A complete metric in the function space D[0, ∞) and its application

1983 ◽  
pp. 60-76 ◽  
Author(s):  
V. V. Kalashnikov
Keyword(s):  
Function Space

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

A Function Space Approach to Spectral Related Problems

2019 ◽  
Vol 10 (6) ◽  
pp. 1220-1222
Author(s):  
T. Venkatesh ◽  
Karuna Samaje
Keyword(s):  
Function Space ◽  
Space Approach

Learning optimal kernel mapping based on function space with dynamic parameters

2013 ◽  
Vol 33 (8) ◽  
pp. 2337-2340
Author(s):  
Zhiying TAN ◽  
Ying CHEN ◽  
Yong FENG ◽  
Xiaobo SONG
Keyword(s):  
Function Space ◽  
Dynamic Parameters ◽  
Optimal Kernel ◽  
Kernel Mapping

ON THE STRUCTURE OF SOME FUNCTION SPACE

1984 ◽  
Vol 10 (1) ◽  
pp. 188
Author(s):  
Kostyrko ◽  
Salat
Keyword(s):  
Function Space

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

scholarly journals Regionalisierung hydrologischer Modelle mit Function Space Optimization

2021 ◽  
Author(s):  
Moritz Feigl ◽  
Mathew Herrnegger ◽  
Robert Schweppe ◽  
Stephan Thober ◽  
Daniel Klotz ◽  
...  
Keyword(s):  
Function Space ◽  
Hydrological Model ◽  
Problem Parameter ◽  
Sind Eine

ZusammenfassungDas Schätzen von räumlich verteilten Parametern hydrologischer Modelle ist ein bereits lang erforschtes und anspruchsvolles Problem. Parameter-Transferfunktionen, die einen funktionellen Zusammenhang zwischen Modellparametern und geophysikalischen Gebietseigenschaften herstellen, sind eine potenzielle Möglichkeit, Parameter ohne Kalibrierung zu schätzen. Function Space Optimization (FSO) ist eine symbolische Regressionsmethode, die automatisiert Transferfunktionen aus Daten schätzen kann. Sie basiert auf einem textgenerierenden neuronalen Netzwerk, das die Suche nach einer optimalen Funktion in ein kontinuierliches Optimierungsproblem umwandelt.In diesem Beitrag beschreiben wir die Funktionsweise von FSO und geben ein Beispiel der Anwendung mit dem mesoscale Hydrological Model (mHM). Ziel der Anwendung ist die Schätzung zweier Transferfunktionen für die Parameter KSat (gesättigte hydraulische Leitfähigkeit) und FieldCap (Feldkapazität). Dafür verwenden wir Daten 7 großer deutscher Einzugsgebieten über einen Zeitraum von 5 Jahren zum Schätzen der Transferfunktionen und weiterer numerischer Parameter. Die resultierenden Funktionen und Parameter werden ohne weitere Kalibrierung auf 222 Validierungsgebiete über eine Validierungsperiode von 35 Jahren angewendet. Mit der Anwendung in diesen „unbeobachteten“ Gebieten können wir die Übertragbarkeit und die zumindest regionale Gültigkeit der Transferfunktionen überprüfen.Die Ergebnisse zeigen, dass bei einer Anwendung in unbeobachteten Gebieten die Modellgüte in einem ähnlichen Wertebereich wie in den Trainingsgebieten liegt und somit weiterhin akzeptabel ist. Die Nash-Sutcliffe Efficiency (NSE) in den Trainingsgebieten über den Validierungszeitraum unterscheidet sich mit einem medianen Wert von 0,73 nicht nennenswert von dem der Validierungsgebiete mit einem medianen NSE von 0,65.Zusammengefasst haben Transferfunktionen das Potenzial, die Vorhersagefähigkeiten, Übertragbarkeit auf andere Gebiete sowie physikalische Interpretierbarkeit bestehender hydrologischer Modelle zu verbessern. Mit FSO wurde zum ersten Mal eine objektive, datengetriebene Methode entwickelt, mit der Transferfunktionen geschätzt werden können.

A SPHERICAL VERSION OF THE KOWALSKI–SŁODKOWSKI THEOREM AND ITS APPLICATIONS

2020 ◽  
pp. 1-26
Author(s):  
SHIHO OI
Keyword(s):  
Banach Algebra ◽  
Function Space ◽  
Affirmative Answer ◽  
List Type ◽  
Private Communication ◽  
Type Number ◽  
Uniform Algebras ◽  
Surjective Isometry ◽  
Complex Linear ◽  
Local Map

Abstract Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties: (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ ); (b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ . Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

scholarly journals Shuttle Ascent Trajectory Optimization with Function Space Quasi-Newton Techniques

AIAA Journal ◽  
1976 ◽  
Vol 14 (10) ◽  
pp. 1369-1376 ◽  
Author(s):  
Ernest R. Edge ◽  
William F. Powers
Keyword(s):  
Function Space ◽  
Trajectory Optimization ◽  
Quasi Newton

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (01) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

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