On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

scholarly journals On the construction of resolving control in the problem of getting close at a fixed time moment

2021 ◽  
Vol 58 ◽  
pp. 73-93
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov ◽  
O.A. Kuvshinov
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Time Moment ◽  
Fixed Time ◽  
Dimensional Euclidean Space ◽  
Controlled System ◽  
Finite Dimensional ◽  
Solvability Set

The problem of getting close of a controlled system with a compact space in a finite-dimensional Euclidean space at a fixed time is studied. A method of constructing a solution to the problem is proposed which is based on the ideology of the maximum shift of the motion of the controlled system by the solvability set of the getting close problem.

scholarly journals On targeting an integral funnel of control system at a target set in the phase space

2020 ◽  
Vol 56 ◽  
pp. 79-101
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov
Keyword(s):  
Exact Solution ◽  
Control System ◽  
Phase Space ◽  
Euclidean Space ◽  
Time Moment ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Finite Dimensional ◽  
Approximate Construction ◽  
Target Set

A control system in finite-dimensional Euclidean space is considered. On a given time interval, we investigate the problem of constructing an integral funnel for which a section corresponding to the last time moment of interval is equal to a target set in a phase space. Since the exact solution of such a funnel is possible only in rare cases, the question of the approximate construction of an integral funnel is being studied.

Minimum norm interpolation in the ℓ1(ℕ) space

2020 ◽  
Vol 19 (01) ◽  
pp. 21-42
Author(s):  
Raymond Cheng ◽  
Yuesheng Xu
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Interpolation Problem ◽  
Dual Space ◽  
Original Problem ◽  
Dual Problem ◽  
Dimensional Euclidean Space ◽  
Minimum Norm ◽  
Finite Dimensional ◽  
Sparse Interpolation

We consider the minimum norm interpolation problem in the [Formula: see text] space, aiming at constructing a sparse interpolation solution. The original problem is reformulated in the pre-dual space, thereby inducing a norm in a related finite-dimensional Euclidean space. The dual problem is then transformed into a linear programming problem, which can be solved by existing methods. With that done, the original interpolation problem is reduced by solving an elementary finite-dimensional linear algebra equation. A specific example is presented to illustrate the proposed method, in which a sparse solution in the [Formula: see text] space is compared to the dense solution in the [Formula: see text] space. This example shows that a solution of the minimum norm interpolation problem in the [Formula: see text] space is indeed sparse, while that of the minimum norm interpolation problem in the [Formula: see text] space is not.

Cell Complexes, Valuations, and the Euler Relation

1970 ◽  
Vol 22 (2) ◽  
pp. 235-241 ◽  
Author(s):  
M. A. Perles ◽  
G. T. Sallee
Keyword(s):  
Euclidean Space ◽  
Hausdorff Metric ◽  
Dimensional Euclidean Space ◽  
Cell Complexes ◽  
Finite Dimensional ◽  
Finite Set ◽  
Euler Relation ◽  
Set Of Points ◽  
The Relationship ◽  
Dimensional Face

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

scholarly journals Small sets in convex geometry and formal independence over ZFC

2005 ◽  
Vol 2005 (5) ◽  
pp. 469-488
Author(s):  
Menachem Kojman
Keyword(s):  
Euclidean Space ◽  
Closed Subset ◽  
Polish Space ◽  
Convex Geometry ◽  
Dimensional Euclidean Space ◽  
Geometric Information ◽  
Polish Spaces ◽  
Finite Dimensional ◽  
Convex Subsets

To each closed subsetSof a finite-dimensional Euclidean space corresponds aσ-ideal of sets𝒥 (S)which isσ-generated overSby the convex subsets ofS. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation ofreducibilitybetween convexity ideals and the connections between convexity ideals and other types of ideals, such as the ideals which are generated over squares of Polish space by graphs and inverses of graphs of continuous self-maps, or Ramsey ideals, which are generated over Polish spaces by the homogeneous sets with respect to some continuous pair coloring. We also attempt to present to nonspecialists the set-theoretic methods for dealing with formal independence as a means of geometric investigations.

scholarly journals Recovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform

2019 ◽  
pp. 1-31
Author(s):  
Jeremy Becnel ◽  
Daniel Riser-Espinoza
Keyword(s):  
Euclidean Space ◽  
Computerized Tomography ◽  
Radon Transform ◽  
Gaussian Measure ◽  
Random Variables ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Computer Implementation ◽  
Reconstruction Algorithms ◽  
Conditional Expectations

The Radon transform maps a function on n-dimensional Euclidean space onto its integral over a hyperplane. The fields of modern computerized tomography and medical imaging are fundamentally based on the Radon transform and the computer implementation of the inversion, or reconstruction, techniques of the Radon transform. In this work we use the Radon transform with a Gaussian measure to recover random variables from their conditional expectations. We derive reconstruction algorithms for random variables of unbounded support from samples of conditional expectations and discuss the error inherent in each algorithm.

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