Dimensional Space Research Articles

New Construction of Mutually Unbiased Bases for Odd-dimensional State Space

Abstract We study the construction of mutually unbiased bases in Hilbert space for composite dimensions d which are not prime powers. We explore the results for composite dimensions which are true for prime power dimensions. And then provide a method for selecting mutually unbiased vectors from the eigenvectors of generalized Pauli matrices to construct mutually unbiased bases. In particular, we present four mutually unbiased bases in C15.

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

A domain knowledge-informed design space exploration methodology for mechanical layout design

Layout designs have a large number of design variables and various physical constraints. Conventional design exploration approaches are time-consuming and may require human intervention. A unique feature about physical layout designs is the availability of domain knowledge, which can be utilised to speed up the design process. In this paper, we propose a generative deep learning-based design space exploration (DSE) methodology that is capable of learning design constraints in the layout design without explicit supervision. Moreover, it can incorporate domain knowledge in the generated layouts, thereby speeding up the design process. This is realised by constructing a layout generation variational autoencoder (LGVAE) model, which uses a latent space as an interface to generate the layouts. By training the LGVAE model, significantly lower-dimensional representations can be learned compared to the original dimensionality of the design space. Therefore, the number of design variables is greatly reduced. We showcase the performance of the proposed DSE approach by solving the heat source layout design problem encountered in thermal management of chips. Experiments demonstrate that the LGVAE model is capable of generating compressed latent representations that capture the characteristics of the input samples, which makes the DSE cost-effective.

CompleteDT: Point cloud completion with information-perception transformers

In this work, we propose a novel point cloud completion network, called CompleteDT. To fully capture the 3D geometric structure of point clouds, we introduce an Information-Perception Transformer (IPT) that can simultaneously capture local features and global geometric relations. CompleteDT comprises a Feature Encoder, Query Generator, and Query Decoder. Feature Encoder extracts local features from multi-resolution point clouds to capture intricate geometrical structures. Query Generator uses the proposed IPT, utilizing the Point Local Attention (PLA) and Point Global Attention (PGA) modules, to learn local features and global correlations, and generate query features that represent predicted point clouds. The PLA captures local information within local points by adaptively measuring weights of neighboring points, while PGA adapts multi-head self-attention by transforming it into a layer-by-layer form where each head learns global features in a high-dimensional space of different dimensions. By dense connections, the module allows for direct information exchange between each head and facilitates the capture of long global correlations. By combining the strengths of both PLA and PGA, the IPT can fully leverage local and global features to facilitate CompleteDT to complete point clouds. Lastly, the query features undergo refining to generate a complete point cloud through the Query Decoder. Our experimental results demonstrate that CompleteDT outperforms current state-of-the-art methods, effectively learning from incomplete inputs and predicting complete outputs.

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

Let M be a hypersurface isometrically immersed in an (n+1)-dimensional semi-Riemannian space of constant curvature, n>3, such that its shape operator A satisfies A3=ϕA2+ψA+ρId, where ϕ, ψ and ρ are some functions on M and Id is the identity operator. The main result of this paper states that on the set U of all points of M at which the square S2 of the Ricci operator S of M is not a linear combination of S and Id, the Riemann-Christoffel curvature tensor R of M is a linear combination of some Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and the tensor S2 of M, i.e., the tensor R satisfies on U some Roter type equation. Moreover, the (0,4)-tensor R⋅S is on U a linear combination of some Tachibana tensors formed by the tensors g, S and S2. In particular, if M is a hypersurface isometrically immersed in the (n+1)-dimensional Riemannian space of constant curvature, n>3, with three distinct principal curvatures and the Ricci operator S with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.

Finite dimension of the ion pathway networks in conducting glasses

Abstract In disordered materials, the ordinary understanding is that charge carriers tend to occupy energetically favorable sites known as ion-conducting channels. Many studies have revealed that the inherent fractal properties of such pathways lead to a sub-diffusive behavior. The linearity or branching of these pathways is crucial for determining how the charge carriers move. It can be thought that as the space dimensionality decreases, the average distance between the highest energy barriers along the conduction paths increases. 
 In this study the finite dimension of those pathways is computed using an extended version of the classical Hausdorff dimension. Also, the Arrhenius behavior of the most mobile lithium ions is proved, confirming that such are responsible for conductivity behavior. The lithium ions mobility behavior in response to temperature changes and the finite dimension allowed to identify the ion diffusion regions fractal features. The reported results demonstrate that as the temperature increases the conducting channels become broadener, facilitating the transfer of electrical charge through the glassy matrix, below the transition temperature. The pathways behavior confirms the increase of the ionic conductivity when the temperature increases as it is experimentally observed.




The theory of screws derived from a module over the dual numbers

AbstractThe theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator $$\mathcal {E}$$ E , $$\mathcal {E}^2=0$$ E 2 = 0 , that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product ($$\mathbb {D}$$ D -module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.

Cephalometric Screening Assessment for Superior Airway Space Narrowing-Added Value of Three-Dimensional Imaging.

Background: Assessing the morphology of the superior airway space is a crucial diagnostic step in the treatment planning of patients with obstructive sleep apnea syndrome (OSAS) or prior to orthognathic surgery. The aim of this study is to evaluate the necessary scope of a two-dimensional cephalometric assessment and the necessity of three-dimensional imaging in the identification of superior airway space narrowing (SASN). Methods: The computed tomography studies of 100 non-obese, non-OSAS patients were evaluated and analyzed retrospectively. Multiplanar reconstructions were created and underwent cephalometric evaluation. The three-dimensional superior airway morphology was segmented and measured for the minimal cross-sectional area (Amin) and volume (V0). Patients were grouped according to Amin < 80 mm2 and V0 < 12 cm3. Cephalometric parameters (CPs) were analyzed according to Amin and V0 with an unpaired t-test, Pearson correlation, and ROC-curve analysis. Results: The CPs regarding sagittal airway space dimensions (IPAS, MPAS, SPAS) and mandibular body length (GoGn) show the strongest correlation to the three-dimensional minimal cross-sectional area (Amin). The ROC-curve analysis classifying for SASN led to an AUC of 0.86 for IPAS, 0.87 for MPAS, 0.88 for SPAS, and 0.63 for GoGn. Three-dimensional imaging may further improve the diagnostic accuracy in the identification of SASN for IPAS below 13.5 mm, MPAS below 10.2 mm, SPAS below 12.5 mm, and GoGn below 90.2 mm. Conclusions: Two-dimensional cephalometric sagittal airway space diameters and mandibular body length are useful initial screening parameters in the identification of superior airway space narrowing. Nevertheless, as the correlation of two-dimensional cephalometric parameters with three-dimensional upper airway space narrowing is varying and highly dependent on acquisition circumstances, indications for three-dimensional imaging, if possible, in the supine position to evaluate upper airway space morphology should be provided generously, especially in patients with low but normal airway space parameters in two-dimensional cephalometry.

Sea Anemone-Inspired Slippery Liquid-Infused Porous Surface (SLIPS) with Bionic Cilia for Responsive 4D Antifouling.

The formation process of biofouling is actually a 4D process with both spatial and temporal dimensions. However, most traditional antifouling coatings, including slippery liquid-infused porous surface (SLIPS), are limited to performing antifouling process in the 2D coating plane. Herein, inspired by the defensive behavior of sea anemones' wielding toxic tentacles, a "4D SLIPS" (FSLIPS) is constructed with biomimetic cilia via a magnetic field self-assembly method for antifouling. The bionic cilia move in 3D space driven by an external magnetic field, thereby preventing the attachment of microorganisms. The FSLIPS releases the gaseous antifoulant (nitric oxide) at 1D time in response to light, thereby achieving a controllable biocide effect on microorganisms. The FSLIPS regulates the movement of cilia via the external magnetic field, and controls the release of NO overtime via the light response, so as to adjust the antifouling modes on demand during the day or night. The light/magnetic response mechanism endow the FSLIPS with the ability to adjust the antifouling effect in the 4D dimension of 1D time and 3D space, effectively realizing the intelligence, multi-dimensionality and precision of the antifouling process.

Urban vitality and built environment from the perspective of spatiotemporal heterogeneity

ABSTRACT The spatiotemporal heterogeneity of the built environment’s effects on urban vitality is analysed through geographically and temporally weighted regression models. It is found that residents’ travel time and patterns lead to temporal heterogeneity, which is manifested by the fluctuation of the environmental effect of the peak in the morning and evening and the difference between weekdays and weekends. This is especially evident in the business district. Spatial heterogeneity is mainly reflected in the difference between the new urban area and the old city area. Single-function green spaces in areas with high functional complexity still have a weak attraction to people. Land use mix, scale and location are rather nonstationary in the dimensions of time and space.

Statistical equivalence of metrics for meteor dynamical association

Meteor showers, originating as a result of the activity of comets or the disruption of large objects, provide a unique window into the composition and dynamics of our Solar System. While modern meteor detection networks have amassed extensive data, distinguishing sporadic meteors from those belonging to specific meteor showers remains challenging. In this study, we statistically evaluate and compare four orbital similarity criteria within five-dimensional parameter space (DSH,DD,DH, and ϱ2) to study dynamical associations using the already classified meteors (manually by a human) in CAMS database as a benchmark. In addition, we assess various distance metrics typically used in Machine Learning with two different vectors: ORBIT, grounded in heliocentric orbital elements, and GEO, predicated on geocentric observational parameters. To estimate their degree of correlation and efficacy, the Kendall rank correlation coefficient and the Top-k accuracy are employed. The statistical equivalence of the Top-1 results is examined using the Kolmogorov–Smirnov test and the percentage of Top-1 agreement is calculated on an event-by-event basis. Additionally, we compute the optimal cut-offs for all methods for distinguishing sporadic background events. Our findings demonstrate the superior performance of the sEuclidean metric in conjunction with the GEO vector. Within the scope of D-criteria, DSH emerged as the preeminent metric, closely followed by ϱ2. The Bray-Curtis metric displayed an advantage compared to the other distance metrics when paired with the ORBIT vector for Top-k accuracy, however, the Cityblock metric is more effective when considering the sporadic background. ϱ2 stands out as the most equivalence to the distance metrics when utilizing the GEO vector and the most compatible with GEO and ORBIT simultaneously, whereas DD aligns more closely when using the ORBIT vector. The stark contrast in DD’s behavior compared to other D-criteria highlights potential inequivalence. Our results suggest that geocentric features provide a more robust basis than orbital elements for meteor dynamical association. Most distance metrics associated with the GEO vector surpass the D-criteria when differentiating the meteoroid background. Accuracy displayed a dependence on solar longitude with a pronounced decrease around 180° matching an apparent increase in the meteoroid background activity, tentatively associated with the transition from the Perseids to the Orionids. Considering lately identified meteor showers, ∼27% of meteors in CAMS would have different associations. This work unveils that Machine Learning distance metrics can rival or even exceed the performance of tailored orbital similarity criteria for meteor dynamical association.

Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves

Let C be a Gorenstein noncomplete intersection monomial curve in the 4-dimensional affine space with defining ideal I(C). In this article, we use the minimal generating set of I(C) to give a criterion for determining whether the tangent cone of C is Cohen-Macaulay. We also show that if the tangent cone of C is Cohen-Macaulay, then the minimal number of generators of the ideal I(C)⁎ is either 5 or an even integer of the form 2d+2, for a suitable integer d. Additionally, we provide a family of Gorenstein noncomplete intersection monomial curves C whose tangent cone is Cohen-Macaulay and the minimal number of generators of I(C)⁎ is large.

Approximate Morse-Sard type results for non-separable Banach spaces

For the Banach spaces E=c0(Γ),ℓp(Γ), where Γ is an arbitrary infinite set and 1<p<∞, we show that for every (non-zero) quotient F of E, every continuous function f:E→F can be uniformly approximated by smooth functions with no critical points, that is for every continuous function ε:E→(0,∞), there exists a Ck smooth function g:E→F such that g′(x) is surjective and ‖f(x)−g(x)‖≤ε(x) for all x∈E (k=∞ if E=c0(Γ) or p is an even integer, k=p−1 if p is an odd integer and k=[p] otherwise). Moreover, for a wide class of (not necessarily separable) infinite dimensional Banach spaces E and a suitable class of quotients F of E, every continuous function f:E→F can be uniformly approximated by Ck smooth functions with no critical points (k depending on the properties of the smoothness of the space E). In particular, for every Banach space E with Ck smooth partitions of unity (k∈N∪{∞}) and an infinite dimensional separable complemented subspace with a Ck smooth and LUR norm, we show that every continuous function f:E→Rn (n∈N) can be uniformly approximated by C1 smooth function with no critical points.

Simple modules for twisted Hamiltonian extended affine Lie algebras

In this paper we consider the twisted Hamiltonian extended affine Lie algebra (THEALA). We classify the irreducible integrable modules for these Lie algebras with finite dimensional weight spaces when the finite dimensional center acts non-trivially. This Lie algebra has a triangular decomposition, which is different from the natural triangular decomposition of twisted full toroidal Lie algebra. Any irreducible integrable module of it is a highest weight module with respect to the given triangular decomposition. In this paper we describe the highest weight space in detail.

3D Clustering of Gene Expression Data from Systemic Autoinflammatory Diseases Using Self-Organizing Maps (Clust3D)

Background and objectiveSystemic autoinflammatory diseases (SAIDs) are characterized by widespread inflammation, but for most of them there is a lack of specific biomarkers for accurate diagnosis. Although a number of machine learning algorithms have been used to analyze SAID datasets, aiding in the discovery of novel biomarkers, there is a growing recognition of the importance of SAID timeseries clustering, as it can capture the temporal dynamics of gene expression patterns. MethodologyThis paper proposes a novel clustering methodology to efficiently associate three-dimensional data. The algorithm utilizes competitive learning to create a self-organizing neural network and adjust neuron positions in time-dependent and high dimensional feature space in order to assign them as clustering centers. The quantitative evaluation of the clustering was based on well-known clustering indices. Furthermore, a differential expression analysis and classification pipeline was employed to assess the capability of the proposed methodology to extract more accurate pathway-specific genes from its clusters. For that, a comparative analysis was also conducted against a heuristic timeseries clustering method. ResultsThe proposed methodology achieved better overall clustering indices scores and classification metrics using genes derived from its clusters. Notable cases include a threefold increase in the Calinski-Harabasz clustering index, a twofold improvement in the Davies–Bouldin clustering index and a ~60% increase in the classification specificity score. ConclusionA novel clustering methodology was developed and applied on several gene expression timeseries datasets from systemic autoinflammatory diseases, and its ability to efficiently produce well separated clusters compared to existing heuristic methods was demonstrated.

Elementary proof of Funahashi's theorem

Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in $n$-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi \cite{Funahashi}. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper.

Twisted spatiotemporal optical vortex beams in dispersive media

We derive a closed-form expression for the mutual coherence function (MCF) of a twisted spatiotemporal optical vortex (STOV) beam after propagating a distance z in a linear dispersive medium. A twisted STOV beam is a partially coherent optical field that possesses a coherent STOV and a stochastic twist coupling its space and time dimensions. These beams belong to a special class of space–time-coupled light fields that carry transverse (to the direction of propagation) orbital angular momentum, making them potentially useful in numerous applications including quantum optics, optical manipulation, and optical communications. After presenting the derivation, we validate our new general MCF by showing that it simplifies to the MCFs for two example beams from the literature. Lastly, we present and analyze space–time beam profiles and complex arguments (phases) of the MCF at multiple z and differing amounts of material dispersion, field correlation or coherence, and beam twist to gain insight into how twisted STOV beams evolve in dispersive media. We conclude with a brief summary.

A Dichotomy for the Dimension of Solenoidal Attractors on High Dimensional Space

A Dichotomy for the Dimension of Solenoidal Attractors on High Dimensional Space

Comprehensive Multiview Representation Learning via Deep Autoencoder-Like Nonnegative Matrix Factorization.

Learning a comprehensive representation from multiview data is crucial in many real-world applications. Multiview representation learning (MRL) based on nonnegative matrix factorization (NMF) has been widely adopted by projecting high-dimensional space into a lower order dimensional space with great interpretability. However, most prior NMF-based MRL techniques are shallow models that ignore hierarchical information. Although deep matrix factorization (DMF)-based methods have been proposed recently, most of them only focus on the consistency of multiple views and have cumbersome clustering steps. To address the above issues, in this article, we propose a novel model termed deep autoencoder-like NMF for MRL (DANMF-MRL), which obtains the representation matrix through the deep encoding stage and decodes it back to the original data. In this way, through a DANMF-based framework, we can simultaneously consider the multiview consistency and complementarity, allowing for a more comprehensive representation. We further propose a one-step DANMF-MRL, which learns the latent representation and final clustering labels matrix in a unified framework. In this approach, the two steps can negotiate with each other to fully exploit the latent clustering structure, avoid previous tedious clustering steps, and achieve optimal clustering performance. Furthermore, two efficient iterative optimization algorithms are developed to solve the proposed models both with theoretical convergence analysis. Extensive experiments on five benchmark datasets demonstrate the superiority of our approaches against other state-of-the-art MRL methods.

Hypergraph Isomorphism Computation.

The isomorphism problem, crucial in network analysis, involves analyzing both low-order and high-order structural information. Graph isomorphism algorithms focus on structural equivalence to simplify solver space, aiding applications like protein design, chemical pathways, and community detection. However, they fall short in capturing complex high-order relationships, unlike hypergraph isomorphism methods. Traditional hypergraph methods face challenges like high memory use and inaccurate identification, leading to poor performance. To overcome these, we introduce a hypergraph Weisfeiler-Lehman (WL) test algorithm, extending the WL test from graphs to hypergraphs, and develop a hypergraph WL kernel framework with two variants: the Hypergraph WL Subtree Kernel and Hypergraph WL Hyperedge Kernel. The Hypergraph WL Subtree Kernel counts different types of rooted subtrees and generates the final feature vector for a given hypergraph by comparing the number of different types of rooted subtrees. The Subtree Kernel identifies different rooted subtrees, while the Hyperedge Kernel focuses on hyperedges' vertex labels, enhancing feature vector generation. In order to fulfill our research objectives, a comprehensive set of experiments was meticulously designed, including seven graph classification datasets and 12 hypergraph classification datasets. Results on graph classification datasets indicate that the Hypergraph WL Subtree Kernel can achieve the same performance compared with the classical Graph Weisfeiler-Lehman Subtree Kernel. Results on hypergraph classification datasets show significant improvements compared to other typical kernel-based methods, which demonstrates the effectiveness of the proposed methods. In our evaluation, our proposed methods outperform the second-best method in terms of runtime, running over 80 times faster when handling complex hypergraph structures. This significant speed advantage highlights the great potential of our methods in real-world applications.