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A DESCRIPTION OF THE BEHAVIOR OF SOME PHASE MOTIONS IN TERMS OF ORDINARY AND STANDARD " LEBESGUE MEASURES " IN R ∞
Abstract
This article presents main results of investigations of the authors which were obtained
during the last five years by the partially support on the Shota Rustaveli National
Science Foundation (Grant no. 31–24). These results are Liouville-type theorems
and describe the behavior of various phase motions in terms of ordinary and
standard “Lebesgue measures” in R∞. In this context, the following three problems
are discussed in this paper:
Problem 1. An existence and uniqueness of partial analogs of the Lebesgue measure
in various function spaces;
Problem 2. A construction of various dynamical systems with domain in function
spaces defined by various partial differential equations;
Problem 3. To establish the validity of Liouville-type theorems for various dynamical
systems with domains in function spaces in terms of partial analogs of the
Lebesgue measure.
... • dynamical system in R ∞ defined by the Black-Scholes equation (cf. [15]); ...
... • dynamical system in R ∞ defined by Fourier differential equation (cf. [15]). The rest of the paper is the following. ...
In the present paper we consider the following Satisfaction Problem of
Consumers Demands (SPCD): {\it The supplier must supply the measurable system
of the measure to the k-th consumer at the moment for . The measure of the supplied measurable system is changed under action
of some dynamical system; What minimal measure of measurable system must take
the supplier at the initial moment t=0 to satisfy demands of all consumers ?
} In this paper we consider Satisfaction Problem of Consumers Demands measured
by ordinary "Lebesgue measures'' in for various dynamical systems
in . In order to solve this problem we use Liouville type theorems
for them which describes the dependence between initial and resulting measures
of the entire system.
... • dynamical system in R ∞ defined by the Black-Scholes equation (cf. [15]); ...
... • dynamical system in R ∞ defined by Fourier differential equation (cf. [15]). The rest of the paper is the following. ...
In the present paper we consider the following Satisfaction
Problem of Consumers Demands (SPCD): The supplier must supply the
measurable system of the measure mk to the k-th consumer at the moment
tk for 1 � k � n. The measure of the supplied measurable system is
changed under action of some dynamical system; What minimal measure
of measurable system must take the supplier at the initial moment t = 0
to satisfy demands of all consumers ? In this paper we consider Satis-
faction Problem of Consumers Demands measured by ordinary “Lebesgue
measures” in R1 for various dynamical systems in R1. In order to solve
this problem we use Liouville type theorems for them which describes the
dependence between initial and resulting measures of the entire system.
Author begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new ″down-and-out″ option are derived. Other results.
In terms of ordinary and standard Lebesgue measures we study the behavior of phase flows in ℝ ∞ defined by Riemann’s alternating zeta function. Also, we give new reformulations of the Riemann’s conjecture in terms of the ordinary (2,2,⋯)-Lebesgue measure μ (2,2,⋯) .
For some infinite-dimensional non-ergodic phase flows, we give generalizations of the well-known Joseph Liouville theorem being a key theorem in classical statistical and Hamiltonian mechanics. In particular, for a normal Hermitian operator A with a convergent trace Tr (A), we construct a quasi-finite translation-invariant Borel measure μ A in the infinite-dimensional separable Hilbert space ℓ 2 such that μ A (e tA (D 0 ))=e t Tr (A) μ A (D 0 ) for an arbitrary Borel subset D 0 in ℓ 2 .
We describe the class of all infinite-dimensional diagonal matrices A and the class of all infinite collections of continuous impulses F such that Mankiewicz generator GM preserves a phase flow (in ℝℕ) defined by a non-homogeneous differential equation dψ/dt = A × ψ + f for A ε A, f ε F. An analogous question is studied for Preiss-Tišer generators in ℝℕ.
We consider a certain generalization of the von Foerster-Lasota differential equation and by using the technique of infinite-dimensional cellular matrices (the so-called Maclaurin differential operators) give its solution in an explicit form. We study the behaviour of corresponding motions in ℝ ∞ in terms of ordinary and standard “Lebesgue measures”.
Let α be an infinite parameter set, and let (α i ) i∈I be its partition such that α i is a non-empty finite subset for every i∈I. For j∈α, let μ j be a σ-finite Borel measure defined on a Polish metric space (E j ,ρ j ). We introduce a concept of standard (α i ) i∈I -product of measures (μ j ) j∈α and investigate its properties. As a consequence, we construct a “standard (α i ) i∈I -Lebesgue measure” on the Borel σ-algebra of subsets of ℝ α for every infinite parameter set α which is invariant under a group generated by shifts. In addition, if card(α i )=1 for every i∈I, then a “standard (α i ) i∈I -Lebesgue measure” m α is invariant under a group generated by shifts and canonical permutations of ℝ α . As a simple consequence, we get that a “standard Lebesgue measure” m ℕ on ℝ ℕ improves the measure proposed by R. L. Baker [Proc. Am. Math. Soc. 132, No. 9, 2577–2591 (2004; Zbl 1064.28015)].
In this paper we give a necessary and sufficient condition for a linear differential operator of infinite order to act invariantly in the space of distributions with compact support. Let P (ξ) be a polynomial in ξ. The solvability and related problems of the differential equation P (D)h = f in the space E = E (R n) of distributions with compact support have been studied by B. Malgrange, L. Hörmander, V.P. Palamodov and others (see, for example, [1-3]). The aim of this paper is to study the same problems for differential operators of infinite order. Let {a α } be a sequence of complex numbers. We put A(D) = α≥0 a α D α and consider the solvability in E of the equation (1) A(D)h = f . To study (1) we have to understand the action of the symbol A(D)h on test functions ϕ ∈ C ∞ (R n). The first thought came to our mind is that: Received October 7, 1995 1991 Mathematics Subject Classification. Primary 46F05, 46F10. Key words and phrases. Linear differential operators of infinite order, Distributions with compact support.
By using a structure of the "Fourier differential operator" in R ∞ , we describe a new and essentially different approach for a solution of the old functional problem posed by R. D. Carmichael in 1936. More precisely, under some natural restrictions, we express in an explicit form the general solution of the linear inhomogeneous differen-tial equation of infinite order with real constant coefficients. In addition, we construct an invariant measure for the corresponding differential equation. Also, we describe a certain approach for a solution of an initial value problem for a special class of linear inhomogeneous partial differential equations of infinite order with real constant coefficients and describe behaviors of corresponding dynamical systems in terms of partial analogs of the Lebesgue measure.
By using a structure of the "Fourier differential operator" in R^∞ , under some general restrictions we describe a new and essentially different approach for a construction of a particular solution of the non-homogeneous differential equation of the higher order with real constant coefficients and give its representation in an explicit form.
By using the technique of infinite-dimensional cellular matrices we obtain solutions of various initial condition problems and study behavior of corresponding mo-tions in R^∞ in the sense of ordinary and standard "Lebesgue measures". words and phrases: Maclaurin's and Fourier's differential operators, Formal solutions, Phase flows, Riemann's conjecture, α-ordinary and α-standard "Lebesgue measures".
We prove change of variable formula for wide class of “Lebesgue measures” on
N R and extend a certain result obtained in [ R. Baker, Lebesgue measure on , Proc. Amer. Math. Soc. 113(4) (1991), 1023-1029 ].
It is shown, in the theory (ZF) & (DC), that the existence of a subset of the real line nonmeasurable in the Lebesgue sense is equivalent to the uniqueness property of a certain invariant measure on the space R N of all real-valued sequences.
We consider two semidynamical systems, (e Tt)t>0 and (Tt)t>0, generated by different partial differential equations of von Foerster-Lasota type. We investigate some of their common prop- erties in the integrable space with the p-exponent. We show that their chaotic or stable behaviour depends on the common value of the parameter γ = λ(0).
We prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.
This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of R n . For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. A uniqueness result for the invariant measure is obtained in the class of all probability measures that are absolutely continuous with respect to Lebesgue's measure. A sufficient condition for the existence of such a measure is also provided.
We introduce the notion of generators of shy sets in Polish topological vector spaces and give their various interesting applications. In particular, we demonstrate that this class contains specific measures which naturally generate implicitly introduced subclasses of shy sets. Moreover, such measures (unlike σ-finite Borel measures) possess many interesting, sometimes unexpected, geometric properties.
We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard
products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue
measures on ℝ∞ and Rogers-Fremlin measures on ℓ
∞, respectively, such that topological weights of quasi-metric spaces associated with these measures are maximal (i.e., 2
c
). We also solve some Fremlin problems concerned with an existence of uniform measures in Banach spaces.
KeywordsOrdinary and standard products of σ-finite measures–invariant extensions–Rogers-Fremlin measures on ℓ
∞
We construct a translation invariant Borel measure lambda on R-infinity = PI-i = 1 infinity R such that for any infinite-dimensional rectangle R = PI-i = 1 infinity (a(i), b(i)), -infinity < a(i) less-than-or-equal-to b(i) < + infinity, if 0 less-than-or-equal-to PI-i = 1 infinity (b(i) - a(i)) < + infinity, then lambda(R) = PI-i = 1 infinity (b(i) - a(i)). Because R-infinity is an infinite-dimensional locally convex topological vector space, the measure lambda can not be sigma-finite.
This book explores a number of new applications of invariant quasi-finite diffused Borel measures in Polish groups for a solution of various problems stated by famous mathematicians (for example, Carmichael, Erdos, Fremlin, Darji and so on). By using natural Borel embeddings of an infinite-dimensional function space into the standard topological vector space of all real-valued sequences, (endowed with the Tychonoff topology) a new approach for the construction of different translation-invariant quasi-finite diffused Borel measures with suitable properties and for their applications in a solution of various partial differential equations in an entire vector space is proposed.
This book explores a new concept of T-shy sets in Radon metric groups which is an extension of the concepts of null sets introduced by J.R. Christensen, J. Mycielski and B. R. Hunt, T. Sauer and J. A. Yorke for Polish groups. This book considers various interesting applications of the theory of infinite-dimensional cellular matrices for a solution of various initial condition problems. It also includes several direct generalizations of well-known classical results (Lebesgue theorem, Weyl theorem, Rademacher theorem, etc) in terms of infinite-dimensional gLebesgue measuresh. (Imprint: Nova).
A sucient condition for entire functions f and g to be such that the series P1 m=0 f(m)(0)g(m)=m! represents an entire function is established; and in that case, the growth of the resulting function is described.
We study asymptotic properties of a nonlinear first-order partial differential equation which describes the reproduction of blood cells. This equation under conditions proposed by Ważewska generates a semigroup of transformations with highly chaotic behaviour of trajectories. We show that this semigroup has invariant measures with arbitrary large dimension.
Differential operators ϕ(Δθ,ω), where ϕ is an exponential type entire function of a single complex variable and Δθ,ω=(θ+ωz)D+zD2, D=∂/∂z, View the MathML source, θ⩾0, View the MathML source, acting in the spaces of exponential type entire function are studied. It is shown that, for ω⩾0, such operators preserve the set of Laguerre entire functions provided the function ϕ also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane View the MathML source. The operator exp(aΔθ,ω), a⩾0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all View the MathML source. An integral representation of exp(aΔθ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δθ,ω.
By regarding the amplitudes of a set of orthogonal modes as the co-ordinates in an infinite-dimensional phase space, the probability distribution for an ensemble of randomly forced two-dimensional viscous flows is determined as the solution of the continuity equation for the phase flow. For a special, but infinite, class of types of random forcing, the exact equilibrium probability distribution can be found analytically from the Navier-Stokes equations. In these cases, the probability distribution is the product of exponential functions of the integral invariants of unforced inviscid flow.
Let be an entire function from the class with simple zeros and let be a bounded convex domain. In this paper particular solutions of the equation (I)are constructed which are analytic in and possess a definite smoothness on the boundary of , for the case in which is analytic in and sufficiently smooth on the boundary. In particular, it is shown that if is an entire function of completely regular growth with proximate order , , , with a positive indicator and a regular set of roots, then for an arbitrary function , analytic in and continuous on , equation (I) has an effectively defined particular solution analytic in and infinitely differentiable at each boundary point of .Bibliography: 14 titles.
We study the chaotic and stable behaviour of the von Foerster–Lasota equation in Orlicz spaces with homogeneous ϕ-function of any positive degree. This work is, in particular, the generalization of the asymptotic properties of the von Foerster– Lasota equation in integrable spaces with exponent p greater than or equal to 1.
x xk; where c;;xk 2 R, 0, n is a nonnegative integer and P1 k=1 1=x 2 k