An alternative proof of Fueter's theorem§

Contragenic functions are defined to be reduced-quaternion-valued harmonic functions which are orthogonal to all monogenic and antimonogenic functions in the $$L^2$$ L 2 norm of a given domain. The parallelism between the spaces of contragenic functions in the interior and exterior of the unit sphere in $$\mathbb {R}^3$$ R 3 is described in detail. Bergman reproducing kernels for the spaces of contragenic functions are given, mirroring the corresponding kernels for the spaces of vector parts of monogenic functions. Numerical examples are given showing the accuracy of truncations of the integral kernels. A striking duality is observed between the basic interior contragenic functions and the vector parts of exterior monogenic functions, and vice versa.

Special Bi-Axial Monogenic Functions

In this paper, we give a construction of Dunkl monogenic and Dunkl harmonic functions starting from holomorphic functions in the plane. This construction has the advantage of not needing Dunkl's intertwining operator or Dunkl spherical harmonics. To this end we study Vekua-type systems and prove a version of Fueter's theorem in the case of finite reflection groups. Important examples, such as a Dunkl monogenic Gaussian distribution or a Cauchy kernel, will be given at the end.

We analyse the finite-time blow-up of solutions of the heat flow for k-corotational maps . For each dimension we construct a countable family of blow-up solutions via a method of matched asymptotics by glueing a re-scaled harmonic map to the singular self-similar solution: the equatorial map. We find that the blow-up rates of the constructed solutions are closely related to the eigenvalues of the self-similar solution. In the case of 1-corotational maps our solutions are stable and represent the generic blow-up.

Several imaging algorithms including patch-based image denoising, image time series recovery, and convolutional neural networks can be thought of as methods that exploit the manifold structure of signals. While the empirical performance of these algorithms is impressive, the understanding of recovery of the signals and functions that live on manifold is less understood. In this paper, we focus on the recovery of signals that live on a union of surfaces. In particular, we consider signals living on a union of smooth band-limited surfaces in high dimensions. We show that an exponential mapping transforms the data to a union of low-dimensional subspaces. Using this relation, we introduce a sampling theoretical framework for the recovery of smooth surfaces from few samples and the learning of functions living on smooth surfaces. The low-rank property of the features is used to determine the number of measurements needed to recover the surface. Moreover, the low-rank property of the features also provides an efficient approach, which resembles a neural network, for the local representation of multidimensional functions on the surface. The direct representation of such a function in high dimensions often suffers from the curse of dimensionality; the large number of parameters would translate to the need for extensive training data. The low-rank property of the features can significantly reduce the number of parameters, which makes the computational structure attractive for learning and inference from limited labeled training data.

Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota–Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and will provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in [F. Batayneh and C. González-Tokman, On the number of invariant measures for random expanding maps in higher dimensions, Discrete Contin. Dyn. Syst. 41 (2021) 5887–5914] on admissible random Jabłoński maps to a more general class of higher-dimensional random maps.

Errata to "Analysis and Applications of Holomorphic Functions in Higher Dimensions"

We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant function under conformal transformations. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions.

Understanding high-dimensional spaces is crucial for various applications in mathematics and physics. The article intended to discover the relations between high-dimensional space and its applications in real life, based on the theories mentioned before. The research will be divided into two large parts and each part contains two objectives, which depend on the depth of knowledge. The first part includes a discussion of the known dimensional space. It will present simple explanations of these dimensional spaces. Next, the differences of adjacent dimensional space will be included. In addition, it will discuss the relationship in the known dimensional space. In the second part, the paragraph will try to transfer 4D or higher dimensions from 3D and existing data, which is the most significant objective. At the end of the research, it will find some applications related to the research in daily life, hoping more people will realize what high dimensional space is and associate it with daily life.

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

Based on general findings from the field of neuroscience and their algorithmic implementations using signal processing, information theory and machine learning techniques, this paper highlights the advantages of modelling a signal in a sparse and high-dimensional feature space. The emphasis is put on the hierarchical organisation, very high dimensionality and sparseness aspects of auditory information, that allow unsupervised learning of meaningful auditory objects from simple linear projections. When the dictionaries are learned using independent component analysis (ICA), it is shown that specific spectro-temporal modulation patterns are learned to optimally represent speech, noise and tonal components. In a noisy isolated-word speech recognition task, sparse and high-dimensional features have shown greater robustness to noise compared to a standard system based on a dense low-dimensional feature space. This brings new ways of thinking in the field of recognition and classification of acoustic signals.

A method of document clustering based on locality preserving indexing (LPI) and support vector machines (SVM) is presented. The document space is generally of high dimensionality, and clustering in such a high-dimensional space is often infeasible due to the curse of dimensionality. In this paper, by using LPI, the documents are projected into a lower-dimension semantic space in which the documents related to the same semantic are close to each other. Then, by using SVM, the vectors in semantic space are mapped by means of a Gaussian kernel to a high-dimensional feature space in which the minimal enclosing sphere is searched. The sphere, when mapped back to semantics space, can separate into several independent components by the support vectors, each enclosing a separate cluster of documents. By combining the LPI and SVM, not only higher clustering accuracies in a more unsupervised effective way, but also better generalization properties can be obtained. Extensive demonstrations are performed on the Reuters-21578 and TDT2 data sets.

Informative path planning is an important and challenging problem in robotics that remains to be solved in a manner that allows for wide-spread implementation and real-world practical adoption. Among various reasons for this, one is the lack of approaches that allow for informative path planning in high-dimensional spaces and non-trivial sensor constraints. In this work we present a sampling-based approach that allows us to tackle the challenges of large and high-dimensional search spaces. This is done by performing informed sampling in the high-dimensional continuous space and incorporating potential information gain along edges in the reward estimation. This method rapidly generates a global path that maximizes information gain for the given path budget constraints. We discuss the details of our implementation for an example use case of searching for multiple objects of interest in a large search space using a fixed-wing UAV with a forward-facing camera. We compare our approach to a sampling-based planner baseline and demonstrate how our contributions allow our approach to consistently out-perform the baseline by 18.0%. With this we thus present a practical and generalizable informative path planning framework that can be used for very large environments, limited budgets, and high dimensional search spaces, such as robots with motion constraints or high-dimensional configuration spaces. [Code] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> Codebase: https://github.com/castacks/tigris [Video] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sup> Video: https://youtu.be/bMw5nUGL5GQ

The use of effective distance functions has been explored for many data mining problems including clustering, nearest neighbor search, and indexing. Recent research results show that if the Pearson variation of the distance distribution converges to zero with increasing dimensionality, the distance function will become unstable (or meaningless) in high dimensional space even with the commonly used Lp metric on the Euclidean space. This result has spawned many subsequent studies. We first comment that although the prior work provided the sufficient condition for the unstability of a distance function, the corresponding proof has some defects. Also, the necessary condition for unstability (i.e., the negation of the sufficient condition for the stability) of a distance function, which is required for function design, remains unknown. Consequently, we first provide in this paper a general proof for the sufficient condition of unstability. More importantly, we go further to prove that the rapid degradation of Pearson variation for a distance distribution is in fact a necessary condition of the resulting unstability. With the result, we will then have the necessary and the sufficient conditions for unstability, which in turn imply the sufficient and necessary conditions for stability. This theoretical result derived leads to a powerful means to design a meaningful distance function. Explicitly, in light of our results, we design in this paper a meaningful distance function, called Shrinkage-Divergence Proximity (abbreviated as SDP), based on a given distance function. It is empirically shown that the SDP significantly outperforms prior measures for its being stable in high dimensional data space and robust to noise, and is thus deemed more suitable for distance-based clustering applications than the priorly