Unbiased Stereological Estimation of d-Dimensional Volume in ℝn from an Isotropic Random Slice Through a Fixed Point

1994 ◽  
Vol 26 (1) ◽  
pp. 1-12 ◽  
Author(s):  
E. B. Vedel Jensen ◽  
K. Kiêu
Keyword(s):  
Fixed Point ◽  
Explicit Formula ◽  
Fundamental Lemma ◽  
Spatial Grid ◽  
Isotropic Random

Unbiased stereological estimators of d-dimensional volume in ℝn are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝn through O hits a fixed point in ℝn is given.

Unbiased Stereological Estimation of d-Dimensional Volume in ℝn from an Isotropic Random Slice Through a Fixed Point

1994 ◽  
Vol 26 (01) ◽  
pp. 1-12
Author(s):  
E. B. Vedel Jensen ◽  
K. Kiêu
Keyword(s):  
Fixed Point ◽  
Explicit Formula ◽  
Fundamental Lemma ◽  
Spatial Grid ◽  
Isotropic Random

Unbiased stereological estimators of d-dimensional volume in ℝ n are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝ n through O hits a fixed point in ℝ n is given.

Variance prediction for pseudosystematic sampling on the sphere

2002 ◽  
Vol 34 (03) ◽  
pp. 469-483
Author(s):  
Ximo Gual-Arnau ◽  
Luis M. Cruz-Orive
Keyword(s):  
Fixed Point ◽  
Prediction Accuracy ◽  
Single Sample ◽  
Particle Model ◽  
Strict Sense ◽  
Biological Cell ◽  
Systematic Design ◽  
Arbitrary Size ◽  
Equal Area ◽  
Isotropic Random

Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.

scholarly journals Extended partial b-metric spaces and some fixed point results

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2837-2850 ◽  
Author(s):  
V. Parvaneh ◽  
Z. Kadelburg
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Space ◽  
Metric Spaces ◽  
Volterra Type ◽  
Fundamental Lemma ◽  
Contractive Mappings ◽  
Type Integral ◽  
Convergence Of Sequences ◽  
Weakly Contractive

In this paper, we introduce the concept of extended partial b-metric space. We demonstrate a fundamental lemma for the convergence of sequences in such spaces. Then we prove some fixed point results for weakly contractive mappings in the setup of ordered extended partial b-metric spaces. An example is given to verify the effectiveness and applicability of our main results. An application of these results to Volterra-type integral equations is provided at the end.

Variance prediction for pseudosystematic sampling on the sphere

2002 ◽  
Vol 34 (3) ◽  
pp. 469-483 ◽  
Author(s):  
Ximo Gual-Arnau ◽  
Luis M. Cruz-Orive
Keyword(s):  
Fixed Point ◽  
Prediction Accuracy ◽  
Single Sample ◽  
Particle Model ◽  
Strict Sense ◽  
Biological Cell ◽  
Systematic Design ◽  
Arbitrary Size ◽  
Equal Area ◽  
Isotropic Random

Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.

scholarly journals Suzuki-Type Fixed Point Results in Metric-Like Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Nabiollah Shobkolaei ◽  
Shaban Sedghi ◽  
Jamal Rezaei Roshan ◽  
Nawab Hussain
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Partial Metric ◽  
Fundamental Lemma ◽  
Partial Metric Spaces ◽  
Convergence Of Sequences

We demonstrate a fundamental lemma for the convergence of sequences in metric-like spaces, and by using it we prove some Suzuki-type…fixed point results in the setup of metric-like spaces. As an immediate consequence of our results we obtain certain recent results in partial metric spaces as corollaries. Finally, three examples are presented to verify the effectiveness and applicability of our main results.

scholarly journals Arithmetic intersection on GSpin Rapoport–Zink spaces

2018 ◽  
Vol 154 (7) ◽  
pp. 1407-1440 ◽  
Author(s):  
Chao Li ◽  
Yihang Zhu
Keyword(s):  
Explicit Formula ◽  
Intersection Number ◽  
Shimura Varieties ◽  
Local Problem ◽  
Fundamental Lemma

We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport–Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan–Gross–Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic–geometric side of the arithmetic fundamental lemma proved by Rapoport–Terstiege–Zhang in the minuscule case.

scholarly journals On the Number of Walks in a Triangular Domain

10.37236/4125 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Paul R.G. Mortimer ◽  
Thomas Prellberg
Keyword(s):  
Fixed Point ◽  
Explicit Formula ◽  
Generating Function ◽  
Triangular Lattice ◽  
Triangular Domain ◽  
Finite Height ◽  
Central Result ◽  
Motzkin Paths

We consider walks on a triangular domain that is a subset of the triangular lattice. We then specialise this by dividing the lattice into two directed sublattices with different weights. Our central result is an explicit formula for the generating function of walks starting at a fixed point in this domain and ending anywhere within the domain. Intriguingly, the specialisation of this formula to walks starting in a fixed corner of the triangle shows that these are equinumerous to two-coloured Motzkin paths, and two-coloured three-candidate Ballot paths, in a strip of finite height.

scholarly journals ON THE PRECISION OF THE NUCLEATOR

2017 ◽  
Vol 36 (2) ◽  
pp. 123 ◽  
Author(s):  
Javier González-Villa ◽  
Marcos Cruz ◽  
Luis M. Cruz-Orive
Keyword(s):  
Monte Carlo ◽  
Fixed Point ◽  
Sampling Design ◽  
Empirical Distribution ◽  
Systematic Sampling ◽  
Practical Relevance ◽  
Pivotal Point ◽  
Basic Goal ◽  
Real Objects ◽  
Isotropic Random

The nucleator is a design unbiased method of local stereology for estimating the volume of a bounded object. The only information required lies in the intersection of the object with an isotropic random ray emanating from a fixed point (called the pivotal point) associated with the object. For instance, the volume of a neuron can be estimated from a random ray emanating from its nucleolus. The nucleator is extensively used in biosciences because it is efficient and easy to apply. The estimator variance can be reduced by increasing the number of rays. In an earlier paper a systematic sampling design was proposed, and theoretical variance predictors were derived, for the corresponding volume estimator. Being the only variance predictors hitherto available for the nucleator, our basic goal was to check their statistical performance by means of Monte Carlo resampling on computer reconstructions of real objects. As a plus, the empirical distribution of the volume estimator revealed statistical properties of practical relevance.

scholarly journals Fixed Points of Contractive Mappings inb-Metric-Like Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Nawab Hussain ◽  
Jamal Rezaei Roshan ◽  
Vahid Parvaneh ◽  
Zoran Kadelburg
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Periodic Point ◽  
Topological Structure ◽  
Fundamental Lemma ◽  
Contractive Mappings ◽  
Different Types ◽  
Convergence Of Sequences

We discuss topological structure ofb-metric-like spaces and demonstrate a fundamental lemma for the convergence of sequences. As an application we prove certain fixed point results in the setup of such spaces for different types of contractive mappings. Finally, some periodic point results inb-metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.

Mental equilibrium: Identifying fixed-point attractors in the stream of thought

2003 ◽  
Author(s):  
Robin R. Vallacher ◽  
Andrzej Nowak ◽  
Matthew Rockloff
Keyword(s):  
Fixed Point
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