scholarly journals ON INVERSE STOCHASTIC RECONSTRUCTION PROBLEM

2021 ◽  
Vol 335 (1) ◽  
pp. 6-13
Author(s):  
M.I. Tleubergenov ◽  
G.K. Vassilina ◽  
A.T. Sarypbek
Keyword(s):  
Integral Manifold ◽  
Sufficient Conditions ◽  
Linear Case ◽  
Inversion Method ◽  
Random Perturbations ◽  
Reconstruction Problem ◽  
Stochastic Problem ◽  
Independent Increments ◽  
Processes With Independent Increments ◽  
Necessary And Sufficient

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

scholarly journals ON INVERSE STOCHASTIC RECONSTRUCTION PROBLEM

2021 ◽  
Vol 335 (1) ◽  
pp. 6-13
Author(s):  
M.I. Tleubergenov ◽  
G.K. Vassilina ◽  
A.T. Sarypbek
Keyword(s):  
Integral Manifold ◽  
Sufficient Conditions ◽  
Linear Case ◽  
Inversion Method ◽  
Random Perturbations ◽  
Reconstruction Problem ◽  
Stochastic Problem ◽  
Independent Increments ◽  
Processes With Independent Increments ◽  
Necessary And Sufficient

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

METHOD OF MEAN VALUE OPERATORS FOR RADON TRANSFORMS IN THE SPACE OF MATRICES

2008 ◽  
Vol 19 (03) ◽  
pp. 245-283 ◽  
Author(s):  
E. OURNYCHEVA ◽  
B. RUBIN
Keyword(s):  
Radon Transform ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Inversion Method ◽  
Radon Transforms ◽  
Inversion Formulas ◽  
Higher Rank ◽  
Necessary And Sufficient

We extend the Funk–Radon–Helgason inversion method of mean value operators to the Radon transform [Formula: see text] of continuous and Lpfunctions which are integrated over matrix planes in the space of real rectangular matrices. Necessary and sufficient conditions of existence of [Formula: see text] for such f and explicit inversion formulas are obtained. New higher-rank phenomena related to this setting are investigated.

Integrability of singular distributions on Banach manifolds

1976 ◽  
Vol 79 (1) ◽  
pp. 117-128 ◽  
Author(s):  
David Chillingworth ◽  
Peter Stefan
Keyword(s):  
Integral Manifold ◽  
Sufficient Conditions ◽  
Infinite Dimensions ◽  
Frobenius Theorem ◽  
Singular Foliation ◽  
Infinite Dimensional ◽  
Banach Manifolds ◽  
Integrable Distribution ◽  
Real Analytic ◽  
Necessary And Sufficient

One of the key results in the work of the second author ((7), (8)) on integrability of systems of vectorfields is the theorem which relates integrability of a distribution to the concept of homogeneity. In this paper, we show that the homogeneity theorem also applies in an infinite-dimensional context, and this allows us to derive infinite-dimensional versions of several further results in (7) and (8), formulated in terms of distributions. In particular, we are able to express necessary and sufficient conditions for homogeneity in terms of Lie brackets (Theorems 3 and 4) and to characterize integrable real-analytic distributions (Theorem 5). As a corollary to our Theorem 2, we recover the standard Frobenius theorem on the integrability of regular distributions. We also discuss briefly a basic problem which arises in infinite dimensions when we view an integral manifold of an integrable distribution as part of a singular foliation.

scholarly journals Recurrence properties of Processes with stationary independent increments

1964 ◽  
Vol 4 (2) ◽  
pp. 223-228 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Increments ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

Let X1, X2,…Xn, … be independent and identically distributed random variables, and write . In [2] Chung and Fuchs have established necessary and sufficient conditions for the random walk {Zn} to be recurrent, i.e. for Zn to return infinitely often to every neighbourhood of the origin. The object of this paper is to obtain similar results for the corresponding process in continuous time.

scholarly journals STOCHASTIC STABILITY OF NON-UNIFORMLY HYPERBOLIC DIFFEOMORPHISMS

2007 ◽  
Vol 07 (03) ◽  
pp. 299-333 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VÍTOR ARAÚJO ◽  
CARLOS H. VÁSQUEZ
Keyword(s):  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Random Perturbations ◽  
Dominated Splitting ◽  
Hyperbolic Diffeomorphisms ◽  
Stable Direction ◽  
Stochastically Stable ◽  
Necessary And Sufficient ◽  
Open Class ◽  
Unstable Direction

We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C2-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.

scholarly journals RECONSTRUCTING MULTISETS OVER COMMUTATIVE GROUPOIDS AND AFFINE FUNCTIONS OVER NONASSOCIATIVE SEMIRINGS

2014 ◽  
Vol 24 (01) ◽  
pp. 11-31 ◽  
Author(s):  
ERKKO LEHTONEN
Keyword(s):  
Finite Fields ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Reconstruction Problem ◽  
Affine Functions ◽  
Necessary And Sufficient

A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of multisets are determined. These results find an application in a different kind of reconstruction problem for functions of several arguments and identification minors: classes of linear or affine functions over nonassociative semirings are shown to be weakly reconstructible. Moreover, affine functions of sufficiently large arity over finite fields are reconstructible.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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