Arbitrary Number Of Dimensions Research Articles

A method to simulate multivariate outliers with known mahalanobis distances for normal and non-normal data

Monte Carlo simulations and theoretical analyses have repeatedly demonstrated the impact of outliers on statistical analysis. Most simulation studies generate outliers using one of two general approaches: by multiplying an arbitrary point by a constant or through a finite mixture. The latter can be extended to multivariate settings by defining the Mahalanobis distance between the centroids of two clusters of points. Nevertheless, when researchers aim to simulate individual data points with population-level Mahalanobis distances, the number of available procedures is very limited. This article generalizes one of the few existing methods to simulate an arbitrary number of outliers in an arbitrary number of dimensions, for both multivariate normal and non-normal data. A small simulation demonstration showcases how this methodology enables new simulation designs that were either unpopular or not possible due to the lack of a data-generating algorithm. A discussion of potential implications highlights the importance of considering multivariate outliers in simulation settings.

Normalized weighted cross correlation for multi-channel image registration

The normalized cross-correlation (NCC) is widely used for image registration due to its simple geometrical interpretation and being feature-agnostic. Here, after reviewing NCC definitions for images with an arbitrary number of dimensions and channels, we propose a generalization in which each pixel value of each channel can be individually weighted using real non-negative numbers. This generalized normalized weighted cross-correlation (NWCC) and its zero-mean equivalent (ZNWCC) can be used, for example, to prioritize pixels based on signal-to-noise ratio. Like a previously defined NWCC with binary weights, the proposed generalizations enable the registration of uniformly, but not necessarily isotropically, sampled images with irregular boundaries and/or sparse sampling. All NCC definitions discussed here are provided with discrete Fourier transform (DFT) formulations for fast computation. Practical aspects of NCC computational implementation are briefly discussed, and a convenient function to calculate the overlap of uniformly, but not necessarily isotropically, sampled images with irregular boundaries and/or sparse sampling is introduced, together with its DFT formulation. Finally, examples illustrate the benefit of the proposed normalized cross-correlation functions.

OGRe: Optimal Grid Refinement Protocol for Accurate Free Energy Surfaces and Its Application in Proton Hopping in Zeolites and 2D COF Stacking.

While free energy surfaces are the crux of our understanding of many chemical and biological processes, their accuracy is generally unknown. Moreover, many developments to improve their accuracy are often complicated, limiting their general use. Luckily, several tools and guidelines are already in place to identify these shortcomings, but they are typically lacking in flexibility or fail to systematically determine how to improve the accuracy of the free energy calculation. To overcome these limitations, this work introduces OGRe, a Python package for optimal grid refinement in an arbitrary number of dimensions. OGRe is based on three metrics that gauge the confinement, consistency, and overlap of each simulation in a series of umbrella sampling (US) simulations, an enhanced sampling technique ubiquitously adopted to construct free energy surfaces for hindered processes. As these three metrics are fundamentally linked to the accuracy of the weighted histogram analysis method adopted to generate free energy surfaces from US simulations, they facilitate the systematic construction of accurate free energy profiles, where each metric is driven by a specific umbrella parameter. This allows for the derivation of a consistent and optimal collection of umbrellas for each simulation, largely independent of the initial values, thereby dramatically increasing the ease-of-use toward accurate free energy surfaces. As such, OGRe is particularly suited to determine complex free energy surfaces with large activation barriers and shallow minima, which underpin many physical and chemical transformations and hence to further our fundamental understanding of these processes.

Hydrodynamics, spin currents and torsion

We construct the canonical constitutive relations for a fluid description of a system with a spin current, valid in an arbitrary number of dimensions in the absence of parity breaking or time reversal breaking terms. Our study encompasses the hydrostatic partition function, the entropy current, Kubo formula, conformal invariance, and the effect of charge. At some stages of the computation we turn on a background torsion tensor which naturally couples to the spin current.

Slowly rotating and accelerating α′ -corrected black holes in four and higher dimensions

We consider the low-energy effective action of string theory at order $\alpha '$, including $R^2$-corrections to the Einstein-Hilbert gravitational action and non-trivial dilaton coupling. By means of a convenient field redefinition, we manage to express the theory in a frame that enables us to solve its field equations analytically and perturbatively in $\alpha ' $ for a static spherically symmetric ansatz in an arbitrary number of dimensions. The set of solutions we obtain is compatible with asymptotically flat geometries exhibiting a regular event horizon at which the dilaton is well-behaved. For the 4-dimensional case, we also derive the stationary black hole configuration at first order in $\alpha '$ and in the slowly rotating approximation. This yields string theory modifications to the Kerr geometry, including terms of the form $a$, $a^2$, $\alpha '$ and $a\alpha '$. In addition, we obtain the first $\alpha'$ correction to the C-metrics, which accommodates accelerating black holes. We work in the string frame and discuss the connection to the Einstein frame, for which rotating black holes have already been obtained in the literature.

A class of exact solutions of the Navier–Stokes equations in three and four dimensions

A few basic, intuitive, properties of the Navier–Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier–Stokes equation in a space having an arbitrary number of dimensions. We then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions all velocity components depend non-trivially on all coordinate directions.

Model-free environmental contours in higher dimensions

This paper presents a method for estimating environmental contours in an arbitrary number of dimensions. The method, referred to as Direct-IFORM, does not require a model for the joint distribution of the variables. It can therefore be applied in higher dimensions without degradation in performance. The method involves multiple univariate analyses under rotations of the axes to estimate return values in various directions, which are used to form a contour. The method accounts for serial correlation in the observations, which removes the positive bias that occurs when this is neglected. An efficient open-source code is provided for estimating Direct-IFORM contours. A four-dimensional example is presented for contours of wind speed, wave height, wave period and wind-wave misalignment direction. The application of the method to the design of offshore wind turbines is discussed.

Instantons: thick-wall approximation

We develop a new method for estimating the decay probability of the false vacuum via regularized instantons. Namely, we consider the case where the potential is either unbounded from below or the second minimum corresponding to the true vacuum has a depth exceeding the height of the potential barrier. In this case, the materialized bubbles dominating the vacuum decay naturally have a thick wall and the thin-wall approximation is not applicable. We prove that in such a case the main contribution to the action determining the decay probability comes from the part of the solution for which the potential term in the equation for instantons can be neglected compared to the friction term. We show that the developed approximation exactly reproduces the leading order results for the few known exactly solvable potentials. The proposed method is applied to generic scalar field potentials in an arbitrary number of dimensions.

Theory of Knowledge Based on the Idea of the Discursive Space

This paper discusses the theory of knowledge based on the idea of dynamical space. The goal of this effort is to comprehend the knowledge that remains beyond the human domain, e.g., of the artificial cognitive systems. This theory occurs in two versions, weak and strong. The weak version is limited to knowledge in which retention and articulation are performed through the discourse. The strong version is general and is not limited in any way. In the weak version, knowledge is represented by the trajectories of discourses in time, in a dynamical space called the discursive space, which has an arbitrary number of dimensions. Given space is used to represent a given part of knowledge. A manifold is introduced to represent knowledge with a wider scope (all knowledge). The strong version is an extrapolation of the weak version to cover all forms of knowledge, not necessarily human or manifesting in language. The use of dynamical space construction allows one to formalize knowledge as such. Such an effort requires us to initially consider knowledge as mainly a social and linguistic phenomenon, which also could be presented as a result of the evolution of the understanding of knowledge that took place in the 20th century.

Classical density functional theory in the canonical ensemble.

Classical density functional theory for finite temperatures is usually formulated in the grand-canonical ensemble where arbitrary variations of the local density are possible. However, in many cases the systems of interest are closed with respect to mass, e.g., canonical systems with fixed temperature and particle number. Although the tools of standard, grand-canonical density functional theory are often used in an ad hoc manner to study closed systems, their formulation directly in the canonical ensemble has so far not been known. In this work, the fundamental theorems underlying classical DFT are revisited and carefully compared in the two ensembles showing that there are only trivial formal differences. The practicality of DFT in the canonical ensemble is then illustrated by deriving the exact Helmholtz functional for several systems: the ideal gas, certain restricted geometries in arbitrary numbers of dimensions, and, finally, a system of two hard spheres in one dimension (hard rods) in a small cavity. Some remarkable similarities between the ensembles are apparent even for small systems with the latter showing strong echoes of the famous exact of result of Percus in the grand-canonical ensemble.

Complexity for Conformal Field Theories in General Dimensions.

We study circuit complexity for conformal field theory states in an arbitrary number of dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian conformal group. Different choices of distance functions can be understood in terms of the geometry of coadjoint orbits of the conformal group. We explicitly relate our circuits to timelike geodesics in anti-de Sitter space and the complexity metric to distances between these geodesics. We extend our method to circuits in other symmetry groups using a group theoretic generalization of the notion of coherent states.

Strongly coupled heavy and light quark thermal motion from AdS/CFT

We give a pedagogical presentation of heavy and light probe quarks in a thermal plasma using the AdS/CFT correspondence. Three cases are considered: external heavy and light quarks undergoing Brownian motion in the plasma, and an external heavy quark moving through the plasma with a constant velocity. For the first two cases we compute the mean-squared transverse displacement of the string’s boundary endpoint. At early times the behaviour is ballistic, while the late time dynamics are diffusive. We extract the diffusion coefficient for both heavy and light quarks in an arbitrary number of dimensions and comment on the relevance for relativistic heavy ion phenomenology.

Ambiguities in the local thermal behavior of the scalar radiation in one-dimensional boxes

This paper reports certain ambiguities in the calculation of the ensemble average $\left<T_\mu{}_\nu\right>$ of the stress-energy-momentum tensor of an arbitrarily coupled massless scalar field in one-dimensional boxes in flat spacetime. The study addresses a box with periodic boundary condition (a circle) and boxes with reflecting edges (with Dirichlet's or Neumann's boundary conditions at the endpoints). The expressions for $\left<T^\mu{}^\nu\right>$ are obtained from finite-temperature Green functions. In an appendix, in order to control divergences typical of two dimensions, these Green functions are calculated for related backgrounds with arbitrary number of dimensions and for scalar fields of arbitrary mass, and specialized in the text to two dimensions and for massless fields. The ambiguities arise due to the presence in $\left<T^\mu{}^\nu\right>$ of double series that are not absolutely convergent. The order in which the two associated summations are evaluated matters, leading to two different thermodynamics for each type of box. In the case of a circle, it is shown that the ambiguity corresponds to the classic controversy in the literature whether or not zero mode contributions should be taken into account in computations of partition functions. In the case of boxes with reflecting edges, it results that one of the thermodynamics corresponds to a total energy (obtained by integrating the non homogeneous energy density over space) that does not depend on the curvature coupling parameter $\xi$ as expected; whereas the other thermodynamics curiously corresponds to a total energy that does depend on $\xi$. Thermodynamic requirements (such as local and global stability) and their restrictions to the values of $\xi$ are considered.

Conflict between non-exclusive groups

How is group conflict over a public-good prize affected when individuals are active members of multiple groups simultaneously? To investigate this question, I introduce a simple model of Tullock group contest: individuals are partitioned into two groups in two dimensions each, the group-level impact function is concave, and effort cost convex. The additional partition dimension does not alter the level of aggregate and individual effort. Asymmetries in group size and effort cost have non-monotonic effects on individual efforts and expected utilities. The formation of an active additional group is beneficial for its members and detrimental to outsiders. I generalise the model to an arbitrary number of symmetric groups in an arbitrary number of dimensions and briefly investigate the full positive range of the Tullock exponent. Additional group dimensions do not alter expected utilities of existing equilibria.

PySpawn: Software for Nonadiabatic Quantum Molecular Dynamics.

The ab initio multiple spawning (AIMS) method enables nonadiabatic quantum molecular dynamics simulations in an arbitrary number of dimensions, with potential energy surfaces provided by electronic structure calculations performed on-the-fly. However, the intricacy of the AIMS algorithm complicates software development, deployment on modern shared computer resources, and postsimulation data analysis. PySpawn is a nonadiabatic molecular dynamics software package that addresses these issues. The program is designed to be easily interfaced with electronic structure software, and an interface to the TeraChem software package is described here. PySpawn introduces a task-based reorganization of the AIMS algorithm, allowing fine-grained restart capability and setting the stage for efficient parallelization in a future release. PySpawn includes a user-friendly and interactive Python analysis module that will enable novice users to painlessly adopt AIMS. As a demonstration of PySpawn's simulation capability and analysis module, we report complete active space self-consistent field-based AIMS simulations of the 1,2-dithienyl-1,2-dicyanoethene molecule, a promising molecular photoswitch.

On the Galilean Duffin–Kemmer–Petiau equation in arbitrary dimensions

We obtain the representations of the Galilean covariant Duffin–Kemmer–Petiau equation in an arbitrary number of dimensions. Their purpose is to facilitate the study of nonrelativistic many-body systems with spinless and spin-one fields. A Galilean covariant formalism exploits the tensor structure of relativistic Lorentz-invariant theories by adding one extra spacelike coordinate and working with light-cone coordinates.

Diffusive Shock Acceleration in N Dimensions

Abstract Collisionless shocks are often studied in two spatial dimensions (2D) to gain insights into the 3D case. We analyze diffusive shock acceleration for an arbitrary number of dimensions. For a nonrelativistic shock of compression ratio , the spectral index of the accelerated particles is this curiously yields, for any N, the familiar (i.e., equal energy per logarithmic particle energy bin) for a strong shock in a monatomic gas. A precise relation between and the anisotropy along an arbitrary relativistic shock is derived and is used to obtain an analytic expression for in the case of isotropic angular diffusion, affirming an analogous result in 3D. In particular, this approach yields in the ultrarelativistic shock limit for N = 2, and for any strong shock. The angular eigenfunctions of the isotropic-diffusion transport equation reduce in 2D to elliptic cosine functions, providing a rigorous solution to the problem; the first function upstream already yields a remarkably accurate approximation. We show how these and additional results can be used to promote the study of shocks in 3D.

Efficient method to create superoscillations with generic target behavior

We introduce a new numerically stable method for constructing superoscillatory wave forms in an arbitrary number of dimensions. The method allows the construction of superoscillatory square-integrable functions that match any desired smooth behavior in their superoscillatory region to arbitrary accuracy.

Mirrors and field sources in a Lorentz-violating scalar field theory

In this paper we consider classical effects in a model for a scalar field incorporating Lorentz symmetry breaking due to the presence of a single background vector vμ coupled to its derivative. We investigate of the interaction energy between stationary steady sources concentrated along parallel branes with an arbitrary number of dimensions, and derive from this study some physical consequences. For the case of the scalar dipole we show the emergence of a nontrivial torque, which is a distinctive sign of the Lorentz violation. We also investigate a similar model in the presence of a semi-transparent mirror. For a general relative orientation between the mirror and the vμ, we are able to perform calculations perturbatively in vμ up to second order, and we also present exact results specific cases. For all these configurations, the propagator for the scalar field and the interaction force between the mirror and a point-like field source are computed. It is shown that the image method is valid in our model for the Dirichlet's boundary condition, and we argue that this is a non-trivial result. We also show the emergence of a torque on the mirror depending on its orientation with respect to the Lorentz violating background: this is a new effect with no counterpart in theories with Lorentz symmetry in the presence of mirrors.

Direct Ellipsoidal Fitting of Discrete Multi-Dimensional Data

Multi-dimensional distributions of discrete data that resemble ellipsoids arise in numerous areas of science, statistics, and computational geometry. We describe a complete algebraic algorithm to determine the quadratic form specifying the equation of ellipsoid for the boundary of such multi-dimensional discrete distribution. In this approach, the equation of an ellipsoid is reconstructed using a set of matrix equations from low-dimensional projections of the input data. We provide a Mathematica program realizing the full implementation of the ellipsoid reconstruction algorithm in an arbitrary number of dimensions. To demonstrate its many potential uses, the direct reconstruction method is applied to quasi-Gaussian statistical distributions arising in elementary particle production at the Large Hadron Collider.