Archive for the ‘Mathematics’ Category

Analytic Memory Dump - A Mathematical Definition

Friday, September 28th, 2012

The previous mathematical definition of memory dump is for raw memory dumps. They are not really useful because they require symbol files. Each symbol file entry conceptually is a correspondence between a memory address and a direct sum or product of letters from some alphabet:

00000000`76e82c40: kernel32!WaitForMultipleObjectsExImplementation

So we propose an analytical definition of a memory dump as a direct sum of disjoint memory areas Mt taken during some time interval (t0, …, tn) where we replace stk having values from Z2 with Stq having values from Zp and cardinality of Zp depending on a platform (32, 64, etc) plus a symbolic description Di for each Stq with cardinality of ”i” set sufficient enough to accommodate the largest symbolic name:

M = Mt where Mt = ∑(Stq+Di)

or simply

M = (Stq+Di)

This can be visualized as a linear memory space such as a virtual memory space when symbol files are applied to modules one after another. However, all this is not necessary, as a symbol from a virtual address can also be mapped to a physical address if necessary. Di, in fact, refers to any symbolic description.

- Dmitry Vostokov @ + -

Facets of Systems Science

Monday, September 17th, 2012

If you liked An Introduction to General Systems Thinking book then you really need this comprehensive introduction which is more formal. Don’t be overwhelmed by the number of pages, you only need to read part 1, the first 218 pages as the rest is a collection of articles you can read selectively later on. For me one of the great features was the coverage of systems literature including some mathematical treatment books (including category theory in addition to famous Rosen’s books such as Anticipatory Systems). I also liked the discussion of critics of general systems theory that points to the fact that it should be called general systems-theory not general-systems theory. Highly recommended.

Facets of Systems Science (IFSR International Series on Systems Science and Engineering)

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Mathematical Physics

Friday, September 14th, 2012

This book came to my attention because it starts with category theory in the first chapters and then moves to traditional contemporary mathematical physics topics such as topology and operators. It also covers groups, vector spaces, their duals, tensors, associative and Lie algebras, representation theory, spectral theorem, distributions, homotopy and homology. The author also provides physical examples along the way such as Fock vector spaces, dynamical systems, Minkowski space and algebra of observables. The flow of this mathematical text is very smooth (proofs can be omitted from reading) and explanations are very intuitive. The latter seems to be the main goal of this text. It is also structured into 56 chapters so it can be possible to casually read this book in 2 months during commuting like I did. One strange thing I noticed though is the avoidance of the manifold terminology: the author only uses the word “manifold” only once and without an explanation what it is about so you may even not notice that.

Mathematical Physics (Chicago Lectures in Physics)

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Classical Mechanics: Point Particles and Relativity

Friday, June 17th, 2011

It was my dream since the school days to learn physics in its entirety. Whereas The Feynman Lectures on Physics: Commemorative Issue, Three Volume Set that I own (I read it at school before university in Russian translation) is a bit light and don’t include the developments of the past 40 - 50 years and Course of Theoretical Physics by Landau was a bit heavy for me at those times (although I read Mechanics volume in Russian and a few beginning chapters from other volumes) I finally found what I need: Theoretical Physics course from Walter Greiner. I have now the first 3 volumes (there are many more volumes including Quantum Electrodynamics, Gauge Theory of Weak Interactions, Quantum Chromodynamics) and just started reading the first one: Classical Mechanics: Point Particles and Relativity (Classical Theoretical Physics). It explains all necessary mathematics, has all derivations, lots of examples and illustrations, and even talks about dark matter (in the first classical mechanics volume). More important I also ordered the original German edition (Theoretische Physik. Klassische Mechanik I. Dynamik und Dynamik der Punktteilchen - Relativität) and reading both in parallel. This improves my German as well.

- Dmitry Vostokov @ -

Dictionary of Debugging: Orbifold Memory Space

Wednesday, February 16th, 2011

Orbifold Memory Space

A multiple virtual/physical memory space view taking into account multiple computers:

The picture can be much more complex if we glue different manifold memory spaces. The space name comes from a mathematical orbifold, a generalization of manifold.

Synonyms: cloud memory space


Also: memory space, memory region, physical memory, virtual memory, manifold memory space, memory mapping.

- Dmitry Vostokov @ + -

Personal Roots of Memory Dump Analysis

Thursday, September 23rd, 2010

When I was a child I experienced dreams where I was carried by a huge wave that was transforming to a torus completely absorbing me up to a breakpoint of my wake up. A year ago I got the book Memory Evolutive Systems because of my interest in applying category theory to memory analysis and debugging and immediately recalled my long-time forgotten childhood dreams while staring at its front cover:

- Dmitry Vostokov @ + -

Godel’s Theorem

Tuesday, December 22nd, 2009

This is a book I bought a few years ago and started reading immediately but put aside and only this summer read it fully from cover to cover. In order to appreciate its content you need some degree of mathematical and computer science maturity. For example, if you have never heard of his theorems and only read Incompleteness: The Proof and Paradox of Kurt Godel or similar popular book then you would have difficulty going through the book and it would appear boring. It is not an entertaining or bedside reading. This is why I put it aside on the first reading although I knew about this theorem since I read “Mathematics: The Loss of Certainty” more than 25 years ago being a schoolboy (in Russian translation). Just before writing this review I ordered “There’s Something About Godel: The Complete Guide to the Incompleteness Theorem” and the latter looks like less heavy reading judged from excerpts from its publisher website. Putting all these reminiscences aside I really enjoyed second reading of “Godel’s Theorem”. It really clarified some points from ¬B->¬A or PA & ¬Con(PA) perspectives and made me curious about fixpoints. I even borrowed the latter term and introduced them for crash dump analysis and debugging: “a dereference fixpoint”. I also liked chapters 4 and 6 about using Godel’s theorems outside mathematics and clarifying misconceptions in Rucker’s and Penrose’s books. However, after a few months I cannot recall anything definite what I read from that book although I felt good that I understood everything while reading so perhaps the book requires the 3rd reading for me :-) I’m going to give it another try after “There’s Something About Godel” and update this review.

Godel’s Theorem: An Incomplete Guide to Its Use and Abuse

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The CRC Encyclopedia of Mathematics

Thursday, November 12th, 2009

The CRC Encyclopedia of Mathematics, Third Edition - 3 Volume Set

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I spotted this book on Amazon US and was thrilled to have the new edition in 3 volumes for easy handling when reading. I also have the previous edition that is even featured on my own book cover (the picture of my previous library book arrangement, the book is highlighted in white rectangle in the lower right corner):

This is a unique volume that sits between The Princeton Companion to Mathematics (that I’m also reading now) and Encyclopedic Dictionary of Mathematics: The Mathematical Society of Japan (that I also own). In fact after reading 3 volumes from cover to cover I can start with 2 volumes of EDM. There is also Springer Encyclopaedia of Mathematics in 11 volumes with various additional supplement volumes that I plan to own as well and it looks to me on the same level as EDM.

After searching for the best price I ordered a copy from Amazon DE and after my purchase in just a few days the price was up by 50%! I can only explain this that more people tried to purchase after I used twitter to announce this encyclopedia (there were 5 copies available on Amazon DE and in just 2 days only 1 left) or there was a mistake in price.

3 volumes arrived and I immediately started reading them, a few pages from each volume every day using mod 3 reading technique, for example, Wed - Vol I, Thu - Vol II, Fri - Vol III, Mon - Vol I, an so on. I prefer paper books for bulk reading instead of electronic version (in this case corresponding website) although if I’m interested in a specific article or a keyword I go to Wolfram MathWorld website to get the latest update and citations. These paperback volumes are just for day-to-day scheduled reading to get ideas and general mathematical education. This is why I don’t need an Index. For example, just after reading the first pages I got the idea of cubic (qubic) memory representation.

I usually put reviews on Amazon after I finish a book from cover to cover but in this case the review would be waiting for at least a year so I write it now based on my first impressions. After some time I plan to update it.

- Dmitry Vostokov @ -

Naming Infinity

Friday, July 3rd, 2009

I read this book from cover to cover while flying on a plane from Dublin to St. Petersburg and back. That was so wonderful reading experience - I couldn’t put the book down during those flights. I recall that I visited the Department of Mathematics a few times when I studied Chemistry in Moscow State University although at that time I knew next to nothing about Russian mathematicians. The book touched me so deeply that I bought the main work of Florensky: The Pillar and Ground of the Truth, the history of Russian philosophy and several books explaining Orthodox Church. This is the best mathematics history book I have ever read, my feelings perhaps comparable to those that I experienced when I finished reading Mathematics: The Loss of Certainty by Morris Kline but that was more than 20 years ago.

Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity

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Memory Dump - A Mathematical Definition

Wednesday, October 24th, 2007

This is the first post in Science of Memory Dump Analysis category where I apply philosophy, systems theory, mathematics, physics and computer science ideas. It was inspired after reading Life Itself book written by Robert Rosen where computers are depicted as direct sums of states. As shown in that book, in the case of machines, their synthetic models (direct sums) are equivalent to analytic models (direct product of observables). Taking every single bit as an observable having its values in Z2 set {0, 1} we can make a definition of an ideal memory dump as a direct product or a direct sum of bits saved instantaneously at the given time:

i si = i si

Of course, we can also consider bytes having 8 bits as observables having their values from Z256 set, etc.

In our case we can simply rewrite direct sum or product as the list of bits, bytes, words or double words, etc:

(…, si-1, si, si+1, …, sj-1, sj, sj+1, …)

According to Rosen we include hardware states (registers, for example) and partition memory into input, output states for particular computation and other states.

Saving a memory dump takes certain amount of time. Suppose that it takes 3 discrete time events (ticks). During the first tick we save memory up to (…, si-1, si) and that memory has some relationship to sj state. During the second tick sj state changes its value and during the 3rd tick we copy the rest of the memory (si+1, …, sj-1, sj, sj+1, …). Now we see that the final memory dump is inconsistent:

(…, si-1, si, si+1, …, sj-1, sj, sj+1, …)

I explained this earlier in plain words in Inconsistent Dump pattern. Therefore we might consider a real memory dump as a direct sum of disjoint memory areas Mt taken during some time interval (t0, …, tn)

M = t Mt where Mt = k stk or simply

M = t k stk

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