THE QUADRATIC AND QUARTIC CHARACTER OF CERTAIN QUADRATIC UNITS I PHILIP A. LEONARD AND KENNETH S. WILLIAMS
|
|
- Eunice Sullivan
- 5 years ago
- Views:
Transcription
1 PACFC JOURNAL OF MATHEMATCS Vl. 7, N., 977 THE QUADRATC AND QUARTC CHARACTER OF CERTAN QUADRATC UNTS PHLP A. LEONARD AND KENNETH S. WLLAMS Let ε m dente the fundamental unit f the real quadratic field Q(Vm). t is ur purpse t evaluate the ratinal quadratic and biquadratic residue symbls f ε m mdul a prime p fr certain values f m. We use the ntatin (εjp) and (ε m /p) 4 thrughut this paper as ratinal quadratic and biquadratic residue symbls, interpreting Vm as an integer mdul p. n 969 Barrucand and Chn [] prved, using the arithmetic f Q(y r^ 9 s that p = c 2 + 8d 2, then " ( L p τ/~2~), that if p = 8n + is prime, Since then a number f similar results have been btained fr certain ther quadratic and quartic symbls using such tls as cycltmy, ratinal biquadratic reciprcity laws, etc. (see Brandler [2], Lehmer [4], [5], [6]). n this paper we apply the ideas f Barrucand and Chn [] in ther biquadratic fields with unique factrizatin, thereby reprving sme knwn results, prving sme cnjectures f E. Lehmer [6] and btaining sme additinal new results. The methd succeeds in the 2 imaginary bicyclic biquadratic fields having class number and which cntain QiV ϊ), QCl/^) r ζ>(τ/~2) as a subfield. t wuld be interesting t knw if similar techniques can be used in the remaining 26 imaginary bicyclic biquadratic fields with class number r t determine ctic symbls. (Fr a cmplete list f the imaginary bicyclic biquadratic fields with class number see Brwn and Parry [3].) We nw sketch the methd used. First the quadratic r quartic symbl under cnsideratin is expressed in terms f the representatin f p by the indefinite frm assciated with the real quadratic subfield f the biquadratic field. This is accmplished using Jacbi's frm f the law f quadratic reciprcity, and the results are given in the table belw. n the case f thse results invlving quartic symbls it is first necessary t bserve that 2e m is a square in the quadratic subfield, and this brings in the symbl (2/p) 4 whse value is well knwn, viz., if p = 8n + is prime s that p a 2 + 6b 2 = c 2 + 8d 2 then (2/p) 4 = (-) 6 = (-l) +d. Next we cnsider a prime
2 2 PHLP A. LEONARD AND KENNETH S. WLLAMS i 2k + <? P A u φ j *Φ cβ u (< 5δ + J dd, 5> u S 8. f i ί % J A A * Si, 6 * 7 <M». T^ <M + CO >
3 THE QUADRATC AND QUARTC CHARACTER 3 l ε 4 S3 β* 5" 3 ih rh a 2 ϋ ϋ.* 7 Jfl* "β -* 3 s + rh "5 + J «t3 ^^ C3 w O w <N c _ * J > > ' as ]
4 4 PHLP A. LEONARD AND KENNETH S. WLLAMS factr f p in the biquadratic field under cnsideratin and by cmputing partial nrms we btain representatins f p by the three quadratic frms assciated with the three quadratic subfields. This infrmatin is then used t derive apprpriate cngruence relatins between the representatins. Finally this infrmatin is cmbined t btain the quadratic r quartic symbl slely in terms f representatins by the psitive definite quadratic frms. f ε m has nrm- we require ( /p) + in rder that (εjp) be unambiguusly defined. Our results are given in the accmpanying table. As the methd f prf is the same fr each field we just give the details fr QO/^lZ, V-ΐ). We have 2ε 4 = 2(5 + 4VΊ4) = (4 + l/ϊϊ) 2, s that (ejp) =. Next as u s ±2λ/ΐiv (mdp) with u, v > we have /4 + T/4\ = /2v\/u + 8tA \ p ) \p A p ) as (A) = = ( - ) (by Jaebi's law) u = ( Sv * ) (as p = Sv 2 (md u + Sv)) Nw let 7r be a prime factr f p in Q{V 2, V 7) s that there are integers A, B, C, D such that with A = B (md 2), C =Z> (md 2), see fr example [7]. Frming relative nrms f π in the three quadratic subfields f Q(i/ 2, τ/^7) (as in []) we can specify: (c, rf) = f (A 2 + 7J5 2-2C 2 - UD 2 ), (AC - 7BD) \4 4
5 THE QUADRATC AND QUARTC CHARACTER 5 (u, v) = (±-(A* + 7B 2 + 2C 2 + UD 2 ), λ(ad + BC)), \4 4 / (, m ) = (^(A C 2-4D 2 ), AB - 2CD)), where ϊ and m are integers such that V + 7m 2 = 4p, ΪΞ m(md 2). Clearly ϊ and m are nt bth dd s that A = B= (md 2). Hence x = (A 2 - B 2 + 2C 2 - UD 2 ), y = ±(AB - 2CD), and SOCΞDΞO (md 2). Setting A = 2A 9 B = 2B 9 C = 2C 9 D = 2D, we btain w = A\ + 75? + 2C\ + UD!, Hence as w is dd exactly ne f A ίf B is even. We just treat the case A t even, B γ dd, say A = 2A 2, B x = 2J5 2 +, as the ther case is exactly similar. We have u^aat + Ί + 2C 2 ί + 6D 2 (md 8), d = D (md 2), y Ξ= A 2 + da (md 2). t nw easily fllws frm the fllwing table that d + y == (md 2) if and nly if u ΞΞ ± (md 8). A 2 (md 2) C^md 2) A(md 2) c? + y{mά 2) u(md 8)
6 6 PHLP A. LEONARD AND KENNETH S. WLLAMS Since (A) = (_!)." and (A) ^ ( +, if u S ± (md8), x P h X u / (-, if w Ξ ±3 (md8), the prf f the result is cmplete. n sme f the ther fields certain cmplexities arse. Fr example if either Q(V r^ϊ) r Q(τ/ 3) is ne f the subfields care had t be taken in identifying the slutins f the crrespnding representatin because f the presence f units Φ ±. Whenever the questin f whether p r 4p is represented by the apprpriate frm there was an increase in the number f cases t be cnsidered. n thse cases in which the parity f n is needed it was handled by relating it t an apprpriate representatin, fr example if p = α = Sn + then n = (a 2 - l)/8 (md 2) was used. Finally we mentin that whenever the number f cases t cnsider became excessive we used Carletn University's Sigma 9 cmputer t treat them. n a frthcming paper we will discuss generalizatins f these results as well as ther results f a similar nature. REFERENCES. P. Barrucand and H. Chn, Nte n primes f the type x 2 +32y 2, class number, and residuacity, J. reine angew. Math., 238 (969), J. Brandler, Residuacity prperties f real quadratic units, J. Number Thery, 5 (973), E. Brwn and C. J. Parry, The imaginary bicyclic biquadratic fields with class number, J. reine angew. Math., 266 (974), E. Lehmer, On the quadratic character f quadratic surds, J. reine angew. Math., 22 (97), # f Q n sme special quartic reciprcity laws, Acta Arith., 2 (972), Q t f Q n the quartic character f quadratic units, J. reine angew. Math., 268/269 (974), K. S. Williams, ntegers f biquadratic fields, Canad. Math. Bull., 3 (97), Received March, 976 and in revised frm July 6, 976. Research by the secnd authr was supprted by Natinal Research Cuncil f Canada Grant N. A ARZONA STATE UNVERSTY AND CARLETON UNVERSTY OTTAWA, ONTARO, CANADA
On small defining sets for some SBIBD(4t - 1, 2t - 1, t - 1)
University f Wllngng Research Online Faculty f Infrmatics - Papers (Archive) Faculty f Engineering and Infrmatin Sciences 992 On small defining sets fr sme SBIBD(4t -, 2t -, t - ) Jennifer Seberry University
More informationAdmissibility Conditions and Asymptotic Behavior of Strongly Regular Graphs
Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department
More informationA proposition is a statement that can be either true (T) or false (F), (but not both).
400 lecture nte #1 [Ch 2, 3] Lgic and Prfs 1.1 Prpsitins (Prpsitinal Lgic) A prpsitin is a statement that can be either true (T) r false (F), (but nt bth). "The earth is flat." -- F "March has 31 days."
More informationBy D. H. Lehmer and Emma Lehmer
MATHEMATCS f cmputatin, VOLUME 8, NUMBER 16, APRL 1974, PAGES 65-635 A New Factrizatin Technique Using Quadratic Frms By D. H. Lehmer and Emma Lehmer Abstract. The paper presents a practical methd fr factring
More informationA Simple Set of Test Matrices for Eigenvalue Programs*
Simple Set f Test Matrices fr Eigenvalue Prgrams* By C. W. Gear** bstract. Sets f simple matrices f rder N are given, tgether with all f their eigenvalues and right eigenvectrs, and simple rules fr generating
More informationPublic Key Cryptography. Tim van der Horst & Kent Seamons
Public Key Cryptgraphy Tim van der Hrst & Kent Seamns Last Updated: Oct 5, 2017 Asymmetric Encryptin Why Public Key Crypt is Cl Has a linear slutin t the key distributin prblem Symmetric crypt has an expnential
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationCHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the
More informationCompetency Statements for Wm. E. Hay Mathematics for grades 7 through 12:
Cmpetency Statements fr Wm. E. Hay Mathematics fr grades 7 thrugh 12: Upn cmpletin f grade 12 a student will have develped a cmbinatin f sme/all f the fllwing cmpetencies depending upn the stream f math
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationThis section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.
Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus
More informationMath 105: Review for Exam I - Solutions
1. Let f(x) = 3 + x + 5. Math 105: Review fr Exam I - Slutins (a) What is the natural dmain f f? [ 5, ), which means all reals greater than r equal t 5 (b) What is the range f f? [3, ), which means all
More informationBootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More informationE Z, (n l. and, if in addition, for each integer n E Z there is a unique factorization of the form
Revista Clmbiana de Matematias Vlumen II,1968,pags.6-11 A NOTE ON GENERALIZED MOBIUS f-functions V.S. by ALBIS In [1] the ncept f a cnjugate pair f sets f psi tive integers is int~dued Briefly, if Z dentes
More informationENSC Discrete Time Systems. Project Outline. Semester
ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding
More informationA solution of certain Diophantine problems
A slutin f certain Diphantine prblems Authr L. Euler* E7 Nvi Cmmentarii academiae scientiarum Petrplitanae 0, 1776, pp. 8-58 Opera Omnia: Series 1, Vlume 3, pp. 05-17 Reprinted in Cmmentat. arithm. 1,
More informationCalculus Placement Review. x x. =. Find each of the following. 9 = 4 ( )
Calculus Placement Review I. Finding dmain, intercepts, and asympttes f ratinal functins 9 Eample Cnsider the functin f ( ). Find each f the fllwing. (a) What is the dmain f f ( )? Write yur answer in
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationLead/Lag Compensator Frequency Domain Properties and Design Methods
Lectures 6 and 7 Lead/Lag Cmpensatr Frequency Dmain Prperties and Design Methds Definitin Cnsider the cmpensatr (ie cntrller Fr, it is called a lag cmpensatr s K Fr s, it is called a lead cmpensatr Ntatin
More informationRevisiting the Socrates Example
Sectin 1.6 Sectin Summary Valid Arguments Inference Rules fr Prpsitinal Lgic Using Rules f Inference t Build Arguments Rules f Inference fr Quantified Statements Building Arguments fr Quantified Statements
More informationIntroduction to Smith Charts
Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary
More information8 th Grade Math: Pre-Algebra
Hardin Cunty Middle Schl (2013-2014) 1 8 th Grade Math: Pre-Algebra Curse Descriptin The purpse f this curse is t enhance student understanding, participatin, and real-life applicatin f middle-schl mathematics
More informationOVERVIEW Properties of Similarity & Similarity Criteria G.SRT.3
OVRVIW Prperties f Similarity & Similarity riteria G.SRT.3 G.SRT.3 Use the prperties f similarity transfrmatins t establish the criterin fr tw triangles t be similar. This bjective has been included in
More informationHomology groups of disks with holes
Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More informationChapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms
Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical
More informationAn Introduction to Complex Numbers - A Complex Solution to a Simple Problem ( If i didn t exist, it would be necessary invent me.
An Intrductin t Cmple Numbers - A Cmple Slutin t a Simple Prblem ( If i didn t eist, it wuld be necessary invent me. ) Our Prblem. The rules fr multiplying real numbers tell us that the prduct f tw negative
More informationPreparation work for A2 Mathematics [2017]
Preparatin wrk fr A2 Mathematics [2017] The wrk studied in Y12 after the return frm study leave is frm the Cre 3 mdule f the A2 Mathematics curse. This wrk will nly be reviewed during Year 13, it will
More informationRevision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax
.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical
More informationThe Equation αsin x+ βcos family of Heron Cyclic Quadrilaterals
The Equatin sin x+ βcs x= γ and a family f Hern Cyclic Quadrilaterals Knstantine Zelatr Department Of Mathematics Cllege Of Arts And Sciences Mail Stp 94 University Of Tled Tled,OH 43606-3390 U.S.A. Intrductin
More informationMath Foundations 20 Work Plan
Math Fundatins 20 Wrk Plan Units / Tpics 20.8 Demnstrate understanding f systems f linear inequalities in tw variables. Time Frame December 1-3 weeks 6-10 Majr Learning Indicatrs Identify situatins relevant
More informationPattern Recognition 2014 Support Vector Machines
Pattern Recgnitin 2014 Supprt Vectr Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recgnitin 1 / 55 Overview 1 Separable Case 2 Kernel Functins 3 Allwing Errrs (Sft
More informationOF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
U. S. FOREST SERVICE RESEARCH PAPER FPL 50 DECEMBER U. S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY OF SIMPLY SUPPORTED PLYWOOD PLATES UNDER COMBINED EDGEWISE BENDING AND COMPRESSION
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More information37 Maxwell s Equations
37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut
More informationA Few Basic Facts About Isothermal Mass Transfer in a Binary Mixture
Few asic Facts but Isthermal Mass Transfer in a inary Miture David Keffer Department f Chemical Engineering University f Tennessee first begun: pril 22, 2004 last updated: January 13, 2006 dkeffer@utk.edu
More informationA Comparison of Methods for Computing the Eigenvalues and Eigenvectors of a Real Symmetric Matrix. By Paul A. White and Robert R.
A Cmparisn f Methds fr Cmputing the Eigenvalues and Eigenvectrs f a Real Symmetric Matrix By Paul A. White and Rbert R. Brwn Part I. The Eigenvalues I. Purpse. T cmpare thse methds fr cmputing the eigenvalues
More informationLHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationA little noticed right triangle
A little nticed right triangle Knstantine Hermes Zelatr Department f athematics Cllege f Arts and Sciences ail Stp 94 University f Tled Tled, OH 43606-3390 U.S.A. A little nticed right triangle. Intrductin
More informationALE 21. Gibbs Free Energy. At what temperature does the spontaneity of a reaction change?
Name Chem 163 Sectin: Team Number: ALE 21. Gibbs Free Energy (Reference: 20.3 Silberberg 5 th editin) At what temperature des the spntaneity f a reactin change? The Mdel: The Definitin f Free Energy S
More informationA Matrix Representation of Panel Data
web Extensin 6 Appendix 6.A A Matrix Representatin f Panel Data Panel data mdels cme in tw brad varieties, distinct intercept DGPs and errr cmpnent DGPs. his appendix presents matrix algebra representatins
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationFunction notation & composite functions Factoring Dividing polynomials Remainder theorem & factor property
Functin ntatin & cmpsite functins Factring Dividing plynmials Remainder therem & factr prperty Can d s by gruping r by: Always lk fr a cmmn factr first 2 numbers that ADD t give yu middle term and MULTIPLY
More information(2) Even if such a value of k was possible, the neutrons multiply
CHANGE OF REACTOR Nuclear Thery - Curse 227 POWER WTH REACTVTY CHANGE n this lessn, we will cnsider hw neutrn density, neutrn flux and reactr pwer change when the multiplicatin factr, k, r the reactivity,
More informationECE 2100 Circuit Analysis
ECE 00 Circuit Analysis Lessn 6 Chapter 4 Sec 4., 4.5, 4.7 Series LC Circuit C Lw Pass Filter Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 00 Circuit Analysis Lessn 5 Chapter 9 &
More information7 TH GRADE MATH STANDARDS
ALGEBRA STANDARDS Gal 1: Students will use the language f algebra t explre, describe, represent, and analyze number expressins and relatins 7 TH GRADE MATH STANDARDS 7.M.1.1: (Cmprehensin) Select, use,
More informationA crash course in Galois theory
A crash curse in Galis thery First versin 0.1 14. september 2013 klkken 14:50 In these ntes K dentes a field. Embeddings Assume that is a field and that : K! and embedding. If K L is an extensin, we say
More information(Communicated at the meeting of September 25, 1948.) Izl=6 Izl=6 Izl=6 ~=I. max log lv'i (z)1 = log M;.
Mathematics. - On a prblem in the thery f unifrm distributin. By P. ERDÖS and P. TURÁN. 11. Ommunicated by Prf. J. G. VAN DER CORPUT.) Cmmunicated at the meeting f September 5, 1948.) 9. Af ter this first
More informationPHYS 314 HOMEWORK #3
PHYS 34 HOMEWORK #3 Due : 8 Feb. 07. A unifrm chain f mass M, lenth L and density λ (measured in k/m) hans s that its bttm link is just tuchin a scale. The chain is drpped frm rest nt the scale. What des
More informationCHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More informationCHAPTER 2 Algebraic Expressions and Fundamental Operations
CHAPTER Algebraic Expressins and Fundamental Operatins OBJECTIVES: 1. Algebraic Expressins. Terms. Degree. Gruping 5. Additin 6. Subtractin 7. Multiplicatin 8. Divisin Algebraic Expressin An algebraic
More informationYear 5 End of Year Expectations Reading, Writing and Maths
Year 5 End f Year Expectatins Reading, Writing and Maths Year 5 Reading Wrd reading Apply their grwing knwledge f rt wrds, prefixes and suffixes (mrphlgy and etymlgy), as listed in Appendix 1 f the Natinal
More informationPhysics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018
Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and
More informationANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels
ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1
More informationREPRESENTATIONS OF sp(2n; C ) SVATOPLUK KR YSL. Abstract. In this paper we have shown how a tensor product of an innite dimensional
ON A DISTINGUISHED CLASS OF INFINITE DIMENSIONAL REPRESENTATIONS OF sp(2n; C ) SVATOPLUK KR YSL Abstract. In this paper we have shwn hw a tensr prduct f an innite dimensinal representatin within a certain
More informationPLEASURE TEST SERIES (XI) - 07 By O.P. Gupta (For stuffs on Math, click at theopgupta.com)
A Cmpilatin By : OP Gupta (WhatsApp @ +9-9650 50 80) Fr mre stuffs n Maths, please visit : wwwtheopguptacm Time Allwed : 80 Minutes Max Marks : 00 SECTION A Questin numbers 0 t 0 carry mark each x x 5
More informationOn Topological Structures and. Fuzzy Sets
L - ZHR UNIVERSIT - GZ DENSHIP OF GRDUTE STUDIES & SCIENTIFIC RESERCH On Tplgical Structures and Fuzzy Sets y Nashaat hmed Saleem Raab Supervised by Dr. Mhammed Jamal Iqelan Thesis Submitted in Partial
More informationCambridge Assessment International Education Cambridge Ordinary Level. Published
Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid
More informationSupport-Vector Machines
Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material
More informationMODULAR DECOMPOSITION OF THE NOR-TSUM MULTIPLE-VALUED PLA
MODUAR DECOMPOSITION OF THE NOR-TSUM MUTIPE-AUED PA T. KAGANOA, N. IPNITSKAYA, G. HOOWINSKI k Belarusian State University f Infrmatics and Radielectrnics, abratry f Image Prcessing and Pattern Recgnitin.
More informationThermodynamics Partial Outline of Topics
Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)
More informationB. Definition of an exponential
Expnents and Lgarithms Chapter IV - Expnents and Lgarithms A. Intrductin Starting with additin and defining the ntatins fr subtractin, multiplicatin and divisin, we discvered negative numbers and fractins.
More informationKepler's Laws of Planetary Motion
Writing Assignment Essay n Kepler s Laws. Yu have been prvided tw shrt articles n Kepler s Three Laws f Planetary Mtin. Yu are t first read the articles t better understand what these laws are, what they
More informationLecture 6: Phase Space and Damped Oscillations
Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:
More informationPreparation work for A2 Mathematics [2018]
Preparatin wrk fr A Mathematics [018] The wrk studied in Y1 will frm the fundatins n which will build upn in Year 13. It will nly be reviewed during Year 13, it will nt be retaught. This is t allw time
More informationLesson Plan. Recode: They will do a graphic organizer to sequence the steps of scientific method.
Lessn Plan Reach: Ask the students if they ever ppped a bag f micrwave ppcrn and nticed hw many kernels were unppped at the bttm f the bag which made yu wnder if ther brands pp better than the ne yu are
More informationNOTE ON APPELL POLYNOMIALS
NOTE ON APPELL POLYNOMIALS I. M. SHEFFER An interesting characterizatin f Appell plynmials by means f a Stieltjes integral has recently been given by Thrne. 1 We prpse t give a secnd such representatin,
More informationPerturbation approach applied to the asymptotic study of random operators.
Perturbatin apprach applied t the asympttic study f rm peratrs. André MAS, udvic MENNETEAU y Abstract We prve that, fr the main mdes f stchastic cnvergence (law f large numbers, CT, deviatins principles,
More informationA Correlation of. to the. South Carolina Academic Standards for Mathematics Precalculus
A Crrelatin f Suth Carlina Academic Standards fr Mathematics Precalculus INTRODUCTION This dcument demnstrates hw Precalculus (Blitzer), 4 th Editin 010, meets the indicatrs f the. Crrelatin page references
More informationMathematics and Computer Sciences Department. o Work Experience, General. o Open Entry/Exit. Distance (Hybrid Online) for online supported courses
SECTION A - Curse Infrmatin 1. Curse ID: 2. Curse Title: 3. Divisin: 4. Department: 5. Subject: 6. Shrt Curse Title: 7. Effective Term:: MATH 70S Integrated Intermediate Algebra Natural Sciences Divisin
More informationRegents Chemistry Period Unit 3: Atomic Structure. Unit 3 Vocabulary..Due: Test Day
Name Skills: 1. Interpreting Mdels f the Atm 2. Determining the number f subatmic particles 3. Determine P, e-, n fr ins 4. Distinguish istpes frm ther atms/ins Regents Chemistry Perid Unit 3: Atmic Structure
More informationFinite Automata. Human-aware Robo.cs. 2017/08/22 Chapter 1.1 in Sipser
Finite Autmata 2017/08/22 Chapter 1.1 in Sipser 1 Last time Thery f cmputatin Autmata Thery Cmputability Thery Cmplexity Thery Finite autmata Pushdwn autmata Turing machines 2 Outline fr tday Finite autmata
More informationKeysight Technologies Understanding the Kramers-Kronig Relation Using A Pictorial Proof
Keysight Technlgies Understanding the Kramers-Krnig Relatin Using A Pictrial Prf By Clin Warwick, Signal Integrity Prduct Manager, Keysight EEsf EDA White Paper Intrductin In principle, applicatin f the
More informationWRITING THE REPORT. Organizing the report. Title Page. Table of Contents
WRITING THE REPORT Organizing the reprt Mst reprts shuld be rganized in the fllwing manner. Smetime there is a valid reasn t include extra chapters in within the bdy f the reprt. 1. Title page 2. Executive
More informationMedium Scale Integrated (MSI) devices [Sections 2.9 and 2.10]
EECS 270, Winter 2017, Lecture 3 Page 1 f 6 Medium Scale Integrated (MSI) devices [Sectins 2.9 and 2.10] As we ve seen, it s smetimes nt reasnable t d all the design wrk at the gate-level smetimes we just
More informationEric Klein and Ning Sa
Week 12. Statistical Appraches t Netwrks: p1 and p* Wasserman and Faust Chapter 15: Statistical Analysis f Single Relatinal Netwrks There are fur tasks in psitinal analysis: 1) Define Equivalence 2) Measure
More informationSUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical model for microarray data analysis
SUPPLEMENTARY MATERIAL GaGa: a simple and flexible hierarchical mdel fr micrarray data analysis David Rssell Department f Bistatistics M.D. Andersn Cancer Center, Hustn, TX 77030, USA rsselldavid@gmail.cm
More informationRangely RE 4 Curriculum Development 5 th Grade Mathematics
Unit Title Dctr We Still Need t Operate... Length f Unit 12 weeks Fcusing Lens(es) Inquiry Questins (Engaging Debatable): Structure Systems Standards and Grade Level Expectatins Addressed in this Unit
More informationCOASTAL ENGINEERING Chapter 2
CASTAL ENGINEERING Chapter 2 GENERALIZED WAVE DIFFRACTIN DIAGRAMS J. W. Jhnsn Assciate Prfessr f Mechanical Engineering University f Califrnia Berkeley, Califrnia INTRDUCTIN Wave diffractin is the phenmenn
More informationThe blessing of dimensionality for kernel methods
fr kernel methds Building classifiers in high dimensinal space Pierre Dupnt Pierre.Dupnt@ucluvain.be Classifiers define decisin surfaces in sme feature space where the data is either initially represented
More informationMODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards:
MODULE FOUR This mdule addresses functins SC Academic Standards: EA-3.1 Classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph. EA-3.2 Use
More informationPhysics 2010 Motion with Constant Acceleration Experiment 1
. Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin
More informationDepartment: MATHEMATICS
Cde: MATH 022 Title: ALGEBRA SKILLS Institute: STEM Department: MATHEMATICS Curse Descriptin: This curse prvides students wh have cmpleted MATH 021 with the necessary skills and cncepts t cntinue the study
More informationMath 9 Year End Review Package. (b) = (a) Side length = 15.5 cm ( area ) (b) Perimeter = 4xside = 62 m
Math Year End Review Package Chapter Square Rts and Surface Area KEY. Methd #: cunt the number f squares alng the side ( units) Methd #: take the square rt f the area. (a) 4 = 0.7. = 0.. _Perfect square
More informationP, A. LEONARD AND I(. S. \YILLI.\MS
P, A. LEONARD AND I(. S. \YILLI.\MS P..4. LEONARD N11 K. S. \VILL.IAMS T.4RLE (part 2) \ I T T I I = I - I --.'ri~l. I, = Idrqest odd dlvlior of 11-111). I = larpcit odd dl\l\or ot 11-21i11. * = ci~njectrlred
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCurseWare http://cw.mit.edu 2.161 Signal Prcessing: Cntinuus and Discrete Fall 2008 Fr infrmatin abut citing these materials r ur Terms f Use, visit: http://cw.mit.edu/terms. Massachusetts Institute
More informationA New Evaluation Measure. J. Joiner and L. Werner. The problems of evaluation and the needed criteria of evaluation
III-l III. A New Evaluatin Measure J. Jiner and L. Werner Abstract The prblems f evaluatin and the needed criteria f evaluatin measures in the SMART system f infrmatin retrieval are reviewed and discussed.
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 4. Function spaces
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 4 Fall 2008 IV. Functin spaces IV.1 : General prperties (Munkres, 45 47) Additinal exercises 1. Suppse that X and Y are metric spaces such that X is cmpact.
More informationTHE FINITENESS OF THE MAPPING CLASS GROUP FOR ATOROIDAL 3-MANIFOLDS WITH GENUINE LAMINATIONS
j. differential gemetry 50 (1998) 123-127 THE FINITENESS OF THE MAPPING CLASS GROUP FOR ATOROIDAL 3-MANIFOLDS WITH GENUINE LAMINATIONS DAVID GABAI & WILLIAM H. KAZEZ Essential laminatins were intrduced
More informationWeathering. Title: Chemical and Mechanical Weathering. Grade Level: Subject/Content: Earth and Space Science
Weathering Title: Chemical and Mechanical Weathering Grade Level: 9-12 Subject/Cntent: Earth and Space Science Summary f Lessn: Students will test hw chemical and mechanical weathering can affect a rck
More informationRelationships Between Frequency, Capacitance, Inductance and Reactance.
P Physics Relatinships between f,, and. Relatinships Between Frequency, apacitance, nductance and Reactance. Purpse: T experimentally verify the relatinships between f, and. The data cllected will lead
More information11. DUAL NATURE OF RADIATION AND MATTER
11. DUAL NATURE OF RADIATION AND MATTER Very shrt answer and shrt answer questins 1. Define wrk functin f a metal? The minimum energy required fr an electrn t escape frm the metal surface is called the
More informationStrategy and Game Theory: Practice Exercises with Answers, Errata in First Edition, Prepared on December 13 th 2016
Strategy and Game Thery: Practice Exercises with Answers, by Felix Munz-Garcia and Daniel Tr-Gnzalez Springer-Verlag, August 06 Errata in First Editin, Prepared n December th 06 Chapter Dminance Slvable
More informationHomework #7. True False. d. Given a CFG, G, and a string w, it is decidable whether w ε L(G) True False
Hmewrk #7 #1. True/ False a. The Pumping Lemma fr CFL s can be used t shw a language is cntext-free b. The string z = a k b k+1 c k can be used t shw {a n b n c n } is nt cntext free c. The string z =
More informationComprehensive Exam Guidelines Department of Chemical and Biomolecular Engineering, Ohio University
Cmprehensive Exam Guidelines Department f Chemical and Bimlecular Engineering, Ohi University Purpse In the Cmprehensive Exam, the student prepares an ral and a written research prpsal. The Cmprehensive
More informationCompressibility Effects
Definitin f Cmpressibility All real substances are cmpressible t sme greater r lesser extent; that is, when yu squeeze r press n them, their density will change The amunt by which a substance can be cmpressed
More information