Dynamics, measure and dimension in the theory of computing
نویسندگان
چکیده
This thesis establishes some results in algorithmic information theory. Martin-Löf in 1966 formalized the notion of what it means for a given sequence to be random, using the theory of computation. Equivalent, alternate definitions justify that this definition is robust. A finite string is called random if there is no short program which outputs the string that is, the Kolmogorov complexity of the string is within an additive constant of the length of the string itself. With the introduction of a suitable notion termed self-delimiting Kolmogorov complexity, an infinite sequence is called random if almost all its finite prefixes are incompressible in the above sense. Algorithmic information theory studies the properties of such random sequences. It is natural to enquire whether a sequence random according to this definition satisfies the well-known probabilistic laws. Early results by van Lambalgen and Vovk established that all random sequences obey Borel-Cantelli Lemmas, the Strong Law of Large Numbers and the Law of the Iterated Logarithm. van Lambalgen conjectured that Birkhoff’s ergodic theorem, however, will resist such effectivization. Nevertheless, V’yugin in a tour-de-force established an effective version of the Ergodic Theorem. V’yugin’s version has a technical limitation, however, that restricts its applicability. Random variables are assumed to be computable, and hence continuous. In the first part of the thesis, we extend V’yugin’s theorem to handle a restricted class of discontinuous functions, and prove that we can apply it to effectivize some results on continued fractions. The concept of dimension gives a quantitative measurement of how non-random sequences or sets are. In subsequent chapters we deal with the concept of dimension, in various resourcebounded settings. We give an alternate characterization of constructive Hausdorff and constructive packing dimension using generalizations of tools used in establishing the effective
منابع مشابه
Analysis of Resting-State fMRI Topological Graph Theory Properties in Methamphetamine Drug Users Applying Box-Counting Fractal Dimension
Introduction: Graph theoretical analysis of functional Magnetic Resonance Imaging (fMRI) data has provided new measures of mapping human brain in vivo. Of all methods to measure the functional connectivity between regions, Linear Correlation (LC) calculation of activity time series of the brain regions as a linear measure is considered the most ubiquitous one. The strength of the dependence obl...
متن کاملUtilization of Soft Computing for Evaluating the Performance of Stone Sawing Machines, Iranian Quarries
The escalating construction industry has led to a drastic increase in the dimension stone demand in the construction, mining and industry sectors. Assessment and investigation of mining projects and stone processing plants such as sawing machines is necessary to manage and respond to the sawing performance; hence, the soft computing techniques were considered as a challenging task due to stocha...
متن کاملSimulation of nanodroplet impact on a solid surface
A novel computational fluid dynamics and molecular kinetic theory (CFD-MK) method was developed to simulate the impingement of a nanodroplet onto a solid surface. A numerical solution of the Navier–Stokes equation using a volume-of-fluid (VOF) technique was used to model nanodroplet deformation. Dynamic contact angle during droplet impact was obtained by molecular kinetic theory. This dynamic c...
متن کاملSimulation of nanodroplet impact on a solid surface
A novel computational fluid dynamics and molecular kinetic theory (CFD-MK) method was developed to simulate the impingement of a nanodroplet onto a solid surface. A numerical solution of the Navier–Stokes equation using a volume-of-fluid (VOF) technique was used to model nanodroplet deformation. Dynamic contact angle during droplet impact was obtained by molecular kinetic theory. This dynamic c...
متن کاملIndirect Adaptive Interval Type-2 Fuzzy PI Sliding Mode Control for a Class of Uncertain Nonlinear Systems
Controller design remains an elusive and challenging problem foruncertain nonlinear dynamics. Interval type-2 fuzzy logic systems (IT2FLS) incomparison with type-1 fuzzy logic systems claim to effectively handle systemuncertainties especially in the presence of disturbances and noises, but lack aformal mechanism to guarantee performance. In contrast, adaptive sliding modecontrol (ASMC) provides...
متن کاملA review of methods for resource allocation and operational framework in cloud computing
The issue of management and allocation of resources in cloud computing environments, according to the breadth of scale and modern technology implementation, is a complicated issue. Issues such as: the heterogeneity of resources, resource dependencies to each other, the dynamics of the environment, virtualization, workload diversity as well as a wide range of management objectives of cloud servi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009