Absolutely Continuous Approximations

2016 ◽  
pp. 107-120
Author(s):  
Vydas Čekanavičius
Keyword(s):  
Absolutely Continuous ◽  
Continuous Approximations

scholarly journals Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1395
Author(s):  
Charles Castaing ◽  
Christiane Godet-Thobie ◽  
Le Xuan Truong
Keyword(s):  
Fractional Order ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Separable Hilbert Space ◽  
Maximal Monotone ◽  
Fractional Flow ◽  
Order Problem ◽  
Absolutely Continuous ◽  
State Dependent ◽  
Fractional Differential

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

The Factorisation of the rectangular distribution

1967 ◽  
Vol 4 (3) ◽  
pp. 529-542 ◽  
Author(s):  
T. Lewis
Keyword(s):  
Bounded Variation ◽  
Distribution Functions ◽  
Absolutely Continuous ◽  
Rectangular Distribution ◽  
Factor Type

Questions of the decomposability of distribution functions into real-valued components of bounded variation were discussed by P. Lévy (1964) in relation to the nature of the components, whether non-decreasing (distribution functions in particular) or absolutely continuous (a.c.) or both. Hanson (1965), in a review of Lévy's paper, raised the question of whether or not a rectangular distribution could be decomposed into two a.c. distributions. In fact, D. G. Kendall had conjectured earlier (Kendall (1960)) that no such decomposition is possible. The object of this paper is to state and prove the truth of Kendall's conjecture. “Decomposition” or “factorisation” will be understood throughout the paper to mean decomposition into distributions. Decompositions of the rectangular distribution into one a.c. and one discrete factor are well known (see, e.g., Lukacs (1960) pp. 128–9), and decompositions in which both factors are singular continuous (s.c.) have been discovered by Kendall and by P. M. Lee; it is shown here that no other combinations of factor-type can exist. References to other work on related decomposability properties are given in the papers by Lévy and Kendall cited above.

QUASI-INVARIANCE FORMULAS FOR COMPONENTS OF QUANTUM LÉVY PROCESSES

2004 ◽  
Vol 07 (01) ◽  
pp. 131-145 ◽  
Author(s):  
UWE FRANZ ◽  
NICOLAS PRIVAULT
Keyword(s):  
Lie Algebras ◽  
Lévy Processes ◽  
Unitary Operator ◽  
Levy Processes ◽  
Absolutely Continuous ◽  
Commutative Subalgebra ◽  
Independent Increments ◽  
Commutative Subalgebras ◽  
Current Representation ◽  
General Method

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,15,16 and the Meixner process.

scholarly journals On the existence of a solution for some distributed optimal control hyperbolic system

2000 ◽  
Vol 23 (5) ◽  
pp. 297-311 ◽  
Author(s):  
Dariusz Idczak ◽  
Stanislaw Walczak
Keyword(s):  
Optimal Control ◽  
Hyperbolic System ◽  
Convex Set ◽  
Linear Time ◽  
Time Varying ◽  
Optimal Process ◽  
Existence Of A Solution ◽  
Distributed Optimal Control ◽  
Absolutely Continuous ◽  
State Variable

We consider a Bolza problem governed by a linear time-varying Darboux-Goursat system and a nonlinear cost functional, without the assumption of the convexity of an integrand with respect to the state variable. We prove a theorem on the existence of an optimal process in the classes of absolutely continuous trajectories of two variables and measurable controls with values in a fixed compact and convex set.

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