Fundamental Manifold Research Articles

Inadequacy of Linear Methods for Minimal Sensor Placement and Feature Selection in Nonlinear Systems: A New Approach Using Secants

Sensor placement and feature selection are critical steps in engineering, modeling, and data science that share a common mathematical theme: the selected measurements should enable solution of an inverse problem. Most real-world systems of interest are nonlinear, yet the majority of available techniques for feature selection and sensor placement rely on assumptions of linearity or simple statistical models. We show that when these assumptions are violated, standard techniques can lead to costly over-sensing without guaranteeing that the desired information can be recovered from the measurements. In order to remedy these problems, we introduce a novel data-driven approach for sensor placement and feature selection for a general type of nonlinear inverse problem based on the information contained in secant vectors between data points. Using the secant-based approach, we develop three efficient greedy algorithms that each provide different types of robust, near-minimal reconstruction guarantees. We demonstrate them on two problems where linear techniques consistently fail: sensor placement to reconstruct a fluid flow formed by a complicated shock–mixing layer interaction and selecting fundamental manifold learning coordinates on a torus.

The Nature of Ultra-diffuse Galaxies in Distant Massive Galaxy Clusters: A370 in the Hubble Frontier Fields

Abstract We report the discovery of ultra-diffuse galaxies (UDGs) in Abell 370 (A370; z = 0.375). We find 46 UDGs in A370 from the images of the Hubble Frontier Fields. Most UDGs are low-luminosity red sequence galaxies, while a few of them are blue UDGs. We estimate the abundance of UDGs in A370, N(UDG) = 644 ± 104. Combining these results with those of Abell S1063 (z = 0.348) and Abell 2744 (z = 0.308), we derive a mean radial number density profile of UDGs in the three clusters. The number density profiles of UDGs and bright galaxies show a discrepancy in the central region of the clusters: the profile of UDGs shows a flattening as clustercentric distance decreases, while that of bright galaxies shows a continuous increase. This implies that UDGs are prone to disruption in the central region of the clusters. The relation between the abundance of UDGs and virial masses of their host systems is described by a power law with an index of nearly one: for M 200 > 1013 M ⊙. We estimate approximately dynamical masses of UDGs using the fundamental manifold method and find that most UDGs have dwarf-like masses (M 200 < 1011 M ⊙). This implies that most UDGs have a dwarf-like origin and a small number of them could be failed L* galaxies. These results suggest that multiple origins may contribute to the formation and evolution of UDGs in massive galaxy clusters.

Two-Wavelet Multipliers on the Dual of the Laguerre Hypergroup and Applications

In this paper, we are interested in the Laguerre hypergroup $$\mathbb {K} = [0,\infty )\times {\mathbb {R}}$$ which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup $$\widehat{\mathbb {K}}$$ which can be topologically identified with the so-called Heisenberg fan, the subset of $${\mathbb {R}}^{2}$$ $$\begin{aligned} \cup _{j\in {\mathbb {N}}}\left\{ (\lambda ,\mu )\in {\mathbb {R}}^{2}:\mu =|\lambda |(2j+\alpha +1), \; \lambda \ne 0\right\} \cup \left\{ (0,\mu )\in {\mathbb {R}}^{2}:\mu \ge 0\right\} , \end{aligned}$$ by means of which the notion of a generalized two-wavelet multiplier is investigated. The boundedness and compactness of the generalized two-wavelet multipliers are studied on $$L^{p}_{\alpha }(\mathbb {K})$$ , $$1 \le p \le \infty $$ . Afterwards, we introduce the generalized Landau–Pollak–Slepian operator and we give its trace formula. We show that the generalized two-wavelet multiplier is unitary equivalent to a scalar multiple of the generalized Landau–Pollak–Slepian operator. As applications, we prove an uncertainty principle of Donoho–Stark type involving $$\varepsilon $$ -concentration of the generalized two-wavelet multiplier operators. Moreover, we study functions whose time–frequency content is concentrated in a region with finite measure in phase space using the phase space restriction operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators.

Are over-massive haloes of ultra-diffuse galaxies consistent with extended MOND?

A sample of Coma cluster ultra-diffuse galaxies (UDGs) are modelled in the context of Extended Modified Newtonian Dynamics (EMOND) with the aim to explain the large dark matter-like effect observed in these cluster galaxies. We first build a model of the Coma cluster in the context of EMOND using gas and galaxy mass profiles from the literature. Then assuming the dynamical mass of the UDGs satisfies the fundamental manifold of other ellipticals, and that the UDG stellar mass-to-light matches their colour, we can verify the EMOND formulation by comparing two predictions of the baryonic mass of UDGs. We find that EMOND can explain the UDG mass, within the expected modelling errors, if they lie on the fundamental manifold of ellipsoids, however, given that measurements show one UDG lying off the fundamental manifold, observations of more UDGs are needed to confirm this assumption.

Two weighted inequalities for fractional integrals on Laguerre hypergroup

ABSTRACTLet be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group, its homogeneous norm and Q its homogeneous dimension. In this paper we prove the two weighted inequality for fractional integrals on . The obtained result is an analog of the Heinig result [Heinig HP. Weighted norm inequalities for classes of operators. Indiana Univ Math J. 1984;33(4):573–582] for fractional integrals on Laguerre hypergroup. Furthermore, the Stein–Weiss inequality for is proved as an application of this result.

Stability and chaos in Kustaanheimo–Stiefel space induced by the Hopf fibration

The need for the extra dimension in Kustaanheimo-Stiefel (KS) regularization is explained by the topology of the Hopf fibration, which defines the geometry and structure of KS space. A trajectory in Cartesian space is represented by a four-dimensional manifold, called the fundamental manifold. Based on geometric and topological aspects classical concepts of stability are translated to KS language. The separation between manifolds of solutions generalizes the concept of Lyapunov stability. The dimension-raising nature of the fibration transforms fixed points, limit cycles, attractive sets, and Poincar\'e sections to higher-dimensional subspaces. From these concepts chaotic systems are studied. In strongly perturbed problems the numerical error can break the topological structure of KS space: points in a fiber are no longer transformed to the same point in Cartesian space. An observer in three dimensions will see orbits departing from the same initial conditions but diverging in time. This apparent randomness of the integration can only be understood in four dimensions. The concept of topological stability results in a simple method for estimating the time scale in which numerical simulations can be trusted. Ideally all trajectories departing from the same fiber should be KS transformed to a unique trajectory in three-dimensional space, because the fundamental manifold that they constitute is unique. By monitoring how trajectories departing from one fiber separate from the fundamental manifold a critical time, equivalent to the Lyapunov time, is estimated. These concepts are tested on N-body examples: the Pythagorean problem, and an example of field stars interacting with a binary.

Fractional maximal function on the dual of Laguerre hypergroup

In this paper, we are interested in the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator, generated by the dual of the Laguerre hypergroup which topologically can be identified with the so-called Heisenberg fan, the subset of : by means of which fractional maximal function is investigated also the necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator on the dual of Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces is obtained.

The fundamental manifold of spiral galaxies: ordered versus random motions and the morphology dependence of the Tully–Fisher relation

(Abridged) We investigate the morphology dependence of the Tully-Fisher relation, and the expansion of the relation into a three-dimensional manifold defined by luminosity, total circular velocity and a third dynamical parameter, to fully characterise spiral galaxies across all morphological types. We use a semi-analytic hierarchical model of galaxy evolution to build the theoretical TF relation. With this tool, we analyse a unique dataset of galaxies for which we cross-match luminosity with total circular velocity and central velocity dispersion. We provide a theoretical framework to calculate such measurable quantities from semi-analytic models. We establish the morphology dependence of the TF relation in both model and data. We analyse the dynamical properties of the model galaxies and determine that the parameter 'sigma/Vc', i.e. the ratio between random and total motions defined by velocity dispersion and circular velocity, accurately characterises the varying slope of the TF relation for different model galaxy types. We apply these dynamical cuts to the observed galaxies and find that such selection produces a differential slope of the TF relation. The TF slope in different ranges of 'sigma/Vc' is consistent with that for the photometric classification in Sa, Sb, Sc. We conclude that 'sigma/Vc' is a good parameter to classify galaxy type, and we argue that such classification based on dynamics more closely mirrors the physical properties of the observed galaxies, compared to visual classification. We also argue that dynamical classification is useful for samples where eye inspection is not reliable or impractical. We conclude that 'sigma/Vc' is a suitable parameter to characterise the hierarchical assembly history that determines the disk-to-bulge ratio, and to expand the TF relation into a three-dimensional manifold, defined by luminosity, circular velocity and 'sigma/Vc'.

Maximal function on the dual of Laguerre hypergroup

In this paper, we are interested in the Laguerre hypergroup \(\mathbb{K} = [0,\infty ) \times \mathbb{R}\) which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup Open image in new window which can be topologically identified with the so-called Heisenberg fan, the subset of ℝ2: $$\bigcup\limits_{j \in \mathbb{N}} {\left\{ {(\lambda ,\mu ) \in \mathbb{R}^2 :\mu = \left| \lambda \right|(2j + \alpha + 1),\lambda \ne 0} \right\} \cup \left\{ {(0,\mu ) \in \mathbb{R}^2 :\mu \geqslant 0} \right\}} ,$$ by means of which the maximal function is investigated. For 1 < p ≤ ∞, the Lp(Open image in new window)-boundedness and weak L1(Open image in new window)-boundedness result for the maximal function is obtained.

TESTING DISTANCE ESTIMATORS WITH THE FUNDAMENTAL MANIFOLD

We demonstrate how the Fundamental Manifold (FM) can be used to cross-calibrate distance estimators even when those "standard candles" are not found in the same galaxy. Such an approach greatly increases the number of distance measurements that can be utilized to check for systematic distance errors and the types of estimators that can be compared. Here we compare distances obtained using SN Ia, Cepheids, surface brightness fluctuations, the luminosity of the tip of the red giant branch, circumnuclear masers, eclipsing binaries, RR Lyrae stars, and the planetary nebulae luminosity functions. We find no significant discrepancies (differences are < 2 sigma) between distance methods, although differences at the ~10% level cannot yet be ruled out. The potential exists for significant refinement because the data used here are heterogeneous B-band magnitudes that will soon be supplanted by homogeneous, near-IR magnitudes. We illustrate the use of FM distances to 1) revisit the question of the metallicity sensitivity of various estimators, confirming the dependence of SN Ia distances on host galaxy metallicity, and 2) provide an alternative calibration of H_0 that replaces the classical ladder approach in the use of extragalactic distance estimators with one that utilizes data over a wide range of distances simultaneously.

Pointwise and integral estimates for the fractional integrals on the Laguerre hypergroup

Let K = [0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper, some pointwise and integral estimates for the fractional integrals in terms of the maximal and fractional maximal functions on the Laguerre hypergroup are obtained. Basing on these results, we prove interpolation theorems for the fractional maximal functions and fractional integrals, and the Sobolev theorem on the Laguerre hypergroup. Mathematics subject classification (2010): Primary 42B20, 42B25, 42B35; Secondary 47G10, 47B37.

Bridging the gap between low- and high-mass dwarf galaxies

While the dark matter content within the most massive giant and smallest dwarf galaxies has been probed -- spanning a range of over one million in mass -- an important observational gap remains for galaxies of intermediate mass. This gap covers K band magnitudes of approximately -16 > M_K > -18 (for which dwarf galaxies have B--K ~ 2). On the high mass side of the gap are dwarf elliptical (dE) galaxies, that are dominated by stars in their inner regions. While the low mass side includes dwarf spheroidal (dSph) galaxies that are dark matter-dominated and ultra compact dwarf (UCD) objects that are star-dominated. Evolutionary pathways across the gap have been suggested but remain largely untested because the `gap' galaxies are faint, making dynamical measurements very challenging. With long exposures on the Keck telescope using the ESI instrument we have succeeded in bridging this gap by measuring the dynamical mass for five dwarf galaxies with M_K ~ -17.5 (M_B ~ --15.5). With the exception of our brightest dwarf galaxy, they possess relatively flat velocity dispersion profiles of around 20 km/s. By examining their 2D scaling relations and 3D fundamental manifold, we found that the sizes and velocity dispersions of these gap galaxies reveal continuous trends from dE to dSph galaxies. We conclude that low-luminosity dwarf elliptical galaxies are dominated by stars, not by dark matter, within their half light radii. This finding can be understood if internal feedback processes are operating most efficiently in gap galaxies, gravitationally heating the centrally-located dark matter to larger radii. Whereas external environmental processes, which can strip away stars, have a greater influence on dSph galaxies resulting in their higher dark matter fractions. Abridged.

STAR CLUSTERS, GALAXIES, AND THE FUNDAMENTAL MANIFOLD

We explore whether global observed properties, specifically half-light radii, mean surface brightness, and integrated stellar kinematics, suffice to unambiguously differentiate galaxies from star clusters, which presumably formed differently and lack dark matter halos. We find that star clusters lie on the galaxy scaling relationship referred to as the Fundamental Manifold (FM), on the extension of a sequence of compact galaxies, and so conclude that there is no simple way to differentiate star clusters from ultra-compact galaxies. By extending the validity of the FM over a larger range of parameter space and a wider set of objects, we demonstrate that the physics that constrains the resulting baryon and dark matter distributions in stellar systems is more general than previously appreciated. The generality of the FM implies 1) that the stellar spatial distribution and kinematics of one type of stellar system do not arise solely from a process particular to that set of systems, such as violent relaxation for elliptical galaxies, but are instead the result of an interplay of all processes responsible for the generic settling of baryons in gravitational potential wells, 2) that the physics of how baryons settle is independent of whether the system is embedded within a dark matter halo, and 3) that peculiar initial conditions at formation or stochastic events during evolution do not ultimately disturb the overall regularity of baryonic settling. We also utilize the relatively simple nature of star clusters to relate deviations from the FM to the age of the stellar population and find that stellar population models systematically and significantly over predict the mass-to-light ratios of old, metal-rich clusters.

Multiresolution Analysis and Haar Wavelets on the Laguerre Hypergroup

Let ℍn be the Heisenberg group. The fundamental manifold of the radial function space for ℍn can be denoted by [0, +∞) × ℝ, which is just the Laguerre hypergroup. In this paper the multiresolution analysis on the Laguerre hypergroup 𝕂 = [0, +∞) × ℝ is defined. Moreover the properties of Haar wavelet bases for are investigated.

(L p , L q ) boundedness of the fractional maximal operator on the Laguerre hypergroup

Let 𝕂=[0, ∞)×ℝ be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator on the Laguerre hypergroup from the spaces L p (𝕂) to the spaces L q (𝕂) and from the spaces L 1(𝕂) to the weak spaces WL q (𝕂).

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let K = [ 0 , ∞ ) × R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces L p ( K ) to the spaces L q ( K ) and from the spaces L 1 ( K ) to the weak spaces W L q ( K ) .

Inversion of the Radon transform on the product Laguerre hypergroup by using generalized wavelets

Let H 1 be the three-dimensional Heisenberg group. The fundamental manifold of the radial function space for H 1 can be denoted by [0, ∞)×R, which is just the Laguerre hypergroup. Naturally, K n =[0, ∞) n ×R n is the product Laguerre hypergroup. In this paper, we give the theory of continuous wavelet analysis and the Radon transform on K n , and devise a subspace 𝒮ℛ(K n ) of 𝒮(K n ) (Schwartz space) on which the Radon transform is a bijection. Also, we give two equivalent characterizations on 𝒮(K n ) for the Radon transform. By using the inverse wavelet transform we establish an inversion formula of the Radon transform on K n in the weak sense.

Low-threshold and continuously tunable Yb:Gd2SiO5 laser

Yb : Gd 2 Si O 5 (Yb:GSO) exhibits a large fundamental manifold splitting. Its long-wavelength emission band around 1088nm, which has the largest emission cross section, encounters the lowest reabsorption losses caused by thermal population of the terminal laser level. As a result, low-threshold and tunable continuous-wave Yb:GSO lasers were demonstrated. A slope efficiency up to 86% and a pumping threshold as low as 127mW were achieved for a continuous-wave Yb:GSO laser at 1092.5nm under the pump of a high-brightness laser diode. A continuous tunability between 1000 and 1120nm was realized with an SF14 prism as the intracavity tuning element.

Local Group Dwarf Galaxies and the Fundamental Manifold of Spheroids

The fundamental manifold (FM), an extension of the fundamental plane formalism, incorporates all spheroid-dominated stellar systems from dwarf ellipticals up to the intracluster stellar populations of galaxy clusters by accounting for the continuous variation of the mass-to-light ratio within the effective radius r_e with scale. Here, we find that Local Group dwarf spheroidal and dwarf elliptical galaxies, which probe the FM relationship roughly one decade lower in r_e than previous work, lie on the extrapolation of the FM. When combined with the earlier data, these Local Group dwarfs demonstrate the validity of the empirical manifold over nearly four orders of magnitude in r_e. The continuity of the galaxy locus on the manifold and, more specifically, the overlap on the FM of dwarf ellipticals like M 32 and dwarf spheroidals like Leo II, implies that dwarf spheroidals belong to the same family of spheroids as their more massive counterparts. The only significant outliers are Ursa Minor and Draco. We explore whether the deviation of these two galaxies from the manifold reflects a breakdown in the coherence of the empirical relationship at low luminosities or rather the individual dynamical peculiarities of these two objects. We discuss some implications of our results for how the lowest mass galaxies form.

The Fundamental Manifold of Spheroids

We present a unifying empirical description of the structural and kinematic properties of all spheroids embedded in dark matter halos. We find that the intracluster stellar spheroidal components of galaxy clusters, which we call cluster spheroids (CSphs) and which are typically 100 times the size of normal elliptical galaxies, lie on a as tight as that defined by elliptical galaxies (rms in effective radius of ~0.07) but having a different slope. The slope, as measured by the coefficient of the log σ term, declines significantly and systematically between the fundamental planes of ellipticals, brightest cluster galaxies (BCGs), and CSphs. We attribute this decline primarily to a continuous change in Me/Le, the mass-to-light ratio within the effective radius re, with spheroid scale. The magnitude of the slope change requires that it arise principally from differences in the relative distributions of luminous and dark matter, rather than from stellar population differences such as in age and metallicity. By expressing the Me/Le term as a function of σ in the simple derivation of the fundamental plane and requiring the behavior of that term to mimic the observed nonlinear relationship between log Me/Le and log σ, we simultaneously fit a two-dimensional to the measured properties of dwarf elliptical and elliptical galaxies, BCGs, and CSphs. The combined data have an rms scatter in log re of 0.114 (0.099 for the combination of ellipticals, BCGs, and CSphs), which is modestly larger than each fundamental plane has alone, but which includes the scatter introduced by merging different studies done in different filters by different investigators. This manifold fits the structural and kinematic properties of spheroids that span a factor of 100 in σ and 1000 in re. While our mathematical form is neither unique nor derived from physical principles, the tightness of the fit leaves little room for improvement by other unification schemes over the range of observed spheroids.