Dimensional Space Research Articles

Novel bisection-method laser scanning strategy for laser powder bed fusion

ABSTRACT Laser scanning strategy is one of the most factors affecting on the temperature gradient for the laser powder bed fusion (L-PBF). In this work, we propose a new Bisection-Method Scanning Strategy (BM-SS) based on dichotomous approximation, to reduce the heat accumulation effect between paths. The results show that BM-SS can effectively optimize the temperature gradient of the sample compared with that of the traditional sequential scanning strategy (TS-SS). The temperature gradient significantly optimizes as the n value increase, and the highest temperature is reduced by about 85K. Further experiments and numerical simulation indicate that BM-SS can effectively increase the cooling time between adjacent paths to reduce the temperature gradient and thermal stress of the sample from both time and space dimensions. The warpage of the sample reduces by up to 5 times without losing the mechanical properties. This BM-SS provides a new insight to improve the temperature gradient during L-PBF.

Non-degenerate Anisocurved Surfaces in Homogeneous 3-Manifolds

In this manuscript we consider non-degenerate surfaces Σ2 immersed in a 3-dimensional homogeneous space L3(κ,τ) endowed with two different metrics, the one induced by the Riemannian metric of E3(κ,τ) and the non-degenerate metric inherited by the Lorentzian one of L3(κ,τ). Therefore, we have two different geometries on Σ2 and we can compare them. In particular, we can consider the Gaussian curvature functions which respect to both metrics and study the geometry of the surfaces satisfying that both Gaussian curvature functions are opposite. We will call these surfaces anisocurved surfaces. In order to obtain our main results we also need to impose some extra assumptions regarding the extrinsic curvatures with respect to both metrics.

Three-dimensional inversion algorithm of magnetotelluric data based on solution space dimensionality reduction technique

The magnetotelluric (MT) method is highly advantageous in geothermal resource exploration because of its deep probing capability, low cost, and sensitivity to key geothermal system elements such as heat sources, reservoirs, and caprocks. To address the challenge of balancing the MT inversion resolution and computational efficiency, we propose a Gauss–Newton optimization inversion algorithm based on the solution space dimensionality reduction technique. This algorithm uses solution space dimensionality reduction to establish algebraic mapping relationships between the edges of grid cells, effectively reducing the degrees of freedom in the linear equation system required for forward and pseudo-forward modeling, thereby saving computational costs. In addition, during the inversion process, a multiple right-hand direct–iterative hybrid solver is employed to solve the dimensionally reduced linear equation s, further enhancing the computational efficiency of the inversion. We then validate and analyze the accuracy and efficiency of the forward and inversion algorithms using various synthetic models. Finally, we apply this inversion strategy to field MT data from the Dabie Mountain geothermal area in eastern China. On the basis of the inversion results, we then analyze the thermal genesis model of the region, providing technical support for the exploration and development of deep geothermal resources.

Spontaneous thought separates into clusters of negative, positive, and flexible thinking

The nature and frequency of spontaneous thoughts play a critical role in cognitive processes like perception, decision-making, attention, and memory. Deficits in these processes are also greatly associated with the development and maintenance of psychopathology. However, the underlying cognitive dynamics of free and stuck spontaneous thought remain unclear, as these often occur in the absence of measurable behaviors. Here, we analyze free word-association data using attractor-state dynamic modeling, which conceptualizes stuck spontaneous thought as navigating a multidimensional semantic space while in the presence of strong attractor locations. Word-association data was collected from an exploratory sample (N1 = 65), a first replication sample (N2 = 79), and, following pre-registration, a second replication sample (N3 = 222). After the data was embedded into a 3-dimensional semantic space and fit by our dynamic model, unsupervised learning consistently grouped data into four clusters across all independent samples. These clusters were characterized by two distinct patterns of stuck negative thinking, a pattern of protective positive thinking, and a pattern of flexible mind-wandering. Our results support a method for modeling spontaneous thought and isolate distinct sub-types that may not be accessible using retrospective self-report methods. We discuss implications for clinical and cognitive science.

Codebase release 1.0 for DanceQ

The complexity of quantum many-body problems scales exponentially with the size of the system, rendering any finite-size scaling analysis a formidable challenge. This is particularly true for methods based on the full representation of the wave function, where one simply accepts the enormous Hilbert space dimensions and performs linear algebra operations, e.g., for finding the ground state of the Hamiltonian. If the system satisfies an underlying symmetry where an operator with a degenerate spectrum commutes with the Hamiltonian, it can be block-diagonalized, thus reducing the complexity at the expense of additional bookkeeping. At the most basic level required for Krylov space techniques (like the Lanczos algorithm), it is necessary to implement a matrix-vector product of a block of the Hamiltonian with arbitrary block-wavefunctions, potentially without holding the Hamiltonian block in memory. An efficient implementation of this operation requires the calculation of the position of an arbitrary basis vector in the canonical ordering of the basis of the block. We present here an elegant and powerful, multi-dimensional approach to this problem for the U(1)U(1) symmetry appearing in problems with particle number conservation. Our divide-and-conquer algorithm uses multiple subsystems and hence generalizes previous approaches to make them scalable. In addition to the theoretical presentation of our algorithm, we provide DanceQ, a flexible and modern – header only – C++20 implementation to manipulate, enumerate, and map to its index any basis state in a given particle number sector as open source software under https://DanceQ.gitlab.io/danceq.

DanceQ: High-performance library for number conserving bases

The complexity of quantum many-body problems scales exponentially with the size of the system, rendering any finite-size scaling analysis a formidable challenge. This is particularly true for methods based on the full representation of the wave function, where one simply accepts the enormous Hilbert space dimensions and performs linear algebra operations, e.g., for finding the ground state of the Hamiltonian. If the system satisfies an underlying symmetry where an operator with a degenerate spectrum commutes with the Hamiltonian, it can be block-diagonalized, thus reducing the complexity at the expense of additional bookkeeping. At the most basic level required for Krylov space techniques (like the Lanczos algorithm), it is necessary to implement a matrix-vector product of a block of the Hamiltonian with arbitrary block-wavefunctions, potentially without holding the Hamiltonian block in memory. An efficient implementation of this operation requires the calculation of the position of an arbitrary basis vector in the canonical ordering of the basis of the block. We present here an elegant and powerful, multi-dimensional approach to this problem for the U(1)U(1) symmetry appearing in problems with particle number conservation. Our divide-and-conquer algorithm uses multiple subsystems and hence generalizes previous approaches to make them scalable. In addition to the theoretical presentation of our algorithm, we provide DanceQ, a flexible and modern – header only – C++20 implementation to manipulate, enumerate, and map to its index any basis state in a given particle number sector as open source software under https://DanceQ.gitlab.io/danceq.

Criteria for smoothness of positroid varieties via pattern avoidance, Johnson graphs, and spirographs

Positroids are certain representable matroids originally studied by Postnikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defining positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson-Lam-Speyer, and Pawlowski studied geometric and cohomological properties of these varieties. In this paper, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids. This allows us to give two formulas for counting the number of smooth positroids along with two q q -analogs. We include enumerative results due to Christian Krattenthaler which follow from these formulas. Furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at key points using an induced subgraph of the Johnson graph. We also give a Bruhat interval characterization of positroids.

On the solution of the Yang–Baxter–like operator equation on Hilbert spaces

This paper deals with solutions of the Yang–Baxter–like operator equation AXA = XAX on the infinite dimensional Hilbert space H , where A is a rank-two bounded linear operator and X is the unknown bounded linear operator. For an operator X, necessary and sufficient conditions such that X is a solution of the equation AXA = XAX are given. With these criteria, it is in fact shown that all solutions of the nonlinear equation AXA = XAX under the corresponding conditions can be derived merely through solving some systems of linear equations.

Massively Parallel Tensor Network State Algorithms on Hybrid CPU-GPU Based Architectures.

The interplay of quantum and classical simulation and the delicate divide between them is in the focus of massively parallelized tensor network state (TNS) algorithms designed for high performance computing (HPC). In this contribution, we present novel algorithmic solutions together with implementation details to extend current limits of TNS algorithms on HPC infrastructure building on state-of-the-art hardware and software technologies. Benchmark results obtained via large-scale density matrix renormalization group (DMRG) simulations on single node multiGPU NVIDIA A100 system are presented for selected strongly correlated molecular systems addressing problems on Hilbert space dimensions up to 4.17 × 1035.

Holographic foliations: Self-similar quasicrystals from hyperbolic honeycombs

Discrete geometries in hyperbolic space are of longstanding interest in pure mathematics and have come to recent attention in holography, quantum information, and condensed matter physics. Working at a purely geometric level, we describe how any regular tessellation of (d+1)-dimensional hyperbolic space naturally admits a d-dimensional boundary geometry with self-similar “quasicrystalline” properties. In particular, the boundary geometry is described by a local, invertible, self-similar substitution tiling, that discretizes conformal geometry. We greatly refine an earlier description of these local substitution rules that appear in the 1D/2D example and use the refinement to give the first extension to higher dimensional bulks; including a detailed account for all regular 3D hyperbolic tessellations. We comment on global issues, including the reconstruction of bulk geometries from boundary data, and introduce the notion of a “holographic foliation”: a foliation by a stack of self-similar quasicrystals, where the full geometry of the bulk (and of the foliation itself) is encoded in any single leaf in a local, invertible way. In the {3,5,3} tessellation of 3D hyperbolic space by regular icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold symmetry which is not the Penrose tiling, and record and comment on a related conjecture of William Thurston. We end with a large list of open questions for future analytic and numerical studies. Published by the American Physical Society 2025

Multidimensional computed measurement for highly accurate PCBA defect detection

Accurate defect detection in industrial automated optical inspection (AOI) is crucial as it directly affects product quality and production efficiency. Although numerous techniques have been developed for industrial defect detection, most of them rely on single-texture image data. This dependence limits the accuracy and robustness of the defect detection due to inadequate optical source information. To overcome the problem of low accuracy owing to the lack of 3D topographic information, a multidimensional information fusion (MIF) module is proposed that fuses texture image and depth image features. The MIF module includes tailored mechanisms to fully extract complementary semantic information from space and channel dimensions. A hierarchical fusion strategy further improves feature integration by enabling higher-layer feature fusion via lower-layer Transformer blocks and efficiently removing redundant features. Afterward, feature extraction is performed on the fused feature map, and output is obtained. To enhance the detection accuracy, the position information mask (PIM) module is introduced for post-processing. The PIM module uses surface mount devices (SMDs) position data from Gerber files to create a position information mask. The mask helps filter out defects that are often misidentified owing to missing design information. The results of the comparative experiments demonstrate that the average accuracy of our method on the printed circuit board assembly (PCBA) defect dataset is 99.93%, which is 5.64% higher than that of conventional YOLOV5. Furthermore, a comprehensive ablation study is conducted to elucidate the contribution of the proposed MIF and PIM modules. It demonstrated that the present model serves as a valuable reference for PCBA surface defect detection.

Autoencoder Reconstruction of Cosmological Microlensing Magnification Maps

Abstract Enhanced modeling of microlensing variations in light curves of strongly lensed quasars improves measurements of cosmological time delays, the Hubble Constant, and quasar structure. Traditional methods for modeling extragalactic microlensing rely on computationally expensive magnification map generation. With large data sets expected from wide-field surveys like the Vera C. Rubin Legacy Survey of Space and Time, including thousands of lensed quasars and hundreds of multiply imaged supernovae, faster approaches become essential. We introduce a deep-learning model that is trained on pre-computed magnification maps covering the parameter space on a grid of κ, γ, and s. Our autoencoder creates a low-dimensional latent space representation of these maps, enabling efficient map generation. Quantifying the performance of magnification map generation from a low dimensional space is an essential step in the roadmap to develop neural network-based models that can replace traditional feed-forward simulation at much lower computational costs. We develop metrics to study various aspects of the autoencoder generated maps and show that the reconstruction is reliable. Even though we observe a mild loss of resolution in the generated maps, we find this effect to be smaller than the smoothing effect of convolving the original map with a source of a plausible size for its accretion disk in the red end of the optical spectrum and larger wavelengths and particularly one suitable for studying the broad-line region of quasars. Used to generate large samples of on-demand magnification maps, our model can enable fast modeling of microlensing variability in lensed quasars and supernovae.

Time‐Multiplexed Reservoir Computing with Quantum‐Dot Lasers: Impact of Charge‐Carrier Scattering Timescale

Reservoir computing with optical devices offers an energy‐efficient approach for time‐series forecasting. Quantum dot lasers with feedback are modeled in this article to explore the extent to which increased complexity in the charge‐carrier dynamics within the nanostructured semiconductor can enhance the prediction performance. By tuning the scattering interactions, the laser's dynamics and response time can be finely adjusted, allowing for a systematic investigation. It is found that both system response time and task requirements need to be considered to find optimal operation conditions. Further, lasers with pronounced relaxation oscillations outperform those with strongly damped dynamics, even if the underlying charge‐carrier dynamics is more complex. This demonstrates that optimal reservoir computing performance relies not only on a high internal phase space dimension but also on the effective utilization of these dynamics through the output sampling process, quantum dot laser, reservoir computing, feedback delay, effective scattering rate, relaxation oscillation.

Transient response and analytical solution of burst phenomenon in long distance water pipeline

The rapid detection and localisation of pipe burst incidents in water pipe systems are crucial for timely repair, minimisation of leakage, and assurance of water supply security. However, previous researches have insufficiently emphasised the identification of pressure characteristics during pipe burst, failing to accurate detection and localisation of pipe burst. To address this problem, an experimental system was designed to elucidate the complete dynamic process of pipe burst pressure fluctuations. Specifically, the continuous transition from steady state to transient state and back to steady state in the hydraulic state of the system after a pipe burst event, was observed both in time and space dimensions. The effects of initial pressure and flow rate at steady state, location and size of pipe burst on pressure characteristics were carefully investigated. Subsequently, a pipe burst model considering unsteady friction effect was developed and validated through experiments. Based on the above study, analytical solution regarding pressure drop during burst events and steady-state pressure at the burst point after burst events were derived to quantify the influence laws of different parameters and validated by the pipe burst model. The obtained results are characterised by the dominance analysis so as to explore the importance ranking of different factors to the pressure drop and steady-state burst pressure in the pipeline system. The analysis solution of pressure drop indicates the following ranking of factors from high to low is: pipe/burst area ratio, initial steady-state pressure at the burst point, wave speed; while the ranking of influencing factors for steady-state burst pressure from high to low is: the boundary pressures, the location of the pipe burst, the initial steady state flow rate, and the burst outflow coefficient. The research findings can provide theoretical and practical guidance for the rapid diagnosis and quantitative analysis of burst pipes.

Low dimensionality of phenotypic space as an emergent property of coordinated teams in biological regulatory networks.

Cell-fate decisions involve coordinated genome-wide expression changes, typically leading to a limited number of phenotypes. Although often modeled as simple toggle switches, these rather simplistic representations often disregard the complexity of regulatory networks governing these changes. Here, we unravel design principles underlying complex cell decision-making networks in multiple contexts. We show that the emergent dynamics of these networks and corresponding transcriptomic data are consistently low-dimensional, as quantified by the variance explained by principal component 1 (PC1). This low dimensionality in phenotypic space arises from extensive feedback loops in these networks arranged to effectively enable the formation of two teams of mutually inhibiting nodes. We use team strength as a metric to quantify these feedback interactions and show its strong correlation with PC1 variance. Using artificial networks of varied topologies, we also establish the conditions for generating canalized cell-fate landscapes, offering insights into diverse binary cellular decision-making networks.

Szegő Limit Theorem for Truncated Toeplitz Operators

We discuss generalizations of the Szegő Limit Theorem to truncated Toeplitz operators. In particular, we consider compressions of Toeplitz operators to an increasing sequence of finite dimensional model spaces. We present two theorems. The first is a new variant of the Szegő Limit Theorem in this setting. The second relates to the variant given by Strouse–Timotin–Zarrabi (J Approx Theory 220:12–30, 2017), and characterizes the sequences for which that result holds.

Uncertainty analysis method for diagnosing multi-point defects in urban drainage systems

Urban drainage system (UDS) plays a key role in city urbanization, where defective pipes can lead to seepage. Previous studies have identified the locations of defects in UDS using inverse optimization models. However, the unique optimal solution neglects uncertainty analysis, which may lead to misdiagnosis. In addition, the multi-point defect diagnosis has heavy computational burden due to high dimensional parameters space. To address these issues, this paper proposes a hybrid method that leverages the genetic algorithm (GA) to identify probable space, and then utilizes the adaptive Metropolis (AM) to provide an estimation of the posterior probability distribution (PPD). Firstly, a multi-population GA is employed for the maximum exploration within the model space. Then, AM algorithm is used to explore the final PPD of each pipe defect parameter. The metrics accuracy (ACC), Matthews correlation coefficient (MCC) and mean absolute error (MAE) are used to evaluate the diagnosis performance. A synthetic UDS case with randomized multi-point seepage scenarios is used to validate the method. Results indicate that the proposed hybrid method is effective in diagnosing multi-point defect, with 0.91, 0.78 of the hybrid method and 0.87, 0.69 of the DiffeRential Evolution Adaptive Metropolis method for the ACC and MCC, respectively. Meanwhile, the diagnosis speed has increased by 32 %. The result PPD passes the 90 % confidence interval validation. The proposed method can provide effective uncertainty analysis to reduce misdiagnosis of the traditional method.

Intrinsic-dimension analysis for guiding dimensionality reduction and data fusion in multi-omics data processing.

Multi-omics data have revolutionized biomedical research by providing a comprehensive understanding of biological systems and the molecular mechanisms of disease development. However, analyzing multi-omics data is challenging due to high dimensionality and limited sample sizes, necessitating proper data-reduction pipelines to ensure reliable analyses. Additionally, its multimodal nature requires effective data-integration pipelines. While several dimensionality reduction and data fusion algorithms have been proposed, crucial aspects are often overlooked. Specifically, the choice of projection space dimension is typically heuristic and uniformly applied across all omics, neglecting the unique high dimension small sample size challenges faced by individual omics. This paper introduces a novel multi-modal dimensionality reduction pipeline tailored to individual views. By leveraging intrinsic dimensionality estimators, we assess the curse-of-dimensionality impact on each view and propose a two-step reduction strategy for significantly affected views, combining feature selection with feature extraction. Compared to traditional uniform reduction pipelines in a crucial and supervised multi-omics analysis setting, our approach shows significant improvement. Additionally, we explore three effective unsupervised multi-omics data fusion methods rooted in the main data fusion strategies to gain insights into their performance under crucial, yet overlooked, settings.

Connection of the Theory of Time with the Friedmann Equation

The proposed work shows a new approach to deriving the Friedmann equation. The approach is based on the fact that the primary medium from which the universe could arise is a time substance consisting of primary tiny particles. This medium is described by a scalar field. The mathematical apparatus used for its study allowed obtaining dual equations, which were interpreted by the author as waves of direct and reverse time directed towards each other. When the waves contacted, an inversion of the reverse wave occurred. It took the direction of the direct wave. As a result of the inversion, dynamic processes arose that led to the appearance of acceleration in 4-dimensional space. Analysis of the formulas for velocity and acceleration led to the derivation of Friedmann equations for a flat universe, obtained by him from Einstein's gravitational equation. Introduction. The concept of a continuous space-time continuum has become firmly established in modern physics. It was introduced by A. Einstein in the special and then General Theory of Relativity (GTR). By tacit agreement, it is generally accepted that the continuum is filled with matter that is in continuous motion, moving at speeds not exceeding the speed of light. Matter has mass, which leads to the curvature of space-time. GTR interprets this curvature as the presence of a gravitational field in the continuum. To describe it, A. Einstein's tensor gravitational equation is used. It is used to analyze curved 4-dimensional space-time. With this approach, the dimension consists of three spatial and one time coordinate. The result is a mathematically complex theory, lacking clarity and poorly understood. In 1922, A.A. Friedman solved the equations of GTR after modifying the second cosmological postulate. This led to a result that meant a change in the size of the Metagalaxy over time, i.e. to its non-stationarity. The solutions obtained formed the basis of the theoretical basis of cosmology. No other approaches to deriving Friedman's equations, except on the basis of GTR, were undertaken. But there is another approach. And the author talks about it in this article. It is based on the theory of time developed by him.

N-Dimensional Lattice Path Enumeration

Let V_n be a set of integer vectors. We enumerate lattice paths that only uses vectors in V_n. Unlike most lattice path enumeration problems, the number of dimensions isn't fixed and the vector set is dependent on the dimension. This requires us to follow a different approach in explicitly expressing the number of lattice paths from origin to any point in $n$-dimensional space. We notice that a special case of this problem corresponds to Fubini numbers, which count the number of weak orderings of a set consisting of $n$ elements. Then, we find the recursive relation of this sequence. Finally, we develop an algorithm that can be used to find the number of paths between any two points that do not touch the lattice points in R. The crucial part of our algorithm is that it doesn't rely on finding all paths and checking each path for usage of restricted points.