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25.9.10

Updates - September 2010

Some updates for September 2010:

1. Currently studying ANT, CD, ET, PRNG and PS on my own

ANT = Algebraic Number Theory (NOT Analytic Number Theory - but I will self-study that too later)

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integers—such as unique factorization—need not hold. The virtue of the primary machinery employed—Galois theory, group cohomology, group representations, and L-functions—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.

Source: Wikipedia

CD = Chaotic Dynamics (part of Chaos Theory)

Chaos theory is a field of study in mathematics, physics, economics and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.

Chaotic behavior can be observed in many natural systems, such as the weather. Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.

Source: Wikipedia

ET = Ergodic Theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics.

A central aspect of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important examples are ergodic theorems of Birkhoff and von Neumann. For the special class of ergodic systems, the time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Stronger properties, such as mixing and equidistribution, have also been extensively studied.

The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems.

Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions).

Source: Wikipedia

PRNG = PseudoRandom Number Generator

A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers that approximates the properties of random numbers. The sequence is not truly random in that it is completely determined by a relatively small set of initial values, called the PRNG's state. Although sequences that are closer to truly random can be generated using hardware random number generators, pseudorandom numbers are important in practice for simulations (e.g., of physical systems with the Monte Carlo method), and are central in the practice of cryptography and procedural generation. Common classes of these algorithms are linear congruential generators, Lagged Fibonacci generators, linear feedback shift registers, feedback with carry shift registers, and generalised feedback shift registers. Recent instances of pseudorandom algorithms include Blum Blum Shub, Fortuna, and the Mersenne twister.

Careful mathematical analysis is required to have any confidence a PRNG generates numbers that are sufficiently "random" to suit the intended use. Robert R. Coveyou of Oak Ridge National Laboratory once titled an article, "The generation of random numbers is too important to be left to chance." As John von Neumann joked, "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."

Source: Wikipedia

PS = Probability and Statistics

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. The concept has been given an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

Source: Wikipedia

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.

Source: Wikipedia

Statistics is the science of data collection, organization, and interpretation; the data is often numerical but may take other forms including relationships between entities. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments. The purpose is to obtain some overall understanding of group characteristics.

A statistician is someone who is particularly well versed in the ways of thinking necessary for the successful application of statistical analysis. Such people have often gained this experience through working in any of a wide number of fields. There is also a discipline called mathematical statistics, which is concerned with the theoretical basis of the subject.

The word statistics can either be singular or plural. When it refers to the discipline, "statistics" is singular, as in "Statistics is an art." When it refers to quantities (such as mean and median) calculated from a set of data, statistics is plural, as in "These statistics are misleading."

Source: Wikipedia

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis. The term "mathematical statistics" is closely related to the term "statistical theory" but also embraces modelling for actuarial science and non-statistical probability theory, particularly in Scandinavia.

Statistics deals with gaining information from data. In practice, data often contain some randomness or uncertainty. Statistics handles such data using methods of probability theory.

Source: Wikipedia

2. Hoping to pass the following DCS subjects in UPOU this semester:

(a) CMSC-E (Data Communications and Networking)
(b) CMSC-F (Database Management Systems)

3. Will be taking the following DCS subjects in UPOU next semester:

(a) CMSC-G (Management Information Systems)
(b) CMSC-H (Object-Oriented Analysis and Design)
(c) CMSC-I (Software Engineering)

4. Choosing between the following five (5) options after finishing DCS (hopefully by next year):

(a) Master in Technology Management @ UP Diliman
(b) Master of Science in Information Technology @ PUP (Distance Education mode)
(c) Professional Masters in Applied Mathematics (Actuarial Science) @ UP Diliman
(d) Ph. D. in Mathematics @ UP Diliman
(e) Master in Information Systems @ UP Open University

Will post further updates on or before December 2010.
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