Nonlinear Space Research Articles

NEW ITERATIVE TRANSFORM METHOD FOR TIME AND SPACE FRACTIONAL (n + 1)-DIMENSIONAL HEAT AND WAVE TYPE EQUATIONS

In this study, we examine and using Laplace Transform as a way to find approximate solutions to multi-dimensional space and time fractional order problems and propose numerical algorithm for solving ([Formula: see text])-dimensional time and space fractional order heat like and time and space fractional wave-like equations. This method is a combination of Laplace Transform and iterative method. The fractional derivative is described in the Caputo sense. The results obtained by proposed scheme are compared with different other schemes. This scheme is found to be very efficient and effective for linear and nonlinear time and space fractional order problems.

Real-time FPGA-based implementation of the AKAZE algorithm with nonlinear scale space generation using image partitioning

The first step in a scale invariant image matching system is scale space generation. Nonlinear scale space generation algorithms such as AKAZE, reduce noise and distortion in different scales while retaining the borders and key-points of the image. An FPGA-based hardware architecture for AKAZE nonlinear scale space generation is proposed to speed up this algorithm for real-time applications. The three contributions of this work are (1) mapping the two passes of the AKAZE algorithm onto a hardware architecture that realizes parallel processing of multiple sections, (2) multi-scale line buffers which can be used for different scales, and (3) a time-sharing mechanism in the memory management unit to process multiple sections of the image in parallel. We propose a time-sharing mechanism for memory management to prevent artifacts as a result of separating the process of image partitioning. We also use approximations in the algorithm to make hardware implementation more efficient while maintaining the repeatability of the detection. A frame rate of 304 frames per second for a 1280 times 768 image resolution is achieved which is favorably faster in comparison with other work.

RMPflow: A Geometric Framework for Generation of Multitask Motion Policies

Generating robot motion for multiple tasks in dynamic environments is challenging, requiring an algorithm to respond reactively while accounting for complex nonlinear relationships between tasks. In this article, we develop a novel policy synthesis algorithm, Riemannian motion policy (RMP)flow, based on geometrically consistent transformations of RMPs. RMPs are a class of reactive motion policies that parameterize non-Euclidean behaviors as dynamical systems in intrinsically nonlinear task spaces. Given a set of RMPs designed for individual tasks, RMPflow can combine these policies to generate an expressive global policy, while simultaneously exploiting sparse structure for computational efficiency. We study the geometric properties of RMPflow and provide sufficient conditions for stability. Finally, we experimentally demonstrate that accounting for the natural Riemannian geometry of task policies can simplify classically difficult problems, such as planning through the clutter on high-degree-of-freedom manipulation systems. Note to Practitioners —Requirements on safety and responsiveness for collaborative robots have driven a need for new ideas in control design that bridge between standard objectives in low-level control (such as trajectory tracking) and high-level behavioral objectives (such as collision avoidance) often relegated to planning systems. Modern results from geometric control, which promise stable controllers that can smoothly and safely transition between many behavioral tasks, therefore, become highly relevant. However, for years, this field has remained inaccessible due to its mathematical complexity. This article aims to: 1) make those ideas accessible to robotics and control experts by recasting them in a concrete algorithmic framework amenable to controller design and 2) to additionally generalize them to better satisfy the specific needs of robotic behavior generation. Our experiments demonstrate that the resulting controllers can engender natural behavior that adapts instantaneously to changing surroundings with zero planning while performing manipulation tasks. The framework is gaining traction within the robotics community, finding increasing application in areas, such as autonomous navigation, tactile servoing, and multi-agent systems. Future research will address learning these controllers from data to simplify that process of design and tuning, which at present can require experience.

An introduction to mixed hemivariational inequality problems on Hadamard manifolds

In this article, we introduce a category of mixed hemivariational inequality problems on Hadamard manifolds. Using Fan-KKM lemma we initiate the existence of solutions of mixed hemivariational inequality problems (MHVIP) when the underlying vector field is monotone. We also establish the proximal point algorithm for solving mixed hemivariational inequality problems on these nonlinear spaces. Our outcomes and schemes may stimulate further exploration in this fascinating and interesting area of analysis.

Breaking reciprocity and preserving-frequency using linear acoustic metamaterials

In this paper, we introduce a linear waveguide implemented by cascading acoustic black holes (ABHs). The asymmetric wave propagation, up to 46 dB, is observed and verified in simulation and experiment. It is shown that, in comparison with the previous nonlinear acoustic diodes, our waveguide can rectify the sound without shifting the impinging sound frequency. The device is simple and easy-to-fabricate without using complex nonlinear materials and space–time modulation. This feature could open a new route for designing acoustic waveguide devices that preserve the key information.

Dual distance adaptive multiview clustering

Adaptively adjusting the graph, by taking the clustering capability into consideration, has become popular in graph-based clustering methods and extended to multiview clustering problem. Existing methods learn the graph from pairwise distances of the data in the original space or a linearly projected space, which requires that those representations can finely reflect the implicit data structure. However, the data structure in high-dimensional space may not always lie on a linear manifold, and this problem becomes more critical in multiview conditions. Aim at this, we propose a multiview clustering method based on adaptive graph and dual distance. Specifically, we fuse the distances computed from nonlinear embedding space and original space. An adaptive graph is then constructed based on the fused distance, and it is more reliable for multiview clustering. In our approach, the clustering result with exact number of clusters can be found without post-processing. We evaluate the proposed method on several multiview datasets, the experimental results show our approach is superior to the state-of-the-art multiview clustering methods.

Nonlinear methods for model reduction

Typical model reduction methods for parametric partial differential equations construct a linear space Vn which approximates well the solution manifold M consisting of all solutions u(y) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n-term approximation, and certain tree-based methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space Vn by a nonlinear space Σn. Little is known in terms of their performance guarantees, and most existing numerical experiments use a parameter dimension of at most two. In this work, we make a step towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation, we give a first comparison of their performance with the performance of standard linear approximation for any compact set. We then study these methods for solution manifolds of parametrized elliptic PDEs. We study a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and give performance guarantees with respect to accuracy of approximation versus m and N.

Tomographic Performance of Multi-Static Radar Formations: Theory and Simulations

3D imaging of Earth’s surface layers (such as canopy, sub-surface, or ice) requires not just the penetration of radar signal into the medium, but also the ability to discriminate multiple scatterers within a slant-range and azimuth resolution cell. The latter requires having multiple radar channels distributed in across-track direction. Here, we describe the theory of multi-static radar tomography with emphasis on resolution, SNR, sidelobes, and nearest ambiguity location vs. platform distribution, observation geometry, and different multi-static modes. Signal-based 1D and 2D simulations are developed and results for various observation geometries, target distributions, acquisition modes, and radar parameters are shown and compared with the theory. Pros and cons of multi-static modes are compared and discussed. Results for various platform formations are shown, revealing that unequal spacing is useful to suppress ambiguities at the cost of increased multiplicative noise. In particular, we demonstrate that the multiple-input multiple-output (MIMO) mode, in combination with nonlinear spacing, outperforms the other modes in terms of ambiguity, sidelobe levels, and noise suppression. These findings are key to guiding the design of tomographic SAR formations for accurate surface topography and vegetation mapping.

GaborNet Visual Encoding: A Lightweight Region-Based Visual Encoding Model With Good Expressiveness and Biological Interpretability.

Computational visual encoding models play a key role in understanding the stimulus–response characteristics of neuronal populations in the brain visual cortex. However, building such models typically faces challenges in the effective construction of non-linear feature spaces to fit the neuronal responses. In this work, we propose the GaborNet visual encoding (GaborNet-VE) model, a novel end-to-end encoding model for the visual ventral stream. This model comprises a Gabor convolutional layer, two regular convolutional layers, and a fully connected layer. The key design principle for the GaborNet-VE model is to replace regular convolutional kernels in the first convolutional layer with Gabor kernels with learnable parameters. One GaborNet-VE model efficiently and simultaneously encodes all voxels in one region of interest of functional magnetic resonance imaging data. The experimental results show that the proposed model achieves state-of-the-art prediction performance for the primary visual cortex. Moreover, the visualizations demonstrate the regularity of the region of interest fitting to the visual features and the estimated receptive fields. These results suggest that the lightweight region-based GaborNet-VE model based on combining handcrafted and deep learning features exhibits good expressiveness and biological interpretability.

Time-Efficient Torque Shaping for Switched Reluctance Machines From Linear Space

In this article, a time-efficient torque shaping technique for the switched reluctance machines is proposed to reduce the torque ripples while optimizing the copper loss. To reduce the computational time for solving the time-consuming bi-objective optimization problem, an efficient and desirable linear space for the SRM is utilized through the torque mapping between the linear space and nonlinear space. The feasible regions and properties of this torque mapping for SRM are studied in detail first. Then, based on this torque mapping, the current reference is generated by solving a convex optimization problem in the linear space and later converted to the current reference in nonlinear space. Compared to the traditional method that only adopts nonlinear space, the proposed current reference generation method requires less than 10% of the time. Furthermore, the proposed method saves more than 70% time consumed by the traditional method for the bi-objective optimization problem but possesses the similar torque ripple reduction performance. The effectiveness of the proposed method has been verified by extensive simulation and experimental tests on a four-phase 5.1 kW 8/6 SRM setup.

Search for adequate closed form wave solutions to space–time fractional nonlinear equations

The nonlinear space–time fractional Phi-4 equation and density dependent fractional reaction–diffusion equation (FRDE) are important models to interpret the fusion and fission phenomena ensued in solid state physics, plasma physics, chemical kinematics, astrophysical fusion plasma, electromagnetic interactions etc. In this study, we search advanced and wide-ranging wave solutions to the formerly reported nonlinear fractional evolution equations in diverse family through the new generalized (G′∕G)-expansion technique. The solutions are developed with trigonometric, hyperbolic, exponential and rational functions including parameters. The technique is a compatible, functional and effective scientific scheme to examine diverse space–time fractional models in physics and engineering concerned with the real life problems.

Rapid charge redistribution leading to core hollowing in a high-intensity ion beam

Recently, the first direct measurement of a full 6D accelerator beam distribution was reported [1]. That work observed a correlation between energy and transverse coordinates, for which the energy distribution becomes hollowed and double-peaked near the transverse core. In this article, a similar structure is shown to emerge in expansion of an initially uncorrelated, high density bunched beam as the result of velocity perturbation from nonlinear space charge forces. This hollowing is obscured when the 6D phase space is projected onto one- and two-dimensional axes. This phenomenon has not been widely recognized in accelerator systems, but parallels can be drawn to observations of laser-ionized nanoclusters and electron sources for diffraction. While this effect provides insight into the origin of the measured core correlation, it does not provide a complete description. A better reproduction of the measured structure can be obtained via self-consistent simulation through the radio-frequency quadrupole. [1] B. Cathey, S. Cousineau, A. Aleksandrov, and A. Zhukov, PRL 121, 064804 (2018).

Fixed point iterations of asymptotically demicontractive and asymptotically hemicontractive mappings in hyperbolic spaces

In this paper, we obtain strong convergence results for asymptotically demicontractive and asymptotically hemicontractive mappings in hyperbolic spaces. We present our results in hyperbolic spaces. This class of spaces contains both linear and nonlinear spaces like CAT(0) spaces, [Formula: see text]-trees, Banach spaces and Hilbert spaces. Thus our results are not only novel but also much more general.

Ball: An R Package for Detecting Distribution Difference and Association in Metric Spaces

The rapid development of modern technology has created many complex datasets in non-linear spaces, while most of the statistical hypothesis tests are only available in Euclidean or Hilbert spaces. To properly analyze the data with more complicated structures, efforts have been made to solve the fundamental test problems in more general spaces (Lyons 2013; Pan, Tian, Wang, and Zhang 2018; Pan, Wang, Zhang, Zhu, and Zhu 2020). In this paper, we introduce a publicly available R package Ball for the comparison of multiple distributions and the test of mutual independence in metric spaces, which extends the test procedures for the equality of two distributions (Pan et al. 2018) and the independence of two random objects (Pan et al. 2020). The Ball package is computationally efficient since several novel algorithms as well as engineering techniques are employed in speeding up the ball test procedures. Two real data analyses and diverse numerical studies have been performed, and the results certify that the Ball package can detect various distribution differences and complicated dependencies in complex datasets, e.g., directional data and symmetric positive definite matrix data.

Reciprocal kernel-based weighted collaborative–competitive representation for robust face recognition

The Gaussian kernel function is widely used to encode the nonlinear correlations of the face images. However, some issues greatly limit its superiority, for example, it is sensitive to the parameter setting because of its definition based on the exponential operation, on the other hand, the Gaussian kernel needs costly computational time. Besides, the hidden information such as the distance information of the samples is conducive to improving the performance of face recognition. To overcome the above problems, we propose a reciprocal kernel-based weighted collaborative–competitive representation for face recognition. Different from other methods, a new reciprocal kernel is designed to realize the nonlinear representation of the samples. Moreover, a new weight based on the reciprocal kernel is imposed on coding coefficients to disclose the hidden information of the samples in the nonlinear space. With the help of the collaborative–competitive method, the proposed method can well achieve the trade-off between collaborative and competitive representation to promote the performance of face recognition. These factors explicitly encourage the proposed method to be a better representation-type classifier. Finally, extensive experiments are conducted on five benchmark datasets, and the experimental results show that the proposed approach outperforms many state-of-the-art approaches.

Dense and Sparse 3D Deformation Signatures for 3D Dynamic Face Recognition

This work analyses dense and sparse 3D Deformation Signatures to represent 3D temporal deformation instances. The signatures are employed in dynamic 3D face recognition, however, they are applicable in other domains. This is demonstrated for dynamic expression recognition. The pushing need for non-intrusive bio-metric measurements made face and its expressions recognition dominant players in domains like entertainment, surveillance and security. The proposed signature can be computed from 2D, 3D or hybrid input by means of robust 3D fitting. It is computed given a non-linear 6D space representation which guarantees by construction physically plausible 3D deformations. A unique deformation indicator is computed per triangle in a triangulated mesh as a ratio derived from scale and in-plane deformation in the canonical space. These indicators are concatenated densely or sparsely to form the signature. It is then used to learn the 3D deformation space from the temporal facial signals. Two dynamic datasets were examined for evaluation. The reported 1-Rank recognition accuracy outperforms the existing literature. Democratising the recognition step results in 100% accuracy as demonstrated by the reported confusion matrices. In an open-world setting in the face recognition context, an accuracy of 100% was achieved in detecting intruders. The signature robustness has been further validated in face expressions recognition from a very challenging highly 3D dynamic dataset.

Warrior Women: Recovering Indigenous Visions across Film and Activism

Warrior Women:Recovering Indigenous Visions across Film and Activism Kiara M. Vigil (bio) On March 28, 1973, the Desert Sun criticized Marlon Brando for sending "an Indian woman" to refuse his award for Best Supporting Actor for The Godfather (Francis Ford Coppola, 1972) from the Academy of Motion Pictures. Celebrities in the audience and television viewers around the world were shocked when Sacheen Littlefeather (Apache) appeared in "fringed-leather and beaded moccasins" to announce Brando would "not accept the Oscar." Together they wanted to draw attention to the long history of mistreatment of Native Americans by the film industry as well as the American Indian Movement's (AIM) occupation of Wounded Knee.1 The alliance between Brando and Littlefeather was an act of protest, a show of solidarity, and a call for change. Over forty years later, I had the honor of interviewing Littlefeather about this moment from her past.2 She spoke candidly about the stakes of her appearance as an actor and as an activist. On stage that evening, Littlefeather worried about the prospect of violence, as a threatened and threatening John Wayne angrily paced backstage, and whether she might be arrested since police officers waited in the wings in case she upset the show's organizers. [End Page 169] I learned about these chilling details, and the tense atmosphere surrounding this moment of activism, through oral history work, and I am indebted to Littlefeather for gifting me with time, stories, and teachings. As we spoke, I was reminded of Kwe scholar Leanne Betasamosake Simpson's As We Have Always Done, in which she writes, of Nishnaabeg intellectualism, that "meaning is derived from both repetition and context."3 Nishnaabeg intelligence cannot be decontextualized, as it must be understood "through a compassionate web of interdependent relationships."4 These relationships value difference and the agency of individuals to generate their own meaning. Teachings emerge out of particular lived experiences and must be understood within this context. As I built relationships with Sacheen Littlefeather and also with Lois Red Elk (Fort Peck Sioux), I remained attuned to how I learned from them as elders as well as actors and activists. I came to this project through another web of relations as well, in search of new understandings about the life and activism of my great-grandfather, the Dakota actor Shooting Star. My research as a Native cultural historian depends upon building ethical relationships with my subjects, whether living or not. I approach research with a deep sense of respect and responsibility for the Native voices and stories I recover. Like Simpson, I am invested in complex learning that takes place within a network of Indigenous intelligence.5 Indigenous knowledge systems are "networked because the modes of communication and interaction between beings occur in complex nonlinear forms, across time and space."6 I highlight the networked contributions of Native people to American society to account for complexity and contradiction and to offer a fuller understanding of settler colonialism and Indigenous constellations of co-resistance since colonizers come to stay, to destroy in order to replace.7 As Patrick Wolfe argues, "invasion is a structure, not an event."8 Therefore, I look for the strategies Indigenous peoples use to resist the imposition of colonial structures onto and into their lives, from the past to the present. For Littlefeather and Red Elk, the warrior women in this essay, there were arenas—social spaces, cultural performances, and work sites—through which they generated knowledge as acts of survivance, Anishinaabe cultural theorist Gerald Vizenor's term for "an active sense of presence, the continuance of Native stories."9 Through their engagement with Indigenous networks of actors and activists, they shaped counter-narratives as acts of refusal. For Audra Simpson, "the politics of refusal" depend upon Indigenous refusals to engage with modern citizenship as defined by the settler nation.10 Both Littlefeather and Red Elk refused to comply with Hollywood's [End Page 170] imagined Indians. They were central in the "native hubs" constituting Los Angeles's pan-tribal community, which were sites of social activity, community building, job training, and political activism for urban Indians living in diaspora.11 Jay Silverheels (Mohawk) and Eddie Little...

Incremental Kernel Principal Components Subspace Inference With Nyström Approximation for Bayesian Deep Learning

As the state-of-the-art technology of Bayesian inference, based on low-dimensional principal components analysis (PCA) subspace inference methods can provide approximately accurate predictive distribution and well calibrated uncertainty. However, the main problem of PCA method is that it is a linear subspace feature extractor, and it cannot effectively represent the nonlinearly high-dimensional parameter space of deep neural networks (DNNs). Firstly, in this paper, in order to solve the main problem of the linear characteristics of PCA in high-dimensional space, we apply kernel PCA to extract higher-order statistical information in parameter space of DNNs. Secondly, to improve the efficiency of subsequent computation, we propose a strictly ordered incremental kernel PCA (InKPCA) subspace of parameter space within stochastic gradient descent (SGD) trajectories. In the proposed InKPCA subspace, we employ two approximation inference methods: elliptical slice sampling (ESS) and variational inference (VI). Finally, to further improve the memory efficiency of computing the kernel matrix, we apply Nyström approximation to determine the suitable size of subsets in the original datasets. The novelty of this paper is that it is the first time to apply the proposed InKPCA subspace with Nyström approximation for Bayesian inference in DNNs, and the results show that it can produce more accurate predictions and well-calibrated predictive uncertainty in regression and classification tasks of deep learning.

Inferring Multilingual Domain-Specific Word Embeddings From Large Document Corpora

The use of distributed vector representations of words in Natural Language Processing has become established. To tailor general-purpose vector spaces to the context under analysis, several domain adaptation techniques have been proposed. They all require sufficiently large document corpora tailored to the target domains. However, in several cross-lingual NLP domains both large enough domain-specific document corpora and pre-trained domain-specific word vectors are hard to find for languages other than English. This paper aims at tackling the aforesaid issue. It proposes a new methodology to automatically infer aligned domain-specific word embeddings for a target language on the basis of the general-purpose and domain-specific models available for a source language (typically, English). The proposed inference method relies on a two-step process, which first automatically identifies domain-specific words and then opportunistically reuses the non-linear space transformations applied to the word vectors of the source language in order to learn how to tailor the vector space of the target language to the domain of interest. The performance of the proposed method was validated via extrinsic evaluation by addressing the established word retrieval task. To this aim, a new benchmark multilingual dataset, derived from Wikipedia, has been released. The results confirmed the effectiveness and usability of the proposed approach.

Solution to linear KdV and nonLinear space fractional PDEs

In this work, the author will briefly discuss applications of the Fourier and Laplace transforms in the solution of certain singular integral equations and evaluation of integrals. By combining integral transforms and operational methods we get more powerful analytical tool for solving a wide class of linear or even non- linear fractional differential or fractional partial differential equation. Numerous examples and exercises occur throughout the paper.