Dimensional Space Research Articles

Chaotic LLM billiards

We study null geodesics of the ten-dimensional LLM geometries. In particular, we show that there are a subset of these null geodesics that are confined to the LLM plane. The effective dynamics of these in-plane geodesics is that of a Hamiltonian system with two degrees of freedom (a phase space of dimension 4). We show that these are chaotic. In the two-coloring of the LLM plane, if they start in the empty region, they cannot penetrate the filled region and viceversa. The dynamical problem is therefore very similar to that of a billiards problem with fixed obstacles. We study to what extent LLM geometries with many droplets may be treated as an incipient black hole and draw analogies with the fuzzball proposal. We argue that for in-plane null geodesics deep in the interior of a region with a lot of droplets, in order to exit towards the AdS boundary they will need to undergo a process that resembles diffusion. This mechanism can account for signals getting lost in the putative black hole for a very long time.

A conditional latent autoregressive recurrent model for generation and forecasting of beam dynamics in particle accelerators

Particle accelerators are complex systems that focus, guide, and accelerate intense charged particle beams to high energy. Beam diagnostics present a challenging problem due to limited non-destructive measurements, computationally demanding simulations, and inherent uncertainties in the system. We propose a two-step unsupervised deep learning framework named as Conditional Latent Autoregressive Recurrent Model (CLARM) for learning the spatiotemporal dynamics of charged particles in accelerators. CLARM consists of a Conditional Variational Autoencoder transforming six-dimensional phase space into a lower-dimensional latent distribution and a Long Short-Term Memory network capturing temporal dynamics in an autoregressive manner. The CLARM can generate projections at various accelerator modules by sampling and decoding the latent space representation. The model also forecasts future states (downstream locations) of charged particles from past states (upstream locations). The results demonstrate that the generative and forecasting ability of the proposed approach is promising when tested against a variety of evaluation metrics.

Early warning signals for bifurcations embedded in high dimensions

Recent work has highlighted the utility of methods for early warning signal detection in dynamic systems approaching critical tipping thresholds. Often these tipping points resemble local bifurcations, whose low dimensional dynamics can play out on a manifold embedded in a much higher dimensional state space. In many cases of practical relevance, the form of this embedding is poorly understood or entirely unknown. This paper explores how measurement of the critical phenomena that generically precede such bifurcations can be used to make inferences about some properties of their embeddings, and, conversely, how prior knowledge about the mechanism of bifurcation can robustify predictions of an oncoming tipping event. These modes of analysis are first demonstrated on a simple fluid flow system undergoing a Hopf bifurcation. The same approach is then applied to data associated with the West African monsoon shift, with results corroborated by existing models of the same system. This example highlights the effectiveness of the methodology even when applied to complex climate data, and demonstrates how a well-resolved spatial structure associated with the onset of atmospheric instability can be inferred purely from time series measurements.

Global stability of the Dirac–Klein–Gordon system in two and three space dimensions

Global stability of the Dirac–Klein–Gordon system in two and three space dimensions

Asymptotic stability of the nonlocal diffusion equation with nonlocal delay

This work focuses on the asymptotic stability of nonlocal diffusion equations in ‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove and ‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if , then the solution converges globally to the trivial equilibrium time‐exponentially. If , then the solution converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when , the solution converges globally to the positive equilibrium time‐exponentially, and when , the solution converges globally to the positive equilibrium time‐algebraically. Here, , and are the coefficients of each term contained in the linear part of the nonlinear term . All convergence rates above are and ‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.

Dynamic relationships and propagation characteristics between meteorological drought and vegetation drought based on a three-dimensional identification algorithm

Drought is a regional phenomenon and progressive in specific dimensions of time and space, with significant continuity and dynamic characteristics on a spatiotemporal scale. The meteorological drought acts as a driving factor for vegetation drought, and studying the response characteristics of vegetation drought to meteorological drought is crucial for comprehending the mechanisms of drought evolution. In this study, based on a three-dimensional spatiotemporal clustering algorithm, meteorological and vegetation drought events in the Yellow River Basin (YRB) from 1982 to 2020 were identified, and the dynamic characteristics of typical drought events were depicted, elucidating the inherent correlations and propagation features between meteorological and vegetation drought events. Additionally, according to the spatiotemporal matching criteria of drought events, the meteorological-vegetation drought event pairs were successfully matched, which can reveal the lag time between meteorological drought and vegetation drought. From a three-dimensional perspective, we revealed the dynamic propagation characteristics of meteorological and vegetation drought events, which could provide an effective way for monitoring and mitigating regional vegetation drought, restoring ecosystem functionality, and promoting sustainable development of the ecological environment.

Operators in the internal space and locality

Realizations of the holographic correspondence in String/M theory typically involve spacetimes of the form AdS × Y where Y is some internal space which geometrizes an internal symmetry of the dual field theory, hereafter referred to as an “R symmetry”. It has been speculated that areas of Ryu-Takayanagi surfaces anchored on the boundary of a subregion of Y, and smeared over the base space of the dual field theory, quantify entanglement of internal degrees of freedom. A natural candidate for the corresponding operators are linear combinations of operators with definite R charge with coefficients given by the “spherical harmonics” of the internal space: this is natural when the product spaces appear as IR geometries of higher dimensional AdS spaces. We study clustering properties of such operators both for pure AdS × Y and for flow geometries, where AdS × Y arises in the IR from a different spacetime in the UV, for example higher dimensional AdS or asymptotically flat spacetime. We show, in complete generality, that the two point functions of such operators separated along the internal space obey clustering properties at scales sufficiently larger than the AdS scale. For non-compact Y, this provides a notion of approximate locality. When Y is compact, clustering happens only when the size of Y is parametrically larger than the AdS scale. This latter situation is realized in flow geometries where the product spaces arise in the IR from an asymptotically AdS geometry at UV, but not typically when they arise near black hole horizons in asymptotically flat spacetimes. We discuss the significance of this result for entanglement and comment on the role of color degrees of freedom.

Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers.

Geoacoustic inversion using Bayesian optimization with a Gaussian process surrogate model.

Geoacoustic inversion can be a computationally expensive task in high-dimensional parameter spaces, typically requiring thousands of forward model evaluations to estimate the geoacoustic environment. We demonstrate Bayesian optimization (BO), an efficient global optimization method capable of estimating geoacoustic parameters in seven-dimensional space within 100 evaluations instead of thousands. BO iteratively searches parameter space for the global optimum of an objective function, defined in this study as the Bartlett power. Each step consists of fitting a Gaussian process surrogate model to observed data and then choosing a new point to evaluate using a heuristic acquisition function. The ideal acquisition function balances exploration of the parameter space in regions with high uncertainty with exploitation of high-performing regions. Three acquisition functions are evaluated: upper confidence bound, expected improvement (EI), and logarithmically transformed EI. BO is demonstrated for both simulated and experimental data from a shallow-water environment and rapidly estimates optimal parameters while yielding results comparable to differential evolution optimization.

Three-dimensional analysis of palatal morphology and PAS in patients with cleft lip and palate prior to orthodontic treatment

BackgroundSince many different conclusions of craniofacial anomalies and their relation to the posterior airway space coexist, this comparative clinical study investigated the palatal morphology concerning volumetric size, posterior airway space dimension and the adenoids of patients with and without a cleft before orthodontic treatment.MethodsThree-dimensional intraoral scans and cephalometric radiographs of n = 38 patients were used for data acquisition. The patients were divided into three groups: unilateral cleft lip and palate (n = 15, 4 female, 11 male; mean age 8.57 ± 1.79 years), bilateral cleft lip and palate (n = 8, 0 female, 8 male; mean age 8.46 ± 1.37 years) and non-cleft control (n = 15, 7 female, 8 male; mean age 9.03 ± 1.02 years). The evaluation included established procedures for measurements of the palatal morphology and posterior airway space. Statistics included Shapiro-Wilk-Test and simple ANOVA (Bonferroni) for the three-dimensional intraoral scans and cephalometric radiographs. The level of significance was set at p < 0.05.ResultsThe palatal volume and cephalometric analysis showed differences between the three groups. The palatal volume, the superior posterior face height and the depth of the bony nasopharynx of patients with cleft lip and palate were significantly smaller than for non-cleft control patients. The superior posterior face height of bilateral cleft lip and palate patients was significantly smaller than in unilateral cleft lip and palate patients (BCLP: 35.50 ± 2.08 mm; UCLP: 36.04 ± 2.95 mm; p < 0.001). The percentage of the adenoids in relation to the entire nasopharynx and the angle NL/SN were significantly bigger in patients with cleft lip and palate than in the non-cleft control. In particular, the palatal volume was 32.43% smaller in patients with unilateral cleft lip and palate and 48.69% smaller in patients with bilateral cleft lip and palate compared to the non-cleft control.ConclusionsSkeletal anomalies relate to the dimension of the posterior airway space. There were differences among the subjects with cleft lip and palate and these without a cleft. This study showed that the morphology of the palate and especially transverse deficiency of the maxilla resulting in smaller palatal volume relates to the posterior airway space. Even the adenoids seem to be affected, especially for cleft lip and palate patients.

An adversarial network reinforcement method for power data time series generation based on knowledge transfer and Rayleigh differential privacy

Abstract In order to solve the problem of sensitive information leakage of power data in machine learning and artificial intelligence processing, this paper proposes a method of civic-text data reinforcement based on time series generation adversarial network, which combines knowledge transfer and Rényi differential privacy protection framework to balance privacy and availability, while improving the accuracy of the model. First, the generative adversarial network generates training data that is similar to the real data but secure through adversarial learning, ensuring that even if the data is stolen, the original information cannot be inferred. Secondly, through the integration of recurrent neural network training and autoencoder, the dynamic and static characteristics of the power load data can be effectively captured, and the high dimension of the learning space can be reduced. In addition, knowledge distillation technology is used to guide the generator of training student models through a powerful teacher network of knowledge transfer, improving the diversity and quality of the generated data. Finally, from the four aspects of privacy, data similarity and diversity, and practicability, the practical application value and potential of the framework for electricity consumption data are comprehensively evaluated.

The future of work of academics in the age of Artificial Intelligence: State-of-the-art and a research roadmap

The Future of Work (FoW) has garnered significant attention among scholars and practitioners, with the advent of Artificial Intelligence (AI) playing an important role in shaping this discourse. Despite the common perception that intelligent machines pose a threat to workers in routine roles, AI technologies are increasingly being utilized for advanced tasks carried out by knowledge workers. Drawing on state-of-the-art research and real-life examples we develop an integrated framework to explore the future of academic work. Our focus is on academics, an essential yet under-researched group of knowledge workers, and we discuss their work in relation to AI across space, time, and task dimensions. Our analysis reveals that the usage of AI technologies can have implications for the research, teaching, and service activities of academics and thereby also for the creation, acquisition, dissemination, and application of knowledge. Based on our framework we develop scenarios and propose a future research roadmap.

Photon number distribution of squeezed light from a silicon nitride microresonator measured without photon number resolving detectors

The measurement of the photon number distribution (PND) allows one to extract metrics of non-classicality of fundamental and technological relevance, but in principle it requires the use of detectors with photon number resolving (PNR) capabilities. In this work we reconstruct the PND of two-mode pulsed squeezed light generated from a silicon nitride microresonator using threshold detectors and variable optical attenuations. The PNDs are characterized up to ∼1.2 photons per pulse, through which we extracted an on-chip squeezing level of 6.2(2) dB and a noise reduction factor of −3.8(2) dB. The PNDs are successfully reconstructed up to a Hilbert space dimension of 6 × 6. The analysis performed on the photon number basis allows us to characterize the influence of a spurious thermal background field that spoils the photon number correlations. We evaluate the impact of self- and cross-phase modulation on the generation efficiency in case of a pulsed pump, and validate the results through numerical simulations of the master equation of the system.

The Alpha Distance Formulae

In this study, we define the concept of tangent in two and three-dimensional alpha spaces concerning the alpha circles and the alpha spheres. Then using this concept, we derive the alpha distance formulae between points, a point and a line, between two lines and a point and a plane of the alpha spaces. Finally, we give simple area and volume formulas in the three dimensional space in terms of the alpha distances.

High-dimensional multi-objective optimization of coupled cross-laminated timber walls building using deep learning

This paper introduces a deep learning-based multi-objective optimization framework, applying for advancing the design of the Cross-Laminated Timber Coupled Wall system. While traditional optimization methods often struggle with the curse of dimensionality in high-dimensional problems, the approach proposed in this study employs an autoencoder to effectively reduce the dimensionality of the design space. Subsequently, a neural network establishes a mapping between input variables and latent spaces, with another neural network forming the crucial link between these latent variables and the output responses. The proposed framework is integrated into the design process of a 20-story Cross-Laminated Timber Coupled Wall system, where uncertainties inherent in connection elements are systematically addressed to optimize the structural parameters. The non-dominated sorting genetic algorithm-II is utilized to estimate optimal design variables by minimizing three conflicting objective functions, thereby generating a Pareto front. This optimized design is then bench-marked against three deterministic models with varying coupling beam shear strengths. A two-dimensional numerical model developed in OpenSees facilitates nonlinear time history analysis using 50 bi-directional ground motion records, representative of the seismicity of Vancouver, Canada. The results of this study not only highlight the efficacy of the deep learning-based framework in enhancing the structural integrity and resilience of high-rise timber structures in seismic regions but also significantly contribute to the evolution of computational approaches in structural engineering.

Premonitory Earthquakes Clustering Process in an Equivalent Dimensions Space before the 2017 Mw 8.2 Tehuantepec, Mexico, Mainshock

Abstract The transformation to equivalent dimensions, an innovative methodology to investigate earthquake clustering, is engaged in tracking the preparatory process of the 2017 Mw 8.4 Tehuantepec, Mexico, mainshock. The studied seismic series comprises earthquakes from 1999 to May 2020 in an area extended three times more than the main rupture length. We parameterized each earthquake by its magnitude, the waiting time for its occurrence, that is, the time since the previous earthquake after which this earthquake occurred, and the epicentral offset from the previous earthquake, that is, the distance of the epicenter of this earthquake from the epicenter of the previous earthquake. After transformation to equivalent dimensions, these three parameters became mutually comparable. The average distance between earthquakes in the space of these three transformed parameters was used to determine the degree of event clustering. We calculated this distance in data windows moving in time, each consisting of 100 events. The average distance exhibited a consistent upward trend from ten to two years before the mainshock. Then, it declined until the mainshock. This precursory up–down signal was highly significant statistically. We showed that the detected time changes of the average distance resulted from the evolution of the earthquake clustering in the space of their parameters.

European Solidarity Takes the Field: The Dimensional Structure of Political Competition in the European Union After the Polycrisis

AbstractMost studies investigating the dimensionality of political space in Europe demonstrate that European Union (EU)‐related issues are orthogonal to the economic left–right dimension. This literature usually employs survey items tapping into general support for the EU. However, the multiple crises that hit Europe after 2008 have polarized even Europhile voters on the more specific issue of European solidarity. This article aims to investigate how voters' attitudes towards European solidarity interplay with their position on the left–right and European integration dimensions, as well as specific economic and cultural issues. Empirical analyses employ original survey data collected in 2019 in 10 EU countries. Results partially support the expectations of regional differences in the dimensionality of the political space, with European solidarity more strongly associated with left–right, EU integration and immigration in Northern EU member states than in Eastern and Southern countries, in which voters are more likely to support solidarity notwithstanding their general political orientations.

Optimal control of a solar sail

SummaryOptimal control of a solar sail orbiting around a fixed center of mass is considered. The sail is modelled as a plane surface with two sides having similar optical properties. The control is assumed to be the attitude of the sail and is represented as an element of the projective plane. Mapping this plane into the three dimensional ambient space to compute the actual force generated by a given attitude results in a generally non‐convex set of admissible control values. A suitable convex relaxation is introduced to study the optimality system associated with maximising the change of the sail orbital parameters in a given direction along one orbit. Existence for both the relaxed and the original problems are deduced. Building on previous works, a refined analysis of the control structure is given, proving rigorously that it contains only switchings, and a global bound of the number of switchings is obtained. In order to compute effective solutions, a multiple shooting approach is retained that is well suited for systems with a sequence of switchings between various control subarcs. Three issues are addressed: initialization of shooting by means a tailored semi‐definite relaxation that takes advantage of good convergence properties of modern convex optimization algorithms; changes in the control structure that are accommodated by coupling shooting with differential continuation; implicit character of the Hamiltonian maximization when using Pontrjagin maximum principle that is taken care of by incorporating the associated equation for the dynamical feedback into the shooting procedure. The method is illustrated on an example from NASA for which the sail inclination is optimally changed over one orbital period.

On the asymptotic behaviour of the fractional Sobolev seminorms: A geometric approach

We study the well-known asymptotic formulas for fractional Sobolev functions à la Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova, in a geometric approach. We show that the key to these asymptotic formulas are Rademacher's theorem and volume growth at infinity respectively. Examples fitting our framework includes Euclidean spaces, Riemannian manifolds, Alexandrov spaces, finite dimensional Banach spaces, and some ideal sub-Riemannian manifolds.

Hilbert-curve assisted structure embedding method

MotivationChemical space embedding methods are widely utilized in various research settings for dimensional reduction, clustering and effective visualization. The maps generated by the embedding process can provide valuable insight to medicinal chemists in terms of the relationships between structural, physicochemical and biological properties of compounds. However, these maps are known to be difficult to interpret, and the ‘‘landscape’’ on the map is prone to ‘‘rearrangement’’ when embedding different sets of compounds.ResultsIn this study we present the Hilbert-Curve Assisted Space Embedding (HCASE) method which was designed to create maps by organizing structures according to a logic familiar to medicinal chemists. First, a chemical space is created with the help of a set of ‘‘reference scaffolds’’. These scaffolds are sorted according to the medicinal chemistry inspired Scaffold-Key algorithm found in prior art. Next, the ordered scaffolds are mapped to a line which is folded into a higher dimensional (here: 2D) space. The intricately folded line is referred to as a pseudo-Hilbert-Curve. The embedding of a compound happens by locating its most similar reference scaffold in the pseudo-Hilbert-Curve and assuming the respective position. Through a series of experiments, we demonstrate the properties of the maps generated by the HCASE method. Subjects of embeddings were compounds of the DrugBank and CANVASS libraries, and the chemical spaces were defined by scaffolds extracted from the ChEMBL database.Scientific contributionThe novelty of HCASE method lies in generating robust and intuitive chemical space embeddings that are reflective of a medicinal chemist’s reasoning, and the precedential use of space filling (Hilbert) curve in the process.Availabilityhttps://github.com/ncats/hcaseGraphical