS-CNN to increase performance. Compared with the data augmentation approach of using rotated images at random angles for training, our proposed method can learn inherent convolution kernels for rotational features. Experiments were conducted on the benchmark datasets: MNIST, FASHION-MNIST, CIFAR-10, and CIFAR-100. Significant improvements over the other models were achieved to show that rotational invariance could be learned. [ABSTRACT FROM AUTHOR]