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COMP1730/6730 Mathematics

创建日期：2024-05-25 17:17:26

所属分类：数学mathematics

标签： COMP1730

Mathematics

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COMP1730/6730 Mathematics

The assignment is an exercise in elementary mathematics which, however, has interesting
connections with computing and information theory. This project will be suitable
to COMP6730 students, or for Honours students studying Science and Mathematics.
Background knowledge (or additional research) regarding the number representation
in different bases will be needed, as well as elementary geometry of periodic lattices. The
estimated size (including the code comments and docstrings) is approximately 250–300
lines of code. The time effort is about 15 hours.
The problem
Consider a rational number Q = MN , M < N which when expanded in a basis β , is
described by a periodic sequence of digits. The digits, βi are bounded by 0≤ βi < β :
Q =
∞∑
i=1
β i
β i
= (0.β1β2β3 . . .)β , 0≤ β i < β , ∀i,
(do not be confused with notations (0.β1β2 . . .)β — it is a generalisation of the
decimal expansion like (0.123456 . . .)10 where the base value 10 is usually omitted).
The sequence βi is periodic: βi+T = βi starting from some number N : i ≥ N . For
some rationals (like 14 or
17
150 , which are expanded in base 10 as follows: 0.250000 . . .
and 0.1133333333333 . . .), the period T=1. These kind of rationals are not very
interesting. Rationals with longer expansion period, for example, are
7
11
= 0.6363636363 . . . , period T = 2,
or
1
990
= 0.001010101010101010 . . . , period T = 2,
or
1
117
151
= 0.77483443708609271523178 . . . , period T = 75,
One may consider a simple problem of finding the period of expansion of a rational
number in the base β . You may take it as a warming up example. But more
interesting numerical studies emerge when we consider a particular expansion base
β = 4.
When the base of expansion is 4, the digits take 4 possible values 0,1,2,3 which
can be considered as the directions of discrete step walks – “East”, “North”,
“West” and “South”. When we accept the interpretation, each rational number will
be represented by a two-dimensional walk on a plane. Each step of the walk has a
length of 1, and the direction of the n-th step corresponds to the n-th digit of the
decimal in base 4. For example, the number 711 has this expansion in the base 4:
7
11
= 0.2202322023 . . . , period T = 5.
The 100 step walk which this number generates looks like this
Figure 1: “Seven-11” walk
here the directions are chosen as {0: "East", 1: "North", 2: "West", 3:
"South"}, and the colour-coding is used to show the “time” of steps (the further the
2
time step from the start the redder is the colour — “the red shift”, like in Universe
expansion).
Sometimes, walks look like random. For example the walks on Fig. 2 are created by
two pairs of relatively small integers, the length is 250,000 in both cases:
Figure 2: Pseudo-random walk on “small” numbers: 36243600697000000001 and
123456789012
1000000000061 .
3
And sometimes, the walks show surprising regularity. For example, for two integers
M = 1049012271677499437486619280565448601617567358491560876166848380843144358447252875551
62924702775955557045371567931305878324772977202177081818796590637365767487981422801328592
0278610192581409571357487047122902674651513128059541953997504202061380373822338959713391954
N = 1612226962694290912940490066273549214229880755725468512353395718465191353017348814314
01750453996944547935301206438332726709700793305262920303509209736004509554561365966493250
7839146477284016238565137429529453089612268152748875615658076162410788075184599421938774835
the rational Q = MN generates a regular walk, which is also periodic, Fig. 3. (Do not
confuse a periodic walk with a periodic expansion — the walk may not be periodic,
but the expansion always is. For the walk to be periodic, it must return to the point
of origin after the number of step which is integer factor of the expansion period.)
Figure 3: A 440 step base-4 walk on the number Q.
There are more examples of such remarkably regular number walks, and some
mathematical theory behind this phenomenon. But we shall not pursue it further (if
you like, one can learn about it in Ref. 1). Instead, we will build programming tools
to explore these behaviours.
The choice of directions is somewhat arbitrary. For consistency, we shall assume
that the East direction corresponds to the value βi = 0, and that other directions are
chosen in the anti-clockwise order (βi = 1 corresponds to North, and so on).
4
It is interesting to note, that base-4 walks can be also generated using RNA/DNA
sequences (which is a subject of study in Project-3). We cannot say whether such
walks have any significance in the bioinformatics research, though.
Another consideration is about reversing the process of generating a walk from the
number. Ask yourself — how were numbers like Q in the above example found? If
we encode a trajectory which generates a target shape and use it as an expansion,
we will be able to compute the rational number, given by the this expansion. In
other words, if we know the shape (and its trajectory), we can calculate the rational
number.
The programming tasks
This is what you will have to do in your program:
1. Define a function which accepts a 2-tuple comprising a numerator and a
denominator, and a base of the numeral system in which to expand the rational
number, and returns the period of the expansion sequence.
2. Define a function which accepts a 2-tuple of integers, a numerator and denom-
inator representing a rational number, the base value and the positive integer
nsteps, and returns the sequence of the expansion digits of this rational of the
total length nsteps.
3. Using the plotting library matplotlib, create a walk which a rational (defined
by a 2-tuple of integers) generates on a square lattice.
4. A sequence representing a rational number expansion in the base β can be
written as a prefix sequence (in the above example
17
150
10 the prefix was
11) concatenated with a period sequence which is repeated infinitely (in the
same example, it was a single element sequence 3). A prefix can be an empty
sequence, a period sequence can consist of zeros (the example 14 above).
Write a function that is passed a sequence prefix, a sequence period and
a base b, which will return a 2-tuple of mutually prime integers to uniquely
represent a rational. Again using the example of rational 17150 , the asked
function should return (17,150) when the prefix (1,1), the period sequence
(3,) and the base 10 are passed for arguments.
5. Compute rationals which generate sample walks – a square, a circle (adopt a
discrete approximation of your choice), a figure “8”, Ying-Yang or some other
of your own choosing (for example, you can generate Homer Simpson which
should give some idea that the topics we discuss in this assignment are related
to the Fourier analysis).

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 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