Nontrivial Metrics Research Articles

Approximation schemes for Min-Sum [formula omitted]-Clustering

We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G=(V,E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of O(log|V|) in polynomial time for general metrics and an approximation ratio 2+ϵ in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC.In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.

Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.

Assessment of Methods for the Real-Time Simulation of Electronic and Thermal Circuits

Time–domain simulation of electronic and thermal circuits is required by a large array of applications, such as the design and optimization of electric vehicle powertrain components. While efficient execution is always a desirable feature of simulation codes, in certain cases like System-in-the-Loop setups, real-time performance is demanded. Whether real-time code execution can be achieved or not in a particular case depends on a series of factors, which include the mathematical formulation of the equations that govern the system dynamics, the techniques used in code implementation, and the capabilities of the hardware architecture on which the simulation is run. In this work, we present an evaluation framework of numerical methods for the simulation of electronic and thermal circuits from the point of view of their ability to deliver real-time performance. The methods were compared using a set of nontrivial benchmark problems and relevant error metrics. The computational efficiency of the simulation codes was measured under different software and hardware environments, to determine the feasibility of using them in industrial applications with reduced computational power.

On systematic and GR effects on muon g − 2 experiments

We derive in full generality the equations that govern the time dependence of the energy ℰ of the decay electrons in a muon g − 2 experiment. We include both electromagnetic and gravitational effects and we estimate possible systematics on the measurements of a ≡ (g − 2)/2, whose experimental uncertainty will soon reach ∆a/a ≈ 10−7. In addition to the standard modulation of ℰ when the motion is orthogonal to a constant magnetic field B, with angular frequency ωa = ea|B|/m, we study effects due to: (1) a non constant muon γ factor, in presence of electric fields E, (2) a correction due to a component of the muon velocity along B (the “pitch correction”), (3) corrections to the precession rate due to E fields, (4) non-trivial spacetime metrics. Oscillations along the radial and vertical directions of the muon lead to oscillations in ℰ with a relative size of order 10−6, for the BNL g − 2 experiment. We then find a subleading effect in the “pitch” correction, leading to a frequency shift of ∆ωa/ωa≈ mathcal{O} (10−9) and subleading effects of about ∆ωa/ωa≈ few × mathcal{O} (10−8–10−9) due to E fields. Finally we show that GR effects are dominated by the Coriolis force, due to the Earth rotation with angular frequency ωT, leading to a correction of about ∆ωa/ωa≈ ωT/(γωa) ≈ mathcal{O} (10−12). A similar correction might be more appreciable for future electron g − 2 experiments, being of order ∆ωa/ωa,el≈ ωT/(ωa,el) ≈ 7 × 10−13, compared to the present experimental uncertainty, ∆ael/ael≈ 10−10, and forecasted to reach soon ∆ael/ael≈ 10−11.

An adaptable agent-based model for guiding multi-species Pacific salmon fisheries management within a SES framework

Informing fishery management decisions using coupled socio-ecological systems (CSES) models requires model construction that captures the systems interactions with high precision. Ecological uncertainty in fishery models is easily reduced using existing scientific literature, but social drivers are often poorly defined or understood. The lack of knowledge about fishermen behavior results in inaccurate models of questionable utility for fishery managers. We designed and constructed a high fidelity agent based model (ABM) using the socio-ecological framework that reduces social system uncertainty by capturing complex behaviors using data-driven bounded rationality and feedback. The resulting generalized ABM of CSES dynamics was instantiated to Pacific salmon fisheries at Kenai river in Upper Cook Inlet, Alaska. The data-driven model construction fuses multiple data-sets for classification of social and ecological fishery regimes into stochastic distributions; the agent behaviors were generalized by evolving parametrized equations using data-driven machine learning; multiple non-trivial metrics on multiple scales verified model's accuracy and predictive capacity. The verified model of CSES dynamics at the Kenai river revealed recent instability in the dipnet fishery coupled dynamics, historic instability in the drift gillnet fishery coupled dynamics due to a compensatory and aggressive fishing strategy, and in the future the model will be used for scenario-based studies to understand the outcomes of alternative management strategies.

Proper size of the visible Universe in FRW metrics with a constant spacetime curvature

In this paper, we continue to examine the fundamental basis for the Friedmann–Robertson–Walker (FRW) metric and its application to cosmology, specifically addressing the question: What is the proper size of the visible universe? There are several ways of answering the question of size, though often with an incomplete understanding of how far light has actually traveled in reaching us today from the most remote sources. The difficulty usually arises from an inconsistent use of the coordinates or an over-interpretation of the physical meaning of quantities such as the so-called proper distance R(t) = a(t)r, written in terms of the (unchanging) co-moving radius r and the universal expansion factor a(t). In this paper, we prove for the five non-trivial FRW metrics with a constant spacetime curvature that, when the expansion began from an initial singularity, the visible universe today has a proper size equal to Rh(t0/2), i.e., the gravitational horizon at half its current age. The exceptions are de Sitter and Lanczos, whose contents had pre-existing positions away from the origin. In so doing, we confirm earlier results showing the same phenomenon in a broad range of cosmologies, including ΛCDM, based on the numerical integration of null geodesic equations through an FRW metric.

CONSERVED QUANTITIES AND DUALITIES FOR PARTICLES IN CURVED SPACE-TIME

We consider particles moving in curved space-time and associated symmetries. We use a generalized Killing equation and search for solutions involving Killing tensors associated with space-time metric. With these tensors some conserved quantities are constructed and they are valid along particles geodesic. In the Hamiltonian formalism for these particles, the conserved quantities can generate a dual description of the metric. We construct nontrivial dual metrics and consider a kind of geometric duality that involves a completely different space-time. From these metrics we calculate geometric invariants and examine the singularity structure of the dual space-time.

Metrics with Galilean conformal isometry

The Galilean conformal algebra (GCA) arises in taking the nonrelativistic limit of the symmetries of a relativistic conformal field theory in any dimensions. It is known to be infinite dimensional in all spacetime dimensions. In particular, the 2d GCA emerges out of a scaling limit of linear combinations of two copies of the Virasoro algebra. In this paper, we find metrics in dimensions greater than 2 which realize the finite 2d GCA (the global part of the infinite algebra) as their isometry by systematically looking at a construction in terms of cosets of this finite algebra. We list all possible subalgebras consistent with some physical considerations motivated by earlier work in this direction and construct all possible higher-dimensional nondegenerate metrics. We briefly study the properties of the metrics obtained. In the standard one higher-dimensional ``holographic'' setting, we find that the only nondegenerate metric is Minkowskian. In four and five dimensions, we find families of nontrivial metrics with a rather exotic signature. A curious feature of these metrics is that all but one of them are Ricci-scalar flat.

Conformally Kähler base metrics for Einstein warped products

A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies ( − a / f ) ∇ d f + r = λ g , for a smooth nonconstant function f and constants λ and a > 0 . If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP 1 bundle whose base manifold admits a Kähler–Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form.

Complete set of inner products for a discrete PT-symmetric square-well Hamiltonian

A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.

Robustness of temporal logic specifications for continuous-time signals

In this paper, we consider the robust interpretation of Metric Temporal Logic (MTL) formulas over signals that take values in metric spaces. For such signals, which are generated by systems whose states are equipped with non-trivial metrics, for example continuous or hybrid, robustness is not only natural, but also a critical measure of system performance. Thus, we propose multi-valued semantics for MTL formulas, which capture not only the usual Boolean satisfiability of the formula, but also topological information regarding the distance, ε , from unsatisfiability. We prove that any other signal that remains ε -close to the initial one also satisfies the same MTL specification under the usual Boolean semantics. Finally, our framework is applied to the problem of testing formulas of two fragments of MTL, namely Metric Interval Temporal Logic (MITL) and closed Metric Temporal Logic (clMTL), over continuous-time signals using only discrete-time analysis. The motivating idea behind our approach is that if the continuous-time signal fulfills certain conditions and the discrete-time signal robustly satisfies the temporal logic specification, then the corresponding continuous-time signal should also satisfy the same temporal logic specification.

A local classification of a class of [formula omitted] metrics with constant flag curvature

We first compute Riemannian curvature and Ricci curvature of ( α , β ) metrics. Then we apply these formulae to discuss a special class ( α , β ) metrics F = α ( 1 + β α ) p ( | p | ⩾ 1 ) which have constant flag curvature. We obtain the sufficient and necessary conditions that F = ( α + β ) 2 α have constant flag curvature. Then we prove that such metrics must be locally projectively flat and complete their local classification. Using the same method we find a necessary condition that flag curvature of F = α 2 α + β is constant and proved that there are no non-trivial Matsumoto metrics. Furthermore, we give a negative answer whether there are non-trivial metrics F = α ( 1 + β α ) p ( | p | ⩾ 1 ) of constant flag curvature when β is closed.

On the biharmonicity of product maps

We introduce the warped product of maps defined between Riemannian warped product spaces and we give necessary and sufficient conditions for warped product maps to be (bi)harmonic. We obtain then some characterizations of nontrivial harmonic metrics and nonharmonic biharmonic metrics on warped product spaces.

Isospectral Pairs of Metrics on Balls, Spheres, and other Manifolds with Different Local Geometries

The first isospectral pairs of metrics are constructed on the most simple simply connected domains, namely, on balls and spheres. This long-standing problem, concerning the existence of such pairs, has been solved by a new method called Among the wide range of such pairs, the most striking examples are provided on the spheres S4k-1, where k > 3. One of these metrics is homogeneous (since it is the metric on the geodesic sphere of a 2-point homogeneous space), while the other is locally inhomogeneous. These examples demonstrate the surprising fact that no information about the isometries is encoded in the spectrum of the Laplacian acting on functions. In other words, the group of isometries, even the local homogeneity property, is lost to the nonaudible in the debate of audible versus nonaudible geometry. In this article we settle a long-standing and intensively studied problem of spectral geometry by constructing the first examples of nontrivial isospectral metrics on the simplest simply connected manifolds, namely, on balls and spheres. This result is achieved by a new isospectral construction technique called anticommutator technique. It provides a whole cornucopia of such examples, not just on balls and spheres, but on other types of manifolds too. The most striking examples are constructed on the geodesic spheres of the hyperbolic quaternionic 2-point homogeneous spaces. The induced metrics on these spheres are obviously homogeneous. While the metrics constructed isospectrally to these are locally inhomogeneous. This shows the weakness of the information about the local geometry encoded in the spectrum of Laplacian acting on functions.

Field theory in superfluid 3He: what are the lessons for particle physics, gravity, and high-temperature superconductivity?

There are several classes of homogeneous Fermi systems that are characterized by the topology of the energy spectrum of fermionic quasiparticles: (i) gapless systems with a Fermi surface, (ii) systems with a gap in their spectrum, (iii) gapless systems with topologically stable point nodes (Fermi points), and (iv) gapless systems with topologically unstable lines of nodes (Fermi lines). Superfluid 3He-A and electroweak vacuum belong to the universality class 3. The fermionic quasiparticles (particles) in this class are chiral: they are left-handed or right-handed. The collective bosonic modes of systems of class 3 are the effective gauge and gravitational fields. The great advantage of superfluid 3He-A is that we can perform experiments by using this condensed matter and thereby simulate many phenomena in high energy physics, including axial anomaly, baryoproduction, and magnetogenesis. 3He-A textures induce a nontrivial effective metrics of the space, where the free quasiparticles move along geodesics. With 3He-A one can simulate event horizons, Hawking radiation, rotating vacuum, etc. High-temperature superconductors are believed to belong to class 4. They have gapless fermionic quasiparticles with a "relativistic" spectrum close to gap nodes, which allows application of ideas developed for superfluid 3He-A.

Non-trivial topologies in quantum gravity

This paper examines recent objections to the proposal that topologically non-trivial metrics could cause pure quantum states to decay into mixed states. It is shown that the Kaluza-Klein examples proposed by Gross in which this did not happen are special cases and that there are other Kaluza-Klein metrics in which it does. The objections of Banks, Peskin and Susskind about energy and momentum conservation arise because they assume that the evolution of density operator can be localized to a few Planck lengths. However, it is shown that energy and momentum are conserved only because of the asymptotic field equations and that this requires a large asymptotic region.

Gravitational calorons

Finite temperature solutions to euclidean gravity (i.e. with ++++ signature) are constructed on the analogy of the “caloron” solutions of the Yang-Mills field. In contrast to the known finite temperature metrics, the imaginary time coordinate (τ) is not a cyclic coordinate here. The metric components are periodic functions of τ. The zero-temperature limit of these solutions gives non-trivial metrics including in particular gravitational instantons corresponding to multicenter metrics of Taub-NUT type. Finite temperature solutions with non self-dual curvature are also found.