We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton–Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step n, we can generate step n+1 in O(log n) operations. Detailed pseudo-code for this algorithm is provided.R
A full closed mathematical model to describe and calculate Kondratiev’s long wave (LW) of economic development is presented for the first time. The innovative process that generates a new long wave in the economy is described as a stochastic Poisson process. The key role in constructing production functions during both the upward and downward trends of the LW is played by the self-similarity property of the innovative process, which is determined by its fractal structure. The role of the switch from an upward wave to a downward one is played by entrepreneurial profit; this article places primary emphasis on calculation of it. The practical effect of the model developed is illustrated through predictive calculations of GDP movement paths and the number of employees in the economy and the dynamics of fixed physical capital formation and growth of labor productivity by the example of the development of the US economy during the coming sixth Kondratiev LW (2018–2050).
We design temporal description logics (TDLs) suitable for reasoning about temporal conceptual data models and investigate their computational complexity. Our formalisms are based on DL-Lite logics with three types of concept inclusions (ranging from atomic concept inclusions and disjointness to the full Booleans), as well as cardinality constraints and role inclusions. The logics are interpreted over the Cartesian products of object domains and the flow of time (ℤ, <), satisfying the constant domain assumption. Concept and role inclusions of the TBox hold at all moments of time (globally), and data assertions of the ABox hold at specified moments of time. To express temporal constraints of conceptual data models, the languages are equipped with flexible and rigid roles, standard future and past temporal operators on concepts, and operators “always” and “sometime” on roles. The most expressive of our TDLs (which can capture lifespan cardinalities and either qualitative or quantitative evolution constraints) turns out to be undecidable. However, by omitting some of the temporal operators on concepts/roles or by restricting the form of concept inclusions, we construct logics whose complexity ranges between NLogSpace and PSpace. These positive results are obtained by reduction to various clausal fragments of propositional temporal logic, which opens a way to employ propositional or first-order temporal provers for reasoning about temporal data models.
We construct a curve in the unstable foliation of an Anosov diffeomorphism such that the holonomy along this curve is defined on all of the corresponding stable leaves.
We strengthen the convergence result in our paper, ibid. 5, No. 6, 1059-1098 (1999; Zbl 0983.62049), proving the local asymptotic mixed normality property in one of the 11 cases considered in that paper.