Arbitrary Metrics Research Articles

Density as an indicator of sustainable urban development: insights from Helsinki?

ABSTRACT Densification has become a dominant strategy for sustainable urban development in recent decades, viewed as a main antidote to urban sprawl and essential for achieving a range of urban qualities from economic to ecological. Despite its prominence as an indicator and driver of sustainable development, measuring density in a meaningful way poses significant challenges. Arbitrary metrics often overlook the complexity of density and its associations, leading to potential unintended outcomes in density-driven planning. Using the Helsinki region as a case study, this article demonstrates the sensitivity of density measurements to minor changes in reference areas and calculation units, highlighting how different types of density can loosely relate to each other. Our findings underscore the importance of precise methodological choices in avoiding misguided planning decisions and policy advice that could compromise sustainability goals, revealing the complex dynamics of infill and edge growth often obscured in regional density measures.

MEMPSEP‐I. Forecasting the Probability of Solar Energetic Particle Event Occurrence Using a Multivariate Ensemble of Convolutional Neural Networks

AbstractThe Sun continuously affects the interplanetary environment through a host of interconnected and dynamic physical processes. Solar flares, Coronal Mass Ejections (CMEs), and Solar Energetic Particles (SEPs) are among the key drivers of space weather in the near‐Earth environment and beyond. While some CMEs and flares are associated with intense SEPs, some show little to no SEP association. To date, robust long‐term (hours‐days) forecasting of SEP occurrence and associated properties (e.g., onset, peak intensities) does not effectively exist and the search for such development continues. Through an Operations‐2‐Research support, we developed a self‐contained model that utilizes a comprehensive data set and provides a probabilistic forecast for SEP event occurrence and its properties. The model is named Multivariate Ensemble of Models for Probabilistic Forecast of Solar Energetic Particles (MEMPSEP). MEMPSEP workhorse is an ensemble of Convolutional Neural Networks that ingests a comprehensive data set (MEMPSEP‐III by Moreland et al. (2024, https://doi.org/10.1029/2023SW003765)) of full‐disc magnetogram‐sequences and in situ data from different sources to forecast the occurrence (MEMPSEP‐I—this work) and properties (MEMPSEP‐II by Dayeh et al. (2024, https://doi.org/10.1029/2023SW003697)) of a SEP event. This work focuses on estimating true SEP occurrence probabilities achieving a 2.5% improvement in reliability and a Brier score of 0.14. The outcome provides flexibility for the end‐users to determine their own acceptable level of risk, rather than imposing a detection threshold that optimizes an arbitrary binary classification metric. Furthermore, the model‐ensemble, trained to utilize the large class‐imbalance between events and non‐events, provides a clear measure of uncertainty in our forecast.

MidRISH: Unbiased harmonization of rotationally invariant harmonics of the diffusion signal

Data harmonization is necessary for removing confounding effects in multi-site diffusion image analysis. One such harmonization method, LinearRISH, scales rotationally invariant spherical harmonic (RISH) features from one site (“target”) to the second (“reference”) to reduce confounding scanner effects. However, reference and target site designations are not arbitrary and resultant diffusion metrics (fractional anisotropy, mean diffusivity) are biased by this choice. In this work we propose MidRISH: rather than scaling reference RISH features to target RISH features, we project both sites to a mid-space. We validate MidRISH with the following experiments: harmonizing scanner differences from 37 matched patients free of cognitive impairment, and harmonizing acquisition and study differences on 117 matched patients free of cognitive impairment. We find that MidRISH reduces bias of reference selection while preserving harmonization efficacy of LinearRISH. Users should be cautious when performing LinearRISH harmonization. To select a reference site is to choose diffusion metric effect-size. Our proposed method eliminates the bias-inducing site selection step.

An evolutionary algorithm for the direct optimization of covariate balance between nonrandomized populations.

Matching reduces confounding bias in comparing the outcomes of nonrandomized patient populations by removing systematic differences between them. Under very basic assumptions, propensity score (PS) matching can be shown to eliminate bias entirely in estimating the average treatment effect on the treated. In practice, misspecification of the PS model leads to deviations from theory and matching quality is ultimately judged by the observed post-matching balance in baseline covariates. Since covariate balance is the ultimate arbiter of successful matching, we argue for an approach to matching in which the success criterion is explicitly specified and describe an evolutionary algorithm to directly optimize an arbitrary metric of covariate balance. We demonstrate the performance of the proposed method using a simulated dataset of 275,000 patients and 10 matching covariates. We further apply the method to match 250 patients from a recently completed clinical trial to a pool of more than 160,000 patients identified from electronic health records on 101 covariates. In all cases, we find that the proposed method outperforms PS matching as measured by the specified balance criterion. We additionally find that the evolutionary approach can perform comparably to another popular direct optimization technique based on linear integer programming, while having the additional advantage of supporting arbitrary balance metrics. We demonstrate how the chosen balance metric impacts the statistical properties of the resulting matched populations, emphasizing the potential impact of using nonlinear balance functions in constructing an external control arm. We release our implementation of the considered algorithms in Python.

A Dually Flat Embedding of Spacetime

A model of spacetime is presented. It has an extension to five dimensions, and in five dimensions the geometry is the dual of the Euclidean geometry w.r.t. an arbitrary positive-definite metric. Dually flat geometries are well-known in the context of information geometry. The present work explores their role in describing the geometry of spacetime. It is shown that the positive-definite metric with its flat 5-d connection can coexist with a pseudometric for which the connection is that of Levi-Civita. The 4-d geodesics are characterized by five conserved quantities, one of which can be chosen freely and is taken equal to zero in the present work. An explicit expression for the parallel transport operators is obtained. It is used to construct a pseudometric for spacetime by choosing an arbitrary possibly degenerate inner product in the tangent space of a reference point, for instance, that of Minkowski. By parallel transport, one obtains a pseudometric for spacetime, the metric connection of which extends to a 5-d connection with vanishing curvature tensor. The de Sitter space is considered as an example.

Blowups and blowdowns of geodesics in Carnot groups

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of Carnot geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. Then as a consequence we get that in every sub-Riemannian manifold any $s$ times iterated tangent of any geodesic is a line, where $s$ is the step of the sub-Riemannian manifold in question. With a similar approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result is also extended to rough geodesics.

Bisecting for Selecting: Using a Laplacian Eigenmaps Clustering Approach to Create the New European Football Super League

Ranking sports teams generally relies on supervised techniques, requiring either prior knowledge or arbitrary metrics. In this paper, we offer a purely unsupervised technique. We apply this to operational decision-making, specifically, the controversial European Super League for association football, demonstrating how this approach can select dominant teams to form the new league. We first use random forest regression to select important variables predicting goal difference, which we use to calculate the Euclidian distances between teams. Creating a Laplacian eigenmap, we bisect the Fiedler vector to identify the natural clusters in five major European football leagues. Our results show how an unsupervised approach could identify four clusters based on five basic performance metrics: shots, shots on target, shots conceded, possession, and pass success. The top two clusters identify teams that dominate their respective leagues and are the best candidates to create the most competitive elite super league.

Relative velocity in pseudo-Riemannian spacetime

We give a coordinate independent definition of relative velocity of test particle in pseudo-Riemannian spacetime as measured along the observer’s line-of-sight in general and several instructive cases. In doing this, the test particle is considered as a luminous object, otherwise, if it is not, we assume that a light source is attached to it, which has neither mass nor volume. Then we utilize the general solution of independent definition of relative velocity of a luminous source in generic pseudo-Riemannian spacetime. As a corollary, we discuss the implications for the Minkowski metric, the test particle and observer at rest in an arbitrary stationary metric, the uniform gravitational field, the rotating reference frame, the Schwarzschild metric, the Kerr-type metrics, and the spatially homogeneous and isotropic Robertson-Walker (RW) spacetime of standard cosmological model. In the last case, it leads to cosmological consequence that the resulting, so-called, kinetic recession velocity of an astronomical object is always subluminal even for large redshifts of order one or more, so that it does not violate the fundamental physical principle of causality. We also calculate the measure of carrying away of a galaxy at redshift z by the expansion of space, which proves, in particular, that cosmological expansion of a flat 3D–space is fundamentally different from a kinematics of galaxies moving in a non-expanding flat 3D-space. So, it is impossible to mimic the true cosmological redshift by a Doppler effect caused by motion of galaxies in a non-expanding 3D-space, flat or curved. We also give a reappraisal of the `standard ́ kinematic interpretation of redshifts in RW spacetime as accumulated Doppler-shifts.

Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tori

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on “curved” NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get Lp-versions of the usual formula for the trace of ΨDOs on NC tori, i.e., for combinations of ΨDOs with Lp-position operators. Next, we get Lp-versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a. Dixmier traces) of products of Lp-position operators with ΨDOs or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators associated with powers of Laplace-Beltrami operators and Lp-potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for our classes of operators on curved NC tori. This supersedes a previous conjecture of [63].

Gauge invariants of linearized gravity with a general background metric

In linearized gravity with distributed matter, the background metric has no generic symmetries, and decomposition of the metric perturbation into global normal modes is generally impractical. This complicates the identification of the gauge-invariant part of the perturbation, which is a concern, for example, in the theory of dispersive gravitational waves (GWs) whose energy–momentum must be gauge-invariant. Here, we propose how to identify the gauge-invariant part of the metric perturbation and the six independent gauge invariants per se for an arbitrary background metric. For the Minkowski background, the operator that projects the metric perturbation on the invariant subspace is proportional to the well-known dispersion operator of linear GWs in vacuum. For a general background, this operator is expressed in terms of the Green’s operator of the vacuum wave equation. If the background is smooth, it can be found asymptotically using the inverse scale of the background metric as a small parameter.

Higher-power harmonic maps and sections

The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for left-invariant vector fields on three-dimensional unimodular Lie groups equipped with an arbitrary left-invariant Riemannian metric.

On the Nonexpansive Operators Based on Arbitrary Metric: A Degenerate Analysis

We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate the connections between firm nonexpansiveness, cocoerciveness and averagedness. The convergence of the associated fixed-point iterations is discussed with particular focus on the case of degenerate metric, since the degeneracy is often encountered when reformulating many existing first-order operator splitting algorithms as a metric resolvent. This work paves a way for analyzing the generalized proximal point algorithm with a non-trivial relaxation step and degenerate metric.

The Pairwise Similarity Partitioning Algorithm: A Method for Unsupervised Partitioning of Geoscientific and Other Datasets Using Arbitrary Similarity Metrics

Abstract A simple yet flexible and robust algorithm is described for fully partitioning an arbitrary dataset into compact, nonoverlapping groups or classes, sorted by size, based entirely on a pairwise similarity matrix and a user-specified similarity threshold. Unlike many clustering algorithms, there is no assumption that natural clusters exist in the dataset, although clusters, when present, may be preferentially assigned to one or more classes. The method also does not require data objects to be compared within any coordinate system but rather permits the user to define pairwise similarity using almost any conceivable criterion. The method therefore lends itself to certain geoscientific applications for which conventional clustering methods are unsuited, including two nontrivial and distinctly different datasets presented as examples. In addition to identifying large classes containing numerous similar dataset members, it is also well suited for isolating rare or anomalous members of a dataset. The method is inductive in that prototypes identified in representative subset of a larger dataset can be used to classify the remainder.

Online k-taxi via Double Coverage and time-reverse primal-dual

We consider the online k-taxi problem, a generalization of the k-server problem, in which k servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger. We show that the classic Double Coverage algorithm has competitive ratio 2^k-1 on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive ratio turns out to be much better and we obtain tight bounds. When the depth is dll k, these bounds are approximately k^d/d!. By standard embedding results, we obtain a randomized algorithm for arbitrary n-point metrics with (polynomial) competitive ratio O(k^cDelta ^{1/c}log _{Delta } n), where Delta is the aspect ratio and cge 1 is an arbitrary positive integer constant. The previous known bound was O(2^klog n). For general (weighted) tree metrics, we prove the competitive ratio of Double Coverage to be Theta (k^d) for any fixed depth d, and in contrast to HSTs it is not bounded by 2^k-1. We obtain our results by a dual fitting analysis where the dual solution is constructed step-by-step backwards in time. Unlike the forward-time approach typical of online primal-dual analyses, this allows us to combine information from both the past and the future when assigning dual variables. We believe this method can also be useful for other problems. Using this technique, we also provide a dual fitting proof of the k-competitiveness of Double Coverage for the k-server problem on trees.

New asymptotically flat static vacuum metrics with near Euclidean boundary data

In our prior work toward Bartnik’s static vacuum extension conjecture for near Euclidean boundary data, we establish a sufficient condition, called static regular, and confirm that large classes of boundary hypersurfaces are static regular. In this paper, we further improve some of those prior results. Specifically, we show that any hypersurface in an open and dense subfamily of a certain general smooth one-sided family of hypersurfaces (not necessarily a foliation) is static regular. The proof uses some of our new arguments motivated from studying the conjecture for boundary data near an arbitrary static vacuum metric.

Asymptotic freedom and safety in quantum gravity

We compute non-perturbative flow equations for the couplings of quantum gravity in fourth order of a derivative expansion. The gauge invariant functional flow equation for arbitrary metrics allows us to extract β-functions for all couplings. In our truncation we find two fixed points. One corresponds to asymptotically free higher derivative gravity, the other is an extension of the asymptotically safe fixed point in the Einstein-Hilbert truncation or extensions thereof. The infrared limit of the flow equations entails only unobservably small modifications of Einstein gravity coupled to a scalar field. Quantum gravity can be asymptotically free, based on a flow trajectory from the corresponding ultraviolet fixed point to the infrared region. This flow can also be realized by a scaling solution for varying values of a scalar field. As an alternative possibility, quantum gravity can be realized by asymptotic safety at the other fixed point. There may exist a critical trajectory between the two fixed points, starting in the extreme ultraviolet from asymptotic freedom. We compute critical exponents and determine the number of relevant parameters for the two fixed points. Evaluating the flow equation for constant scalar fields yields the universal gravitational contribution to the effective potential for the scalars.

Beta functions of ( 3+1 )-dimensional projectable Hořava gravity

We derive the full set of beta functions for the marginal essential couplings of projectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity in ($3+1$)-dimensional spacetime. To this end we compute the divergent part of the one-loop effective action in static background with an arbitrary spatial metric. The computation is done in several steps: reduction of the problem to three dimensions, extraction of an operator square root from the spatial part of the fluctuation operator, and evaluation of its trace using the method of universal functional traces. This provides us with the renormalization of couplings in the potential part of the action which we combine with the results for the kinetic part obtained previously. The calculation uses symbolic computer algebra and is performed in four different gauges yielding identical results for the essential beta functions. We additionally check the calculation by evaluating the effective action on a special background with spherical spatial slices using an alternative method of spectral summation. We conclude with a preliminary discussion of the properties of the beta functions and the resulting renormalization group flow, identifying several candidate asymptotically free fixed points.

Ricci flow on manifolds with boundary with arbitrary initial metric

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

Customizable Route Planning in Road Networks (Extended Abstract)

Computing driving directions in road networks is a fundamental problem. Although it can be solved in essentially linear time by Dijkstra's algorithm, this is not fast enough to enable interactive queries on large-scale inputs. Instead, modern algorithms typically work in two stages: first an offline preprocessing routine computes some auxiliary data, which is then used to answer exact queries in real time. The past decade has seen a surprisingly diverse set of techniques that follow this approach, mostly relying on the fact that road networks tend to have a strong hierarchy. These methods work very well when minimizing driving times, but are much less efficient with other cost functions. We present a practical algorithm that has no such drawbacks, and can compute shortest paths on continental road networks with arbitrary metrics (cost functions). Our customizable route planning approach works in three stages. The first, metric-independent preprocessing, uses graph partitioning to define the topology of a multilevel overlay graph, which is the same regardless of the cost function. The second stage, customization, uses the metric to compute the actual costs of the overlay arcs. Finally, the query stage uses the output of the first two stages to compute shortest paths in real time (milliseconds). The first stage uses a recent partitioning algorithm based on the notion of natural cuts, which are sparse regions separating much denser areas. It may take a few minutes (or even hours), but only needs to be run (or updated) when new road segments are built. Metric changes (which are much more frequent) require running only customization, which takes a second or less even on continental road networks. Since it does not rely on strong hierarchies, CRP is robust to metric changes. Unlike most other methods, it can also handle turn costs (and restrictions) quite naturally, with little effect on performance and space usage. It is thus ideal for a real-world routing engine, and is indeed in use by Bing Maps. This extended abstract includes results first published at SEA 2011 and SEA 2013.

Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ⩽n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.