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| -rw-r--r-- | Ethernet | 107 | ||||
| -rw-r--r-- | math/algebra | 71 | ||||
| -rw-r--r-- | math/books to read | 33 | ||||
| -rw-r--r-- | math/good books | 5 | ||||
| -rw-r--r-- | math/integers | 5 | ||||
| -rw-r--r-- | math/logic | 18 | ||||
| -rw-r--r-- | math/numbers | 54 | ||||
| -rw-r--r-- | math/sets | 45 | ||||
| -rw-r--r-- | sheets/rst (renamed from restructured text cheatsheet) | 0 |
9 files changed, 329 insertions, 9 deletions
@@ -1,14 +1,15 @@ ======== Ethernet ======== +:Author: Bryan Newbold <bnewbold@mit.edu> -.. warning:: Under progress! +.. contents:: History ======= -Bob Metcalfe first described the ethernet system in a memo while working at -the Xerox Palo Alto Research Center in 1973 ([oreilly]_). It was based on -the earlier *Aloha* system developed by Norman Abramson at the University of +Bob Metcalfe first described the ethernet system in a memo while working at the +Xerox Palo Alto Research Center in 1973 ([oreilly]_). It was based on the +earlier *Aloha* system developed by Norman Abramson at the University of Hawaii. *Aloha* was a radio network for inter-island communications. Structure @@ -34,16 +35,114 @@ Checksum 32 The preamble originally served as a "warm up" period for hardware to stabilize and prepare for the rest of the frame. + The last 8 bits are the preamble are a particular "start frame delimiter" + pattern to help hardware notice the beginning of signal transmission. + .. topic:: Source and Destination addresses Manufacturers are given a 24 bit id (first half), and are expected to ensure the uniqueness of the second half of each device's address. + If the first bit of the destination is high (1), the destination is + multicast; if low (0), then unicast. Under IEEE, if the second bit is high, + global administration rules should be followed, and if low, local + administration rules should be followed. + The IEEE-SA (Standards Association) delegates addresses. .. topic:: Checksum The checksum (or Frame Check Sequence) is a Cyclic Redundancy Checksum. +Inter-Frame Gap +~~~~~~~~~~~~~~~~~ +The Inter-Frame Gap (IFG) is a pause in between signal frames; it is 96 +bit-times long. + +.. _collision: + +Collisions +============= +Collision management is the heart of Ethernet; a collision occurs when two +devices try to transmit at the same time and their signals overlap and the +transmission is unreadable. This problem is made worse the further apart +stations are because it takes longer for signal fronts to propagate, so there +is a longer overlap period when the first device has started transmitting but +the second hasn't noticed and may start transmitting on its own. The network +size (in distance, not nodes) is determined by the maximum collision period +(also called slot time). This period is 512 bit-times, so usually the faster a +network is, the smaller it should be (in distance); at gigabit speeds the +slot is expanded, see Speeds_. + +Under Ethernet, if any device detects a collision, it sends out a 32 bit +"collision enforcement jam signal" to ensure that all devices notice the +collision. Then each device that wishes to transmit waits a random amount of +bit-times r before transmitting again. If there is another collision, a new +r is randomly chosen with a larger maximum; up to 15 retries are allowed before +the devices will give up and drop the frame. The constraints for the random +backoff time r are `$0\leq r \leq 2^k$`:latex:, where r is in bit times and +k is the number of retry attempts up to 10 (for 11th to 15th retries, use +k=10). + +Speeds +========= +At 10mb, bit time is 100ns (nano-seconds); at 100mb, 10ns; at 1gb, 1ns. + +At gigabit speeds, the slot time would limit network size to 20 meters, so +it gets padded with a null "carrier extension" signal to 4096 bits (512 bytes). +On high speed half-duplex gigabit links (which are rare!), frame bursting is +a technique to allow several small data frames to be transmitted at the same +time: one frame is sent regularly to capture the channel, then up to 65536 +bit times of short non-carrier extended (only IFG) frames are sent, then +a single last regular frame is sent. + +Phrases +========= + +Auto-negotiation + Optional protocol for direct links (eg, from switch to switch) to + determine network speed. Can be reinitiated by dropping and restarting + the link. + +802.1 + IEEE working group for higher level networking. + +802.2 + IEEE working group for "logical link control" (LLC), which is related + to the type/length frame field. + +802.3 + For _`CSMA/CD` LANs + +802.5 + For Token Ring LANs + +CSMA/CD Protocol + Stands for Carrier Sense Multiple Access/Collision Detect. The two main + implementations are DIX (DEC-Intel-Xerox) and IEEE. (Can't remember how + this works...) + +VLAN + Optional 4-byte field in between "source" and "type" fields. The first + two bytes are a valid type field; this increases the max length to + 1522 bits. Only used by switches. + +Channel Capture + This phenomena is a result of _collision management. After a few + sequential collisions, the device that ultimately transmits first gets + its backoff reset to zero, while all other devices retain a high backoff; + this means the first device is much much more likely to retransmit again + faster than any of the others, and if it has a lot of frames to send it + will dominate the carrier until it is done transmitting. + +Jabber + When a broken station/device keeps a carrier open and prevents other + transmissions. + +PAUSE + A PAUSE frame is a MAC control (higher level, type 0x8808) frame which + causes switches to stop for up to 65536 slot times to allow a switch or + station to clear its buffers. + .. [oreilly] `Ethernet: The Definitive Guide`:title:, by Charles Spurgeon. O'Reilly, 2000 diff --git a/math/algebra b/math/algebra new file mode 100644 index 0000000..b337a2e --- /dev/null +++ b/math/algebra @@ -0,0 +1,71 @@ +==================== +Algebra +==================== + +.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ + +.. list-table:: Closure of binary operators on given sets of numbers + * Operation + - :latex:`$+$` + - :latex:`$\times$` + - :latex:`$-$` + - :latex:`$\divide$` + - :latex:`$^$` + - :latex:`$\sqrt{\text{ }}$` + * Positive Integers + - Y + - Y + - N + - N + - Y + - N + * Positive rationals + - Y + - Y + - N + - Y + - Y + - N + * Rationals (and zero) + - Y + - Y + - Y + - Y + - Y + - N + * Reals wrt positive integers + - Y + - Y + - Y + - Y + - Y + - Y + * Complex numbers + - Y + - Y + - Y + - Y + - Y + - Y + +Definitions +============= + +*involution* + to raise a number to a given power + +*evolution* + to take a given root of a number + +*associative* + :latex:`$(a+b)+c=a+(b+c)$` + +*comutative* + :latex:`$a+b=b+c$` + +*distributive* + :latex:`$(a+b)c=ac+bc$` + +.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976 + +.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve. diff --git a/math/books to read b/math/books to read new file mode 100644 index 0000000..ba75d8e --- /dev/null +++ b/math/books to read @@ -0,0 +1,33 @@ +=============================================== +Math books that look interesting +=============================================== + +`On formally undecidable propositions of Principa Mathematica and related systems`:title:, by Kurt Godel. + +`Computability and Unsolvability`:title:, by Martin Davis. + +`Mathematical Foundations of Information Theory`:title:, by A.I. Khinchin. + +`Calculus of Variations with Applications to Physics and Engineering`:title:, by Robert Weinstock. + +`Relativity, Thermodynamics, and Cosmology`:title:, by Richard Tolman. + +`Mathematics Applied to Continuum Mechanics`:title:, by Lee Segel. + +`Optimization Theory and Applications`:title:, by Donald Pierre. + +`The Variational Principles of Mechanics`:title:, by Cornelius Lanczos. + +`Tensor Analysis for Physicists`:title:, by J.A. Schonten. + +`Investigations on the Theory of Brownian Movement`:title:, by Albert Einstein. + +`Great Experiments in Physics`:title:, ed. by ???. + +`Curvature and Homology`:title:, by Samuel Goldberd. + +`The Philosophy of Mathematics`:title:, by Stephan Korner. + +`The Various and Ingenious Machines of Agostino Ramelli`:title:, by A. Ramelli (!). + +`Experiments in Topology`:title:, by Stephan Barr. diff --git a/math/good books b/math/good books new file mode 100644 index 0000000..bc3efe5 --- /dev/null +++ b/math/good books @@ -0,0 +1,5 @@ +========================================== +Recommended Math Reading +========================================== + +BLANK diff --git a/math/integers b/math/integers new file mode 100644 index 0000000..b946ce8 --- /dev/null +++ b/math/integers @@ -0,0 +1,5 @@ +==================== +Integers +==================== + +definition via Peano's Postulates: meserev 1-4 diff --git a/math/logic b/math/logic new file mode 100644 index 0000000..65ceba9 --- /dev/null +++ b/math/logic @@ -0,0 +1,18 @@ +======================= +Mathematical Logic +======================= + +.. note:: + Incomplete; in progress + +definition of induction: meserev 1-4 + +Proofs +=========== + +Indirect Proof: "reductio ad absurdum" + Show a paradox or impossibility in all cases by assuming the proposition + is false; then the proposition is true. + +Proof by elimination + Propose a complete set of propositions and remove all but one. diff --git a/math/numbers b/math/numbers new file mode 100644 index 0000000..541d174 --- /dev/null +++ b/math/numbers @@ -0,0 +1,54 @@ +======================== +Numbers +======================== + +.. note:: + incomplete + +.. note:: + Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ + +.. contents:: + +*incommensurable* + objects are incommensurable when their ratio isn't rational + +Real Numbers +================== +The *real numbers* are defined via Dedakind cuts in [rudin]_, or [meserve]_ +(1-12). + +Complex Numbers +================== +The *complex numbers* are constructed as an ordered pair of real numbers. + +Algebraic and Transendental Numbers +=============================================== +*Algebraic numbers* are solutions of polynomials, such as x in +:latex:`$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$`, where all a are +real numbers. *Transcendental numbers* are not solutions to any such +polynomials. + +All real numbers are either algebraic or transcendental. + +Some algebraic numbers aren't real (such as :latex:`$i = \sqrt{-1}$`). They +can be rational or irrational. All transcendental numbers are irrational; +some are not real. + +Exersize: is the square root of 5 algebraic or transcendental? + +e +======== +:latex:`$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$` + +Infinities +================== +*aleph-zero* (:latex:`$\aleph_0$`) is the countably infinite set. + +Positive integers, integers, and rational numbers are all countably infinite. + +It is unproven that the real numbers are *aleph-one* (:latex:`$\aleph_1$`). + +.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976 + +.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve. @@ -2,11 +2,46 @@ Sets ==================== -.. warning:: Under progress! +.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ -.. note:: Most of the definitions and notation in the section are based on [rudin]_ +Basics +============= +If every element :latex:`$a \in A$` is also :latex:`$a \in B$`, then we call +A a *subset* of B and write :latex:`$A \subset B$`. If there are elements of B +which are not elements of A, then we call A a *proper subset* of B. -If :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`; otherwise :latex:`$A \neq B$`. +If :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`; +otherwise :latex:`$A \neq B$`. -.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. - McGraw-Hill, 1976 +The null or empty set, which has no elements, is a subset of all others. + +A relation on a space of sets S is something that can be definted as either +true or false (holding or not holding) for any binary pair in S. + +Binary Operators +================== +Binary operators defined on a set apply to any two elements of that set; order +may or may not be important. A set is *closed* with regards to a binary +operator if it contains the result of the binary operator. A set is *uniquely +defined* with regards to a binary operator if the result of the operator on two +elements of the set is unique from the results from all other pairs of +elements. + +Some equivalence relations are +:latex:`$\identity$` (NOTE: = with three lines) (*identity*); +:latex:`$\congruence$` (NOTE: = with tilde on top) (*congruence*; eg of +geometric figures); and +:latex:`$~$` (NOTE: tilde) (*similarity*; eg of geometric figures). + +Some properties of equivalence relations are + +*reflexive* + if :latex:`$a=a$` is true for all a +*symetric* + if :latex:`$a=b$` implies :latex:`$b=a$` +*transitive* + if :latex:`$a=b$` and :latex:`$b=c$` implies :latex:`$a=c$` + +.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976 + +.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve. diff --git a/restructured text cheatsheet b/sheets/rst index e105cba..e105cba 100644 --- a/restructured text cheatsheet +++ b/sheets/rst |