Set Of Finite Measure Research Articles

Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion

In this paper, we study the heat content for the Schrödinger operator related to a subordinate Brownian motion and we also establish its small time asymptotic behavior for suitable potentials V. The case V=c1Ω for c>0 and Ω a Borel set of finite measure is investigated in detail.

Uncertainty principles for the continuous wavelet transform associated with a Bessel type operator on the half line

This paper presents uncertainty principles pertaining to generalized wavelet transforms associated with a second-order differential operator on the half line, extending the concept of the Bessel operator. Specifically, we derive a Heisenberg-Pauli-Weyl type uncertainty principle, as well as other uncertainty relations involving sets of finite measure

Time-frequency analysis of (k,a)-generalized wavelet transform and applications

The (k, a)-generalized wavelet transform is a novel addition to the class of wavelet transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short period of time. Since the study of time-frequency analysis is both theoretically interesting and practically useful, in this article, we investigated several subjects of time-frequency analysis for the (k, a)-generalized wavelet transform. First, we analyze the concentration of this transform on sets of finite measure. In particular, we prove Donoho–Stark and Benedicks-type uncertainty principles. We prove several versions of Heisenberg-type uncertainty principles for this transformation. Furthermore, involving the reproducing kernel and spectral theories, we investigate the time frequency and study the scalogram for the same wavelet transform. Finally, we provide Shapiro’s mean dispersion type theorems at the end.

The problem of determining multiple coefficients in an ultrahyperbolic equation

Abstract In this article, we discuss an inverse problem of determining unknown coefficients of the first-order derivatives in an ultrahyperbolic equation. By a finite set of measurements, we prove the uniqueness of solution of the problem in semi-geodesic coordinates under some conditions on the principal coefficients of the equation. Our main tool is a Carleman estimate.

Sound rating of catalog air handlers

Publishing sound power data for air handlers involves providing data across millions of variations related to operating conditions and unit configurations. AHRI 260 provides both the measurement and modeling guidelines used by most HVACR companies in the United States. However, many building designers are unfamiliar with the details of how a finite set of measurements taken in accordance with AHRI 260 becomes published sound rate data used to design quiet buildings. A case study based on a real product release is used to clearly define this process.

Constrained sensor placement and state reconstruction in power systems from partial system observations

Constrained sensor placement is a significant problem relevant to various applications in power system monitoring. This paper proposesa general data-driven technique for simultaneously observing (reconstructing) the dominant system oscillatory behavior from a limited number of measurements and optimally selecting sensor locations when several sensors are fixed or pre-assigned. First, a novel framework is introduced based on the proper orthogonal decomposition (POD) technique and a greedy sampling algorithm to extract a low-dimensional subspace from a finite set of measurements. Based on this framework, a fast algorithm is developed to solve the sensor optimization problem on a reduced basis. This approach improves the ability of the POD technique to capture the most relevant features of the observed dynamics and projects the measured data onto a subspace of a few optimal coordinates. The proposed method is robust to measurement noise and can be used to identify the significant temporal and spatial scales in the observed system response. These features make the approach of particular interest for control applications, prediction, compression, and visualization of system oscillatory behavior. Simulation studies on two test systems show the applicability and effectiveness of the proposed method.

Benedicks–Amrein–Berthier theorem for the Heisenberg motion group

It is well known that ( H n ⋊ U ( n ) , U ( n ) ) $(\mathbb {H}^n\rtimes U(n),U(n))$ is a Gelfand pair, an exact analogue of the Heisenberg group result due to Narayanan and Ratnakumar is not possible for the Heisenberg motion group. We prove that if an integrable function on the Heisenberg motion group is supported on a set of finite measure, and its Weyl transform is non-zero only for finitely many Fourier-Wigner pieces and have finite rank, then the function must be zero. In the end, a quantitative interpretation of this result is described through strong annihilating pair for the Weyl transform.

Statistical learning for fluid flows: Sparse Fourier divergence-free approximations

We reconstruct the velocity field of incompressible flows given a finite set of measurements. For the spatial approximation, we introduce the Sparse Fourier divergence-free approximation based on a discrete L2 projection. Within this physics-informed type of statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. We regularize our minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the incompressibility (divergence-free) constraint becomes a finite set of linear algebraic equations. We couple our spatial approximation with the truncated singular-value decomposition of the flow measurements for temporal compression. Our computational framework thus combines supervised and unsupervised learning techniques. We assess the capabilities of our method in various numerical examples arising in fluid mechanics.

Quantitative Uncertainty Principle for Sturm-Liouville Transform

In this paper we consider the Sturm-Liouville transform ℱ(f) on ℝ+. We analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given.

Spatially Discrete Control of Scalar Linear Distributed Plants of Parabolic and Hyperbolic Types

A spatially discrete control law is proposed for a class of systems described by scalar linear differential equations of parabolic and hyperbolic types with unknown parameters and disturbances. A finite set of discrete measurements (with respect to the spatial variable) of the plant state is available. The control law depends on a function that depends on the spatial variable and on a finite set of measurements of the plant state. Examples of this function, which allows realizing the control signal only at certain intervals in the spatial variable and providing lower control costs than some other analogs, are given. The exponential stability of the closed-loop system and robustness with respect to interval uncertain parameters of the plant and exogenous bounded disturbances are proved. Numerical modeling examples confirm the results of calculations and show the efficiency of the algorithm compared with some existing analogs.

Shapiro and local uncertainty principles for the multivariate continuous shearlet transform

ABSTRACT Quantitative Shapiro's dispersion uncertainty principle and umbrella theorem are proved for the multivariate continuous shearlet transform introduced earlier in Dahlke et al. [The continuous shearlet transform in arbitrary space dimensions. Preprint Nr. 2008-7, Philipps-Universität Marburg; 2008; The continuous shearlet transform in arbitrary space dimensions. J Fourier Anal Appl. 2010;16:340–364]. Also, we extend local uncertainty principles for a set of finite measure to .

Heisenberg type uncertainty principle for the Gabor spherical mean transform

We prove a Heisenberg type uncertainty principle for the Gabor spherical mean transform, and we study its generalization. Next, we extend local uncertainty principle for sets of finite measure to the latter transform.

Large deviations for intersection measures of some Markov processes

AbstractConsider an intersection measure of p independent (possibly different) m‐symmetric Hunt processes up to time t in a metric measure space E with a Radon measure m. We derive a Donsker–Varadhan type large deviation principle for the normalized intersection measure on the set of finite measures on E as , under the condition that t is smaller than life times of all processes. This extends earlier work by W. König and C. Mukherjee [16], in which the large deviation principle was established for the intersection measure of p independent N‐dimensional Brownian motions before exiting some bounded open set . We also obtain the asymptotic behavior of logarithmic moment generating function, which is related to the results of X. Chen and J. Rosen [7] on the intersection measure of independent Brownian motions or stable processes.Our results rely on assumptions about the heat kernels and the 1‐order resolvents of the processes, hence include rich examples. For example, the assumptions hold for with when the processes enjoy (sub‐)Gaussian type or jump type heat kernel estimates, where is determined by the Hausdorff dimension of E and the so‐called walk dimensions of the processes.

Convergence of metric two-level measure spaces

We extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space (X,r,ν) is a Polish metric space (X,r) equipped with a two-level measure ν∈Mf(Mf(X)), i.e. a finite measure on the set of finite measures on X. We introduce a topology on the set of (equivalence classes of) m2m spaces induced by certain test functions (i.e. the initial topology with respect to these test functions) and show that this topology is Polish by providing a complete metric. The framework introduced in this article is motivated by possible applications in biology. It is well suited for modeling the random evolution of the genealogy of a population in a hierarchical system with two levels, for example, host–parasite systems or populations which are divided into colonies. As an example we apply our theory to construct a random m2m space modeling the genealogy of a nested Kingman coalescent.

Towards continuous-to-continuous 3D imaging in the real world

Imaging systems are often modeled as continuous-to-discrete mappings that map the object (i.e. a function of continuous variables such as space, time, energy, wavelength, etc) to a finite set of measurements. When it comes to reconstruction, some discretized version of the object is almost always assumed, leading to a discrete-to-discrete representation of the imaging system. In this paper, we discuss a method for single-photon emission computed tomography (SPECT) imaging that avoids discrete representations of the object or the imaging system, thus allowing reconstruction on an arbitrarily fine set of points.

Quantifying the Impact of Linear Regression Model in Deriving Bio-Optical Relationships: The Implications on Ocean Carbon Estimations.

Linear regression is widely used in applied sciences and, in particular, in satellite optical oceanography, to relate dependent to independent variables. It is often adopted to establish empirical algorithms based on a finite set of measurements, which are later applied to observations on a larger scale from platforms such as autonomous profiling floats equipped with optical instruments (e.g., Biogeochemical Argo floats; BGC-Argo floats) and satellite ocean colour sensors (e.g., SeaWiFS, VIIRS, OLCI). However, different methods can be applied to a given pair of variables to determine the coefficients of the linear equation fitting the data, which are therefore not unique. In this work, we quantify the impact of the choice of “regression method” (i.e., either type-I or type-II) to derive bio-optical relationships, both from theoretical perspectives and by using specific examples. We have applied usual regression methods to an in situ data set of particulate organic carbon (POC), total chlorophyll-a (TChla), optical particulate backscattering coefficient (bbp), and 19 years of monthly TChla and bbp ocean colour data. Results of the regression analysis have been used to calculate phytoplankton carbon biomass (Cphyto) and POC from: i) BGC-Argo float observations; ii) oceanographic cruises, and iii) satellite data. These applications enable highlighting the differences in Cphyto and POC estimates relative to the choice of the method. An analysis of the statistical properties of the dataset and a detailed description of the hypothesis of the work drive the selection of the linear regression method.

Localization operators, time frequency concentration and quantitative-type uncertainty for the continuous wavelet transform associated with spherical mean operator

We consider the continuous wavelet transform [Formula: see text] associated with the spherical mean operator. We investigate the localization operators for [Formula: see text], in particular, we prove that they are in the Schatten-von Neumann class. Next, we analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given. Finally, we prove many versions of quantitative uncertainty principles for [Formula: see text].

Random triangles in planar regions

In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue measurable set of finite measure) contains a given fixed point O. These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point O. The formulae provide another way to approach the Sylvester’s four-point problem. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: \(\frac{2}{27}+20\frac{\ln 2}{81}\) (Halasz and Kleitman in Stud Appl Math 53:225–237, 1974; Prekopa in Period Math Hung 2:259–282, 1972). We compute this probability in the case of a regular polygon and its center of mass for the point O. Other families of regions are studied. For the family of Limacons \(r=a+\cos t, a>1\), and O the origin of the polar coordinates, the probability is \(\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3\pi ^2}\).

Continuous wavelet transform and uncertainty principle related to the Weinstein operator

ABSTRACTIn this paper, we define and study the continuous wavelet transform associated with the Weinstein operator. Next, we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg's inequality for wavelet transform. Last, we prove a quantitative version of Shapiro's mean dispersion theorem for the continuous wavelet transform.

New results on the continuous Weinstein wavelet transform

We consider the continuous wavelet transform mathcal{S}_{h}^{W} associated with the Weinstein operator. We introduce the notion of localization operators for mathcal {S}_{h}^{W}. In particular, we prove the boundedness and compactness of localization operators associated with the continuous wavelet transform. Next, we analyze the concentration of mathcal{S}_{h}^{W} on sets of finite measure. In particular, Benedicks-type and Donoho-Stark’s uncertainty principles are given. Finally, we prove many versions of Heisenberg-type uncertainty principles for mathcal{S}_{h}^{W}.