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MAT 246 computational
Creation date：2024-05-20 16:55:32
Belonging category：数学mathematics
Tag MAT 246
computational

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MAT 246

Please note:
1. Ideally the PS must be written in the spaces provided, and page by page (a picture or scan of individual pages) to be
uploaded to the appropriate pages of a Crowdmark file (link to be posted.) If you wish, you can write your answers on
separate pieces of paper and upload them properly at each page of the Crowdmark. Please note that any mismatch in
uploading the pages of the solutions may cause in parts of the solutions not getting marked.
2. Please provide your final, polished solutions in the spaces provided. If you have no access to a printer you can write your
solutions on a blank sheet of paper and upload a picture of scan. Remember, it is an art to write a short yet complete
solution; one learns a lot from practicing this art. Please write a complete, even though long solutions. Then inspect and
edit; while editing, ask yourselves:
- am I proving common believes and facts accepted in the literature? If yes, know which facts are common belief and
which ones need an argument.
- can I just quote some of the facts that I tried to prove? This requires a good familiarity with the literature of the
subject (in our case, the textbook and the slides, and earlier questions in the same PS).
- am I introducing too many cases in my proof, and presenting parallel arguments? Change your type of argument if
possible (direct proof versus proof by cases, or proof by contradiction.)
Editing your first complete solution is a very good way of learning. Make sure you have a good solution, then return to
it and reflect on the above questions.
3. Problem set questions contain ideas complementary to the textbook and lecture materials, opportunity for reflection
and deepening on the subject. The level of difficulty of the questions is elementary so that each student has chance to
individually think about the problems. So, please don’t let your “kind" friends take this opportunity away from you!
4. Not all questions will be marked, nor they are all of equal weight, and it is not known in advance which questions will be
marked, and how heavy they will be weighed. Therefore if a question is skipped one may lose a larger portion of the PS
mark than one would expect.
5. To receive full mark on the computational questions all the computation details must be provided. For questions involving
proofs there is no need to prove textbook or “slides" proofs but one can simply quote them. When a theorem is applied
the relevance of it to the proof must be made clear.
6. Collaboration in this work is not allowed. You may discuss the idea of a the solutions only, but each person must write
their own solutions completely independently, without knowing or consulting the structure and details of the argument of
another person.
So please be careful not to share any written hint with your friends, because some people have photographic memory,
and then your solutions may become subject to plagiarism.
7. By participating in this problem set you agree that you are aware of the university regulations regarding plagiarism.
Individual students are expected to organize and write their own solutions independently. During the online teaching,
and while students are connected through email, any assistance to your fellow classmate in the form of a written note
might be directly copied and it may count as a form of plagiarism. Markers are carefully monitoring the solutions, and
any incidence of plagiarism, and all the parties involved will be dealt with according to university regulations. Please
carefully read the item regarding plagiarism in our course outline, and consult the university policies regarding plagiarism
(not knowing the rules is not a good excuse for not obeying them):
http://www.artsci.utoronto.ca/newstudents/transition/academic/plagiarism
Page 1 of 3
MAT 246, PS2 first draft Due: Fri. Mar. 3 Time 10:00 pm. EST
-
On PS1, unfortunately over %10 of the class did not properly acknowledged the statement below. As
a result they lost their PS1 mark. Please note that an acknowledgement is when you sign your name at
the bottom of a statement, as a sign of agreement with what the statement is suggesting.
00 If I sign my name on top of a page, what am I acknowledging?
0. Failure to answer this question may result in a mark of 0 on the entire PS2. And it is not the name,
signature and date, but it is the acknowledgement .. signing some statement that is intended in this part:
Please sign below as an acknowledgment that you have read the cover page ; (unsigned papers shall not be marked.)
Print Name (first, last) : , Signature: Date:
1. (a) We like to solve 51x ≡ 1 (mod 100). Note that 100 is not prime, so we cannot automatically apply
cancellation law, as corollaries of chapter 4 guaranteed. Translate this equation to a Diophantine equation,
apply Euclidean Algorithm to solve it and determine a solution to the original equation.
(b) RSA: read the discussion of RSA on top of page 54, and consider N = 253 and the encrypter E = 9. With
this protocol, a person wants to send the message of M = 13, what would they actually send us (R)? And in
another occasion someone sends us the message 37. What was their intended message M?
2. a) Read Lemma 7.2.9 and its proof. This proof goes against our vision of transition between chapters because it
is using a proof technique from chapter 4! Use Lemma 0 (from the lecture slides week 5 or 6) to give a different
proof (a very short proof) of this lemma that uses gcd(s, u) = 1 (Hint: read lemma 7.2.2).
(b) prove if a and b are integers and m > 1, and a ≡ b (mod m) and that gcd(a,m) = 1, then gcd(b,m) = 1 as
well.
(c) Prove that if a is multiplicative inverse of b in mod m then both a and b are relatively prime to m.
3. Consider two natural numbers a and b that are relatively prime. Define the sets Sa and Sb to be the set of all
natural numbers (between 1 and a, including 1) that are relatively prime to a and b respectively. Similarly let
Sab that is the set of all numbers less than ab which are relatively prime to ab.
a) Prove that any number of the form ar + bs, where s ∈ Sa and r ∈ Sb must be relatively prime to ab.
b) Conclude from part (a) that φ(a)φ(b) ≤ φ(ab).
4. (a) Prove the following statement in two ways, by applying PMI and by applying PCMI: : with the quantifiers
over natural numbers, ∀n∃i∃j such that n = 2i−1(2j − 1), that is, any natural number can be written as the
product of powers 2 and the left over, which is an odd number.
(b) Prove for each n the choice of (i, j) must be unique.
Page 2 of 3
MAT 246, PS2 first draft Due: Fri. Mar. 3 Time 10:00 pm. EST
(c) How are the facts in (a) and (b) used to prove the function f : N×N −→ N defined by f(i, j) = 2i−1(2j − 2)
is a bijection between N × N and N? What is the interpretation of this bijection in terms of the infinite hotel
puzzle?
5. (a) Given that g : A×A −→ A is a bijection, show that the function G : A×A×A −→ A is also a bijection:
G(m,n, k) = g(g(m,n), k) for all (m,n, k) ∈ A×A×A
(b) Use the idea of part (a) and the previous question, and an argument using mathematical induction to prove for
each n ∈ N there must exist a bijection Fn
Fn : N× N× · · · × N −→ N
between the Cartesian product of number of n copies of N and N.
6. Greek Constructions, Golden ratio, Golden rectangle, Golden Triangle, Construction of 36◦:
This question involves reading ahead, some accessible material from chapter 12, and some Greek Constructions. (Please
consult Tutorial Week 6 exercise on Greek Construction and read the slides on Greek Construction, posted on the
LECTURES module.)
(a) Read theorem 12.2.15 and its proof. Replace the proof with a proof using Pythagorean identity applied to
triangle MDC. This proof is shorter and does not need theorems 11.3.9 and 11.3.11.
(b) Golden Ratio: For positive real numbers a > b the ratio ab is said to be Golden ratio, if
a
b =
a+b
a . A rectangle
with sides a and b is a golden rectangle. Denote the golden ratio ab by g, and prove that g must be a root of
the polynomial x2 − x− 1. Determine the value of g.
(c) Golden Rectangle: Start with a precisely drawn square ABEF (read the vertices counterclockwise with A
being the lower left vertex) with each side of length 2. Let O be the mid-point of AB. Show that the line
segment OE is
√
5, and use Greek Construction to extend the original square to a golden rectangle that you call
it ACDF (counterclockwise). Make sure your Greek Construction steps are precisely drawn and documented.
(d) Golden Triangle and Construction of 36◦: Consider the isosceles triangle with the angles 36◦ and 72◦ . Let the
base be of size b, and let the sides be of size a each. Prove that the ratio ab is a golden ratio (as in part (b)).
(Hint: see Theorem 12.4.10.) Use this information, and your Golden rectangle in part (c) to construct a angle of
36 at the vertex A. This consrtruction must be executed in three steps of Greek Construction. Demonstrate
your construction, and indicate your steps and explain which rule was used in each step.
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EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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