Model Of Euclidean Space Research Articles

Inhomogeneous long-range percolation in the weak decay regime

We study a general class of percolation models in Euclidean space including long-range percolation, scale-free percolation, the weight-dependent random connection model and several other previously investigated models. Our focus is on the weak decay regime, in which inter-cluster long-range connection probabilities fall off polynomially with small exponent, and for which we establish several structural properties. Chief among them are the continuity of the bond percolation function and the transience of infinite clusters.

Autoregressive optimal transport models.

Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu & Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425-436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.

Extended mean-field approach for chimera states in random complex networks.

Identical oscillators in the chimera state exhibit a mixture of coherent and incoherent patterns simultaneously. Nonlocal interactions and phase lag are critical factors in forming a chimera state within the Kuramoto model in Euclidean space. Here, we investigate the contributions of nonlocal interactions and phase lag to the formation of the chimera state in random networks. By developing an extended mean-field approximation and using a numerical approach, we find that the emergence of a chimera state in the Erdös-Rényi network is due mainly to degree heterogeneity with nonzero phase lag. For a regularly random network, although all nodes have the same degree, we find that disordered connections may yield the chimera state in the presence of long-range interactions. Furthermore, we show a nontrivial dynamic state in which all the oscillators drift more slowly than a defined frequency due to connectivity disorder at large phase lags beyond the mean-field solutions.

Оптимизация в конечномерном евклидовом пространстве

In this paper, optimization models in Euclidean space are divided into four complexity classes. Ef-fective algorithms have been developed to solve the problems of the first two classes of complexity. These are the primal-dual interior-point methods. Discrete and combinatorial optimization problems of the third complexity class are recommended to be converted to the fourth complexity class with continuous change of variables. Effective algorithms have not been developed for problems of the third and fourth complexity classes, with the exception of a narrow class of problems that are unimodal. The general optimization problem is formulated as a minimum (maximum) objective function in the presence of constraints. The complexity of the problem depends on the structure of the objective function and its feasible region. If the functions that determine the optimization model are quadratic or polynomial, then semidefinite programming can be used to obtain estimates of so-lutions in such problems. Effective methods have been developed for semidefinite optimization problems. Sometimes it’s enough to develop an algorithm without building a mathematical model. We see such an example when sorting an array of numbers. Effective algorithms have been devel-oped to solve this problem. In the work for sorting problems, an optimization model is constructed, and it coincides with the model of the assignment problem. It follows from this that the sorting problem is unimodal. Effective algorithms have not been developed to solve multimodal problems. The paper proposes a simple and effective algorithm for the optimal allocation of resources in mul-tiprocessor systems. This problem is multimodal. In the general case, for solving multimodal prob-lems, a method of exact quadratic regularization is proposed. This method has proven its compara-tive effectiveness in solving many test problems of various dimensions. Keywords: Euclidean space, optimization, unimodal problems, multimodal problems, complexity classes, numerical methods.

Calculation for Multidimensional Topological Relations in 3D Cadastre Based on Geometric Algebra

Maintaining topological consistency is a crucial issue for 3D cadastral modeling as this helps to represent the cadastral boundary clearly and accurately. As a result, 3D cadastral data models are mainly built on the basis of topological models that allow topology to be expressed clearly. However, topological models in Euclidean space cannot directly represent objects’ geometric information. As geometric information is important in 3D spatial analysis, 3D cadastral data models based on topological models cannot realize topological calculation and analysis. Previous research has proved that geometric and topological information for cadastral objects can be integrated and represented by conformal geometric algebra (CGA) expressions. This paper aims to realize 3D topological analysis in the cadastral field using CGA’s advantages in geometric relations computation. A calculation framework is designed on the basis of the outer product to achieve the purpose of multidimensional unity for 3D cadastral topological analysis in this paper. A calculation framework of topological relations between a boundary point (or a boundary line) and a cadastral parcel is developed. A total of 13 types of topological relations between a boundary point and a cadastral parcel and 48 types of topological relations between a boundary line and a cadastral parcel are obtained. The study indicates that the advantages of CGA in multidimensional unified representation and calculation can be used to solve problems encountered by topological models in Euclidean space.

3D Topological Error Detection for Cadastral Parcels Based on Conformal Geometric Algebra

Topological models in Euclidean space are difficult for spatial analysis because of the lack of direct representation of geometric three-dimensional (3D) object information. A 3D cadastral data model in the form of boundary representation based on conformal geometric algebra (CGA) is proposed to realize the integrated representation of geometric and topological information based on previous research. On the basis of the 3D cadastral data model based on CGA and basic geometric operators, self-defined geometric algebraic operators are designed using the advantages of geometric algebra in spatial topological calculation. A computation framework for cadastral parcel topological error detection is put forward based on those self-defined geometric operators. A case study is designed to verify the feasibility of topological error detection methods proposed in this paper. This study is an expansion of research for a 3D cadastral data model based on CGA.

Face super-resolution via bilayer contextual representation

Learning-based face super-resolution (SR), an important application field in computational imaging technology, offers a paradigm to enhance the quality and resolution of low-resolution (LR) facial images with the prior from training samples. The core of face SR is to learn the complex relationship between LR and high-resolution (HR) face images. Although most of the existing SR methods have achieved good results, the nonlinear nature of contextual prior is often ignored by locally linear assumption. In order to explore the accurate mapping relationship between LR and HR images, we propose a novel face SR method using bilayer contextual representation. In the first layer, we use context-patch to enrich the prior of image representation and shrink contextual dictionary in solution domain to induce local linear representation model in Euclidean space, thereby improving a set of LR images. In the second layer, kernel function provides a nonlinear mapping model for these enhanced images in Hilbert space, which maintains the consistency of the underlying manifold structure to capture a more accurate mapping between LR and HR. Furthermore, contextual residual learning is used to boost the reconstruction performance. The experimental results on FEI, CAS-PEAL-R1 and LDHF face databases show that the proposed approach preserves more image details and effectively improves the quality of the reconstructed image comparing with some state-of-the-art face SR methods.

Global Solutions to Elliptic and Parabolic $${\Phi^4}$$ Φ 4 Models in Euclidean Space

We prove the existence of global solutions to singular SPDEs on $${\mathbb{R}^{\rm d}}$$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions d = 4, 5 and in the parabolic setting for d = 2, 3. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $${\Phi^{4}_d}$$ Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the $${\Phi^{4}_{d-2}}$$ Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism.

Visibility in the vacant set of the Brownian interlacements and the Brownian excursion process

We consider the Brownian interlacements model in Euclidean space, introduced by Sznitman (2013). We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the vacant set of the Brownian excursion process in the unit disc and show that it undergoes a phase transition regarding visibility to infinity as in Benjamini et al. (2009). Additionally, we determine the critical value and that there is no visibility to infinity at the critical intensity.

Pressure behavior of shale-gas flow in dual porous medium based on fractal theory

The shale gas reservoir is a complex subject with a multiscale nanopore and fracture system, and the gas flow mechanism indicates an evident difference from the conventional gas reservoir. We have introduced fractal theory to characterize the multiscale distribution of pores and fractures, and we have developed a single-phase radial flow model considering nonequilibrium adsorption to describe the flow characteristics in the shale gas reservoir. The numerical solution of the flow model in Euclidean space is obtained by inversing the analytical solution derived in Laplace space through the Stehfest numerical inversion method, and the log-log curve of the dimensionless bottom-hole pressure (BHP) and its derivative versus dimensionless time are analyzed. The log-log curve of the dimensionless BHP has two distinct straight-line segments: The unit slope line reflects early well-storage effect, and the straight line with slope [Formula: see text] reflects reservoir fractal characteristics. The slope of the straight line will become smaller with the increasing fractal dimension. The adsorption coefficient mainly affects the middle and late period of the log-log curves, and more shale gas will desorb from the matrix with the increasing adsorption coefficient. The wellbore storage coefficient has a significant negative correlation with dimensionless BHP especially at the early and transitional stages. The skin factor mainly affects the transition section; a smaller skin factor generally leads to the earlier appearance of the transition section. In addition, a smaller interporosity flow coefficient also results in an earlier transition stage appearance. The lower storativity ratio means a higher dimensionless BHP and an earlier appearance of the transition stage.

Solving Partial Least Squares Regression via Manifold Optimization Approaches.

Partial least squares regression (PLSR) has been a popular technique to explore the linear relationship between two data sets. However, all existing approaches often optimize a PLSR model in Euclidean space and take a successive strategy to calculate all the factors one by one for keeping the mutually orthogonal PLSR factors. Thus, a suboptimal solution is often generated. To overcome the shortcoming, this paper takes statistically inspired modification of PLSR (SIMPLSR) as a representative of PLSR, proposes a novel approach to transform SIMPLSR into optimization problems on Riemannian manifolds, and develops corresponding optimization algorithms. These algorithms can calculate all the PLSR factors simultaneously to avoid any suboptimal solutions. Moreover, we propose sparse SIMPLSR on Riemannian manifolds, which is simple and intuitive. A number of experiments on classification problems have demonstrated that the proposed models and algorithms can get lower classification error rates compared with other linear regression methods in Euclidean space. We have made the experimental code public at https://github.com/Haoran2014.

Model of Four-Dimensional Sub-Proton Euclidean Space with Real Time for Valence Quarks. Lagrangian Mechanics

The model of Euclidean space with imaginary time used in sub-hadron physics uses only part of it since this part is isomorphic to Minkowski space and has the velocity limit 0 ≤ ||vEi|| ≤ 1. The model of four-dimensional Euclidean space with real time (Е space), in which 0 ≤ ||vE|| ≤ ∞ is investigated. The vectors of this space have Е-invariants, equal or analogous to the invariants of Minkowski space. All relations between physical quantities in Е-space, after they are mapped into Minkowski space, satisfy the principles of SRT and are Lorentz-invariant, and the velocity of light corresponds to infinite velocity. Results obtained in the model are different from the physical laws in Minkowski space. Thus, from the model of the Lagrangian mechanics of quarks in a centrally symmetric attractive potential it follows that the energy-mass of a quark decreases with increase of the velocity and is equal to zero for v = ∞. This made it possible to establish the conditions of emission and absorption of gluons by quarks. The effect of emission of gluons by high-energy quarks was discovered experimentally significantly earlier. The model describes for the first time the dynamic coupling of the masses of constituent and current quarks and reveals new possibilities in the study of intrahardon space. The classical trajectory of the oscillation of quarks in protons is described.

Propagation of chaos, Wasserstein gradient flows and toric Kähler–Einstein metrics

Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general nonequilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasiconvex N-particle interaction energy. We show that a deterministic macroscopic evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at negative temperature, the corresponding limiting evolution equation yields a driftdiffusion equation, coupled to the Monge-Ampere operator, whose static solutions correspond to toric Kahler-Einstein metrics. This drift-diffusion equation is the gradient flow on the Wasserstein space of probability measures of the K-energy functional in Kahler geometry and it can be seen as a fully nonlinear version of various extensively studied dissipative evolution equations and conservation laws, including the Keller-Segel equation and Burger's equation. In a companion paper, applications to singular pair interactions in one dimension are given.

Towards Secure Approximate <inline-formula> <tex-math notation="LaTeX">${k}$ </tex-math> </inline-formula>-Nearest Neighbor Query Over Encrypted High-Dimensional Data

As many services have been subcontracted to the cloud, data in such services (i.e., management applications, signal processing, and so on) are stored on cloud in encrypted form for protecting user private information. In many emerging data and signal processing applications, approximate k-nearest neighbor (ANN) query is a prerequisite component for massive high-dimensional data (such as time series, biometric, signal, multimedia data, and so on). Unfortunately, existing secure ANN (SANN) methods encounter the limitation of dimensionality. To further resolve ANN query over high-dimensionally encrypted data, in this paper, we propose an effective SANN model in Euclidean space. In overview, a secure greedy partition method is carefully designed by applying locality sensitive hashing coding and an optimized linear order. Based on that, a novel partition-based SANN ( $\text {SANN}_{\text {P}}$ ) and a multi-division version $\text {mSANN}_{\text {P}}$ resolve SANN query by sequentially scanning a candidate set, which is produced by matching a cloak query with a map index. Our proposed solutions guarantee security and accuracy simultaneously, and reduce communication cost significantly. Meanwhile, the greedy partition method is proved to be indistinguishable secure under chosen-plaintext attack , which is the foundation of security for the proposed solutions. Through extensive experimental studies on four data sets, the proposed mechanisms outperform the state-of-the-art approaches and provide effective and tradeoff between result accuracy and communication cost.

Euclidean bridge to the relativistic constituent quark model

${\bf Background}$ Knowledge of nucleon structure is today ever more of a precision science, with heightened theoretical and experimental activity expected in coming years. At the same time, a persistent gap lingers between theoretical approaches grounded in Euclidean methods (e.g., lattice QCD, Dyson-Schwinger Equations [DSEs]) as opposed to traditional Minkowski field theories (such as light-front constituent quark models). ${\bf Purpose}$ Seeking to bridge these complementary worldviews, we explore the potential of a Euclidean constituent quark model (ECQM). This formalism enables us to study the gluonic dressing of the quark-level axial-vector vertex, which we undertake as a test of the framework. ${\bf Method}$ To access its indispensable elements with a minimum of inessential detail, we develop our ECQM using the simplified quark $+$ scalar diquark picture of the nucleon. We construct a hyperspherical formalism involving polynomial expansions of diquark propagators to marry our ECQM with the results of Bethe-Salpeter Equation (BSE) analyses, and constrain model parameters by fitting electromagnetic form factor data. ${\bf Results}$ From this formalism, we define and compute a new quantity --- the Euclidean density function (EDF) --- an object that characterizes the nucleon's various charge distributions as functions of the quark's Euclidean momentum. Applying this technology and incorporating information from BSE analyses, we find the dressing effect on the proton's axial-singlet charge to be small in magnitude and consistent with zero. ${\bf Conclusions}$ The scalar quark $+$ diquark ECQM is a step toward a realistic quark model in Euclidean space, and urges additional refinements. The small size we obtain for the impact of the dressed vertex on the axial-singlet charge suggests that models without this effect are on firm ground to neglect it.

Product Grassmann Manifold Representation and Its LRR Models

It is a challenging problem to cluster multi- and high-dimensional data with complex intrinsic properties and non-linear manifold structure. The recently proposed subspace clustering method, Low Rank Representation (LRR), shows attractive performance on data clustering, but it generally does with data in Euclidean spaces. In this paper, we intend to cluster complex high dimensional data with multiple varying factors. We propose a novel representation, namely Product Grassmann Manifold (PGM), to represent these data. Additionally, we discuss the geometry metric of the manifold and expand the conventional LRR model in Euclidean space onto PGM and thus construct a new LRR model. Several clustering experimental results show that the proposed method obtains superior accuracy compared with the clustering methods on manifolds or conventional Euclidean spaces.

Knot topology in QCD

We consider topological structure of classical vacuum solutions in quantum chromodynamics. Topologically non-equivalent vacuum configurations are classified by non-trivial second and third homotopy groups for coset of the color group SU(N) (N=2,3) under the action of maximal Abelian stability group. Starting with explicit vacuum knot configurations we study possible exact classical solutions. Exact analytic non-static knot solution in a simple CP1 model in Euclidean space–time has been obtained. We construct an ansatz based on knot and monopole topological vacuum structure for searching new solutions in SU(2) and SU(3) QCD. We show that singular knot-like solutions in QCD in Minkowski space–time can be naturally obtained from knot solitons in integrable CP1 models. A family of Skyrme type low energy effective theories of QCD admitting exact analytic solutions with non-vanishing Hopf charge is proposed.

Euclidean Geometric Objects in the Clifford Geometric Algebra of {Origin, 3-Space, Infinity}

This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and (useful) parametrizations of the 3D Euclidean object models are explicitly demonstrated in order to show how 3D Euclidean information on positions, orientations and radii can be extracted.

Spatial Statistics

When the distribution of X∈ℝp depends only on its distance to some θ0∈ℝp, we discuss results from Hössjer and Croux and Neeman and Chang on rank score statistics. Similar results from Neeman and Chang are also given when X and θ0 are constrained to lie on the sphere in ℝp. Results from Ko and Chang on M estimation for spatial models in Euclidean space and the sphere are also discussed. Finally we discuss a regression type model: the image registration problem. We have landmarks ui on one image and corresponding landmarks Vi on a second image. It is desired to bring the two images into closest coincidence through a translation, rotation and scale change. The techniques and principles of this paper are summarized through extensive discussion of an example in three-dimensional image registration and a comparison of the L1 and L2 registrations. Two principles are important when working with spatial statistics: (1) Assumptions, such as that the distribution of X depends only on its distance to θ0, introduce symmetries to spatial models which, if properly used, greatly simplify statistical calculations. These symmetries can be expressed in a more general setting by using the notion of statistical group models. (2) When working with a non-Euclidean parameter space Θ such as the sphere, techniques of elementary differential geometry can be used to minimize the distortions caused by using a coordinate system to reexpress Θ in Euclidean parameters.

Extended Gauge Theories in Euclidean Space with Higher Spin Fields

The extended Yang–Mills gauge theory in Euclidean space is a renormalizable (by power counting) gauge theory describing a local interacting theory of scalar, vector, and tensor gauge fields (with maximum spin 2). In this article we study the quantum aspects and various generalizations of this model in Euclidean space. In particular, the quantization of the pure gauge model in a common class of covariant gauges is performed. We generalize the pure gauge sector by including matter fermions in the adjoint representation of the gauge group and analyze its N=1 and N=2 supersymmetric extensions. We show that the maximum half-integer spin contained in these fermion fields in dimension 4 is 3/2. Moreover, we develop an extension of this theory which will include internal gauge symmetries and the coupling to bosonic matter fields. The spontaneous symmetry breaking of the extended gauge symmetry is also analyzed.