QUASICOMPONENTS AND SHAPE THEORY
|
|
- Noah Harrington
- 5 years ago
- Views:
Transcription
1 Vlume 13, 1988 Pages QUASICOMPONENTS AND SHAPE THEORY by Jerzy Dydak and Manuel Alns Mrón Tplgy Prceedings Web: Mail: Tplgy Prceedings Department f Mathematics & Statistics Auburn Uniersity, Alabama 36849, USA tplg@auburn.edu ISSN: COPYRIGHT c by Tplgy Prceedings. All rights resered.
2 TOPOLOGY PROCEEDINGS Vlume QUASICOMPONENTS AND SHAPE THEORY Jerzy Dydak and Manuel Alns Mrn K. Brsuk [BJ (p. 214) cnstructed a functr A: SH ~ TOP frm the shape categry f cmpacta t the CM tplgical categry such that A(X) is the space f cmpnents f X. Mreer, fr eery fundamental sequence f = {fk,x,y} frm X t Y and fr eery cmpnent C f X, {fk,c,a(c)} is a fundamental sequence frm C t A[fJ(C). Seeral authrs (see [Ba l,2,3 J, [GJ and [SJ) tried t generalize this result t nn-cmpact spaces (lcally cmpact r metrizable). In the present paper we shw that the crrect setting fr pssible generalizatins is the shape analgue f quasicmpnents. Gien a space X let ~X be the set f all quasicmpnents f X with the qutient tplgy. PX: X ~ ~x is the prjectin and S: HTOP ~ Sh dentes the shape functr (see [D-SJ) frm the hmtpy categry t the shape categry. Fr any map a: X ~ Y and~-x E ~X there is a ~nique element f 4Y cntaining a(x O ). That means the map pya: X -+ 6Y factrs thrugh 6X, s there is a cntinuus map ~ (a) : 6X ~ ~y such that 6(a)px = pya.. Mreer, a hmtpic t b ilnplies ~(a) = ~(b) Thus we hae a functr 6: HTOP ~ TOP frm the hmtpy categry t the tplgical categry. If ne wants t get a functr ~: SH ~ TOP frm the shape categry, the natural way is
3 74 Dydak and Mrn t take the Cech system {XU,PUV,COV} f X (XU are the,neres f numerable cerings U f X) and define ~(X) as the inerse limit l~m{~xu'~(pu),cov} (with the inerse limit tplgy) f the system {~Xu'~(PUV),COV}. Therem 1. If eery pen cer f ~X admits an pen refinement cnsisting f mutually disjint sets, then the natural map ~X ~ ~(X) is a hmemrphism. Prf. The map ~X ~ ~(X) is always ne-t-ne. Indeed, if F and G are tw different quasicmpnents f X, then there is an pen-clsed set U in X with U cntaining F and X-U cntaining G. The cering U = {U,X-U} determines Xu such that F and G are sent t tw different pints f ~XU XU. Claim. If C is a clsed set in ~X, then the image f C in ~(X) is equal t rr{~pu(c): U E CO V} n ~(X). Prf f Clatm~ Obiusly the image f C in ~(X) is cntained in rr{~pu(c):u E COV} n 1(x). Suppse {C ' U E COV} E rr{~pu(c): U E COV} n 1(x). Then u {pu-l(c )' u E COV} is a system f pen-clsed sets in X. U If its intersectin has a mutual pint with C, we are dne. S let us assume n {pu -1 (C U )' U E COV} C X-C. Ntice -1 that the sets X-PU (CUl, U E COV, are pen-clsed in X. Thus there is a refinement W f {X - C} U {X - pu-l(c U )' U E COV} cnsisting f mutually disjint pen-clsed sets. By the definitin f the Cech system f X, W E COV, Xw is the nere f Wand PW: X ~ X w is an enumeratin f W.
4 TOPOLOGY PROCEEDINGS Vlume In ur case X is a O-dimensinal cmplex and PW: X + X w w maps each element U f W nt the ertex U f X. Obsere that ~xw = x and C must be a ertex U f X W On the w w -1 ther hand U is cntained in X-PV (C V ' fr sme V. Thus PV(C ) is disjint with C V ' a cntradictin. W Ntice that fr each Xu the set ~Pu(C) the discrete space ~xu (here PU: X + Xu are the prjectins). Since rr{~pu(c): y is clsed in U E COV} is clsed, we infer ~X + ~(X) is clsed. Taking C = X we get that ~X + ~(X) is nt. y Crllary 1. The natural map ~X + y ~(X) is a hmemrphism if ne f the filwing anditins is satisfied: a. ~X is paraampaat and dim~x = 0, b. X is laally ampaat metrizable and eaah ampnent f X is ampaat. Prf. Case a) fllws frm Therem 3 in [E] (p. 278). In case b) X is a tplgical sum f cmpact metrizable spaces (see [Ba 2 ], p. 258). Remark. In [M] (prpsitin 1.4) it is shwn that each quasicmpnent f X is cnnected prided X is nrmal, PX: X + ~X is clsed and ind(~x) =. Therem 2. Suppse f: X + Y is a shape mrphism, Y is paraampaat, dim~y = 0 and Py: Y + ~Y is alsed. If B is a alsed subset f ~Y and A C ~X n 1(f)-1(B), then there exists a unique shape mrphism
5 76 Oydak and Mrn -1-1 f:px (A) ~ Py (B) such that S(j)f O = f S(i)3 where -1-1 i: Px (A) ~ X and j: py (B) ~ Yare inclusins. Therem 2 is a simple cnsequence f the fllwing: Lemma 1. Suppse B is a subset f a paracmpact spaae Y suah that eaah neighbrhd f B in Y cntains an pen-alsed neighbrhd f B in Y. Let {YU,PUV~GOV} 'Y be the Cech system f Y. Fr eaah aering U E COV cnsider the subamplex N(UIB) (equal t the nere f U restricted t B) f Y U and let B U be the unin f all ampnents f pints f N(UIB) in Y U Then the natural pr-hmtqpy map B ~ {B U ' U E COV} satisfies the antinuity anditin. Prf. COV is L~e set f all numerable cerings f Y, Y is the nere N(U) f U and PU: Y ~ u Y is an U enumeratin f U (see [0-5J, pp ). The induced maps B ~ B will be dented by que We need t shw tw U prperties: a. Fr each map f: B ~ K E ANR(M) there is U E :COV and f U : B U ~ K with f hmtpic t fuqu. b. If fu,h U : B U ~. K are maps such that fuqu is hmtpic t huqu' then there is V ~ U with fuqu is hmtpic t huqvu' where q-vu: B-V ~ BU Since we can change K up t hmtpy type we may assume K is a cmplete metric space, and therefre it is an ANE fr paracmpact spaces (see [O-KJ).
6 TOPOLOGY PROCEEDINGS Vlume If f: B ~ K, we extend f t f': V ~ K, where V is an pen-clsed set cntaining B, and then we extend f' t f": Y ~ K. Since Y ~ {Y ' U E A} satisfies the cntinuity U cnditin, there is U E COV and gu: Y ~ K with fn hmu tpic t guqu. If fu,h U : B U ~ Nw take f U = gulb. K are maps such that fuqu is hmtpic t huqu' chse an pen-clsed neighbrhd V f B in Y with qu(v) cntained in BU. By extending the hmtpy between fuqu and huqu we may assume that thse tw maps are hmtpic as maps frm V t K. Let W (ul) u (UIY - V). Define a map s: Y ~ K (t: Y W W ~ K) as fupwu n N(Ul) (gupwu n N(UIV» and cnstant n N(UIY - V). Then spw is hmtpic t tpw' s there is V ~ Wwith spvw is hmtpic t tpvw. we are dne. Since N(Ul) = B W ' A simple cnsequence f Therem 2 is the fllwing: CrZZary. Suppse f: X ~ Y is a shape ismrphism f paracmpaat spaces. If bth maps Px: X ~ ~X, Py: y ~ 6Y are clsed and dim6x = dim6y = 0, then fr eery X E ~X there is a shape equialence fa: X ~ Y ~(f) (X ) such that S(j)f O ~ j: YOc. Yare inczusins. fs(i), where~: XO~ X and Here is a partial cnerse t the abe Crllary: Therem 3. Suppse f: X ~ Y is a azsed map f paracmpaat spaces such that fr eery X E ~X the
7 78 Oydak and Mr6n restrictin f O : X ~ YO = ~(XO) f f is a shape equialence and ~(f): ~(X) ~ ~(Y) is a bijectin. If Py: Y ~ ~Y is clsed and di~y = 0, then f is a shape equialence. Prf. Replacing Y by the mapping cylinder f f we may assume that f is the inclusin f a clsed set X int Y. We need t shw that fr each map a: X ~ K E ANR(M) there is an extensin a': Y ~ K. Since we can change K up t hmtpy type we may assume K is a cmplete metric space, and therefre it is an ANE fr paracm~act spaces (see [O-K]). Fr each X E ~x we can extend alx t YO E ~y cntaining X O Subsequently we extend a frm X t an pen-clsed neighbrhd V f YO in Y. Nw, chse a refinement f {V} cnsisting f mutually disjint pen sets and it is clear hw t define an extensin f a. If a,b: Y ~ K are tw maps such that alx and blx are hmtpic, we prceed similarly as abe t shw that a and b are hmtpic. Remark. Therem 3 is a generalizatin f Therem in [D-5] (p. 62). Therem 4. Suppse X is a paracmpact space such that p : X ~ 6X is clsed and di~x =. If fr eery X X E ~X the defrmatin dimensin def-dim(x ) is Zess than O r equaz t n, then def-dim(x) ~ n. Prf. It suffices t shw that fr eery simplicial cmplex K (with the metric tplgy) any map f: X ~ hmtpic t a map g with g(x) C K(n) K is
8 TOPOLOGY PROCEEDINGS Vlume Claim. If def-dim(y) ~ n, then any map U K(P) x [ g: Y ~ p=l p,) is hmtpic t a map which alues lie in un K(P) x [p 00). p=l ' Prf f Claim. Since the identity map U OO K (p) x [p 00) ~ u K(P) x [p,) p=l w ' p=l is a hmtpy euqia1ence (here K w means K with the weak tplgy) we can factr 9 up t hmtpy thrugh L = U;=l Kw(P) x [p,) (L is cnsidered with the natural CW structure) and using the fact that def-dim(y) < n we can push 9 int L(n) which is cntained in L up K(P) x [p 00) n p=l ' Suppse g: X ~ K. Since the prjectin ~: L' = U;=l K(P) x [p,) ~ K is a hmtpy equialence (~ induces ismrphisms f hmtpy grups at each pint), there is g': X ~ LV with g' ~ TIg. Gien any quasicmpnent X f X, g'lx is hmtpic t g with g (X ) cntained in Ln. Since L' is an ANE fr paracmpact spaces (see [H], p. 63), we can find an pen clsed neighbrhd V f X in X such that g'l is hmtpic t a map w~th alues in Ln. Finally we can find a cering V f X cnsisting f mutually disjint pen sets such that g' I is hmtpic t a map with alues in L fr eery V n in V. Nw, it is clear that g' is hmtpic t a map gil with alues in L. Then g"~ is hmtpic t g and its n alues are in K(n) Remark. pred by S. Nwak [N]. In the cmpact metrizable case Therem 4 was
9 80 Dydak and Mrn Therem 5. Suppse X is a paracmpact space such that PX: X ~ ~X is clsed and di~x =. If F is a clsed cering f X such that PX -1 (px(r» F and eery B E F is mable (unifrmly mable)~ then X is mabze (unifrmly mable). Prf. We will pre nly the unifrmly mable case. The pr~f f the mable case is similar. Let {Xu,PUV,COV} be the Cech system f X. By Lemma 1, the natural pr-hmtpy map B ~ {B ' U E CO'V} satisfies the U cntinuity cnditin fr each B E F. Thus fr each cering U E COV and fr each B E F there is a cering V{B) E COV, V(B) ~ U, and a shape mrphism gb: BV(B) ~ X such that S[PU]gB = S[PV(B)U]. Chse an pen refinement W f {(PV(B» -1 (BV(B»: B E F} cnsisting f mutually disjint pen sets. Nw, we define an pen refinement V f U as fllws: gien W in W we chse B E F with -1 W C (PV{B» (BV{B» and declare the elements f V intersecting W t be precisely W n V{B). Ntice that N(W n V) is cntained in BV(B)' s that we can piece tgether gbis t get g: N(V) = X ~ X with S[PU]g S[PVU]. Remarks. Therem 5 is related t Therem 3.1 f [M]. In [D-S-S] there is an example f a nn-mable metrizable space X such that each f its quasicmpnents is mable. The same example shws that ne cannt drp bth f the hyptheses (px clsed and dim~x = 0) in Therem 4.
10 TOPOLOGY PROCEEDINGS Vlume References [BO] K. Brsuk, Thery f shape~ Plish Scientific Publishers, Warsaw [Ba ] B. J. Ball, Shapes f saturated subsets f cmpacta~ l Cllq. Math. 24 (1974), [Ba ] B. J. Ball, Quasicmpactificatins and shape thery~ 2 Pacific J. Math. 84 (1979), [Ba ] B. J. Ball, Partitining shape-equialent spaces~ 3 Bull. Acad. Pl. Sci. 29 (1981), [D-K] J. Dydak and G. Kzlwski, A generalizatin f the Viet;;'-;{s--Begle therem~ (1988), Prc. Amer. Math. Sc. 102 [D-S-S] [D-S] J. Dydak and J. Segal, Shape thery: An intrductin~ 1978, ' Lecture Ntes in Math. 688, Springer Verlag, J. Dydak, J. Segal and Stanislaw Spiez, A nnmable space with mable cmpnents~ Prceedings f ~he Arner. Math. Sc. (t appear) [E] R. Engelking, Outline f general tplgy~ Nrth [G] Hlland Publishing C., Amsterdam S. Gdlewski, On cmpnents f MANR-spaces~ Fund. Math. 114 (1981), [H] S. T. Hu, Thery f retracts~ WayneuState Uniersity [M] [N] Press, Detrit, M. A. Mrn, Upper semicntinuus decmpsitins and mability in metric spaces~ 35 (1987), Bull. Acad. Pl. Sci. S. Nwak, Sme prperties f fundamentaz dimensin~ Fund. Math. 85 (1974), [S] J. M. R. Sanjurj, On a therem f B. J. Ball~ Acad. Pl. Sci. 33 (1985), Bull. Uniersity f Tennessee Knxille, TN and
11 82 Dydak and M~6n E. T. S. de Ingeniers de Mntes Uniersidad Plitecnica de Madrid Ciudad Uniersitaria Madrid-28040, SPAIN
Homology 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 informationA new Type of Fuzzy Functions in Fuzzy Topological Spaces
IOSR Jurnal f Mathematics (IOSR-JM e-issn: 78-578, p-issn: 39-765X Vlume, Issue 5 Ver I (Sep - Oct06, PP 8-4 wwwisrjurnalsrg A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces Assist Prf Dr Munir Abdul
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 4. Function spaces
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 4 Fall 2014 IV. Functin spaces IV.1 : General prperties Additinal exercises 1. The mapping q is 1 1 because q(f) = q(g) implies that fr all x we have f(x)
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 informationCOVERS OF DEHN FILLINGS ON ONCE-PUNCTURED TORUS BUNDLES
prceedings f the american mathematical sciety Vlume 105, Number 3, March 1989 COVERS OF DEHN FILLINGS ON ONCE-PUNCTURED TORUS BUNDLES MARK D. BAKER (Cmmunicated by Frederick R. Chen) Abstract. Let M be
More informationFINITE BOOLEAN ALGEBRA. 1. Deconstructing Boolean algebras with atoms. Let B = <B,,,,,0,1> be a Boolean algebra and c B.
FINITE BOOLEAN ALGEBRA 1. Decnstructing Blean algebras with atms. Let B = be a Blean algebra and c B. The ideal generated by c, (c], is: (c] = {b B: b c} The filter generated by c, [c), is:
More informationTHE DEVELOPMENT OF AND GAPS IN THE THEORY OF PRODUCTS OF INITIALLY M-COMPACT SPACES
Vlume 6, 1981 Pages 99 113 http://tplgy.auburn.edu/tp/ THE DEVELOPMENT OF AND GAPS IN THE THEORY OF PRODUCTS OF INITIALLY M-COMPACT SPACES by R. M. Stephensn, Jr. Tplgy Prceedings Web: http://tplgy.auburn.edu/tp/
More informationSEMILATTICE STRUCTURES ON DENDRITIC SPACES
Vlume 2, 1977 Pages 243 260 http://tplgy.auburn.edu/tp/ SEMILATTICE STRUCTURES ON DENDRITIC SPACES by T. B. Muenzenberger and R. E. Smithsn Tplgy Prceedings Web: http://tplgy.auburn.edu/tp/ Mail: Tplgy
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 informationMAKING DOUGHNUTS OF COHEN REALS
MAKING DUGHNUTS F CHEN REALS Lrenz Halbeisen Department f Pure Mathematics Queen s University Belfast Belfast BT7 1NN, Nrthern Ireland Email: halbeis@qub.ac.uk Abstract Fr a b ω with b \ a infinite, the
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 informationChapter 2 GAUSS LAW Recommended Problems:
Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating
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 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 information- 6 - UNIQUENESS OF A ZARA GRAPH ON 126 POINTS AND NON-EXISTENCE OF A COMPLETELY REGULAR TWO-GRAPH ON. A. Blokhuis and A.E.
- 6 - UNIQUENESS OF A ZARA GRAPH ON 126 POINTS AND NON-EXISTENCE OF A COMPLETELY REGULAR TWO-GRAPH ON 288 POINTS by A. Blkhuis and A.E. Bruwer VecUc.a.:ted :t J.J. SeA-de.. n :the ec L6-tn 015 11M ftet.ulemen:t.
More informationAbstract. x )-elationj and. Australasian Journal of Combinatorics 3(1991) pp
Abstract x )-elatinj and f this paper frm part f a PhD thesis submitted the authr t the "",,,,,,.,.,,,,,t,, f Lndn. The authr the supprt f the Cmmnwealth :::;c.hjlarsrup Cmmissin. Australasian Jurnal f
More informationOn 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 informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationSEMI STABILITY FACTORS AND SEMI FACTORS. Daniel Hershkowitz and Hans Schneider
Cntemprary Mathematics Vlume 47, 1985 SEMI STABILITY FACTORS AND SEMI FACTORS * Daniel Hershkwitz and Hans Schneider ABSTRACT. A (semistability) factr [semifactr] f a matrix AE r: nn is a psitive definite
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
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 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 informationCONSTRUCTING STATECHART DIAGRAMS
CONSTRUCTING STATECHART DIAGRAMS The fllwing checklist shws the necessary steps fr cnstructing the statechart diagrams f a class. Subsequently, we will explain the individual steps further. Checklist 4.6
More informationSection 6-2: Simplex Method: Maximization with Problem Constraints of the Form ~
Sectin 6-2: Simplex Methd: Maximizatin with Prblem Cnstraints f the Frm ~ Nte: This methd was develped by Gerge B. Dantzig in 1947 while n assignment t the U.S. Department f the Air Frce. Definitin: Standard
More informationLim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)?
THESE ARE SAMPLE QUESTIONS FOR EACH OF THE STUDENT LEARNING OUTCOMES (SLO) SET FOR THIS COURSE. SLO 1: Understand and use the cncept f the limit f a functin i. Use prperties f limits and ther techniques,
More informationON TRANSFORMATIONS OF WIENER SPACE
Jurnal f Applied Mathematics and Stchastic Analysis 7, Number 3, 1994, 239-246. ON TRANSFORMATIONS OF WIENER SPACE ANATOLI V. SKOROKHOD Ukranian Academy f Science Institute f Mathematics Kiev, UKRAINE
More informationCONSTRUCTING BUILDINGS AND HARMONIC MAPS
CONSTRUCTING BUILDINGS AND HARMONIC MAPS LUDMIL KATZARKOV, ALEXANDER NOLL, PRANAV PANDIT, AND CARLOS SIMPSON Happy Birthday Maxim! Abstract. In a cntinuatin f ur previus wrk [17], we utline a thery which
More informationCONSTRUCTING BUILDINGS AND HARMONIC MAPS
CONSTRUCTING BUILDINGS AND HARMONIC MAPS LUDMIL KATZARKOV, ALEXANDER NOLL, PRANAV PANDIT, AND CARLOS SIMPSON Happy Birthday Maxim! Abstract. In a cntinuatin f ur previus wrk [21], we utline a thery which
More informationNAME: Prof. Ruiz. 1. [5 points] What is the difference between simple random sampling and stratified random sampling?
CS4445 ata Mining and Kwledge iscery in atabases. B Term 2014 Exam 1 Nember 24, 2014 Prf. Carlina Ruiz epartment f Cmputer Science Wrcester Plytechnic Institute NAME: Prf. Ruiz Prblem I: Prblem II: Prblem
More informationDepartment of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets
Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0
More informationThe influence of a semi-infinite atmosphere on solar oscillations
Jurnal f Physics: Cnference Series OPEN ACCESS The influence f a semi-infinite atmsphere n slar scillatins T cite this article: Ángel De Andrea Gnzález 014 J. Phys.: Cnf. Ser. 516 01015 View the article
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 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 informationChapter 3: Cluster Analysis
Chapter 3: Cluster Analysis } 3.1 Basic Cncepts f Clustering 3.1.1 Cluster Analysis 3.1. Clustering Categries } 3. Partitining Methds 3..1 The principle 3.. K-Means Methd 3..3 K-Medids Methd 3..4 CLARA
More informationSOME CONSTRUCTIONS OF OPTIMAL BINARY LINEAR UNEQUAL ERROR PROTECTION CODES
Philips J. Res. 39, 293-304,1984 R 1097 SOME CONSTRUCTIONS OF OPTIMAL BINARY LINEAR UNEQUAL ERROR PROTECTION CODES by W. J. VAN OILS Philips Research Labratries, 5600 JA Eindhven, The Netherlands Abstract
More informationUniversity Chemistry Quiz /04/21 1. (10%) Consider the oxidation of ammonia:
University Chemistry Quiz 3 2015/04/21 1. (10%) Cnsider the xidatin f ammnia: 4NH 3 (g) + 3O 2 (g) 2N 2 (g) + 6H 2 O(l) (a) Calculate the ΔG fr the reactin. (b) If this reactin were used in a fuel cell,
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 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 informationSPH3U1 Lesson 06 Kinematics
PROJECTILE MOTION LEARNING GOALS Students will: Describe the mtin f an bject thrwn at arbitrary angles thrugh the air. Describe the hrizntal and vertical mtins f a prjectile. Slve prjectile mtin prblems.
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 informationIntroduction to Spacetime Geometry
Intrductin t Spacetime Gemetry Let s start with a review f a basic feature f Euclidean gemetry, the Pythagrean therem. In a twdimensinal crdinate system we can relate the length f a line segment t the
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 informationModule 4: General Formulation of Electric Circuit Theory
Mdule 4: General Frmulatin f Electric Circuit Thery 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated
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 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 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 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 informationCLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS VIA NECKLACE DIAGRAMS
CLASSIFICATION OF TOTALLY REAL ELLIPTIC LEFSCHETZ FIBRATIONS VIA NECKLACE DIAGRAMS NERMİN SALEPCİ Abstract. We shw that ttally real elliptic Lefschetz fibratins admitting a real sectin are classified by
More informationA Transition to Advanced Mathematics. Mathematics and Computer Sciences Department. o Work Experience, General. o Open Entry/Exit
SECTION A - Curse Infrmatin 1. Curse ID: 2. Curse Title: 3. Divisin: 4. Department: MATH 245 A Transitin t Advanced Mathematics Natural Sciences Divisin Mathematics and Cmputer Sciences Department 5. Subject:
More informationf t(y)dy f h(x)g(xy) dx fk 4 a. «..
CERTAIN OPERATORS AND FOURIER TRANSFORMS ON L2 RICHARD R. GOLDBERG 1. Intrductin. A well knwn therem f Titchmarsh [2] states that if fel2(0, ) and if g is the Furier csine transfrm f/, then G(x)=x~1Jx0g(y)dy
More informationDUAL F-SIGNATURE OF COHEN-MACAULAY MODULES OVER QUOTIENT SURFACE SINGULARITIES
DUL F-SIGNTURE OF COHEN-MCULY MODULES OVER QUOTIENT SURFCE SINGULRITIES YUSUKE NKJIM. INTRODUCTION Thrughut this paper, we suppse that k is an algebraically clsed field f prime characteristic p >. Let
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 informationA NONLINEAR STEADY STATE TEMPERATURE PROBLEM FOR A SEMI-INFINITE SLAB
A NONLINEAR STEADY STATE TEMPERATURE PROBLEM FOR A SEMI-INFINITE SLAB DANG DINH ANG We prpse t investigate the fllwing bundary value prblem: (1) wxx+wyy = 0,0
More information45 K. M. Dyaknv Garsia nrm kfk G = sup zd jfj d z ;jf(z)j = dened riginally fr f L (T m), is in fact an equivalent nrm n BMO. We shall als be cncerned
Revista Matematica Iberamericana Vl. 5, N. 3, 999 Abslute values f BMOA functins Knstantin M. Dyaknv Abstract. The paper cntains a cmplete characterizatin f the mduli f BMOA functins. These are described
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 informationLast Updated: Oct 14, 2017
RSA Last Updated: Oct 14, 2017 Recap Number thery What is a prime number? What is prime factrizatin? What is a GCD? What des relatively prime mean? What des c-prime mean? What des cngruence mean? What
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 informationLecture 13: Markov Chain Monte Carlo. Gibbs sampling
Lecture 13: Markv hain Mnte arl Gibbs sampling Gibbs sampling Markv chains 1 Recall: Apprximate inference using samples Main idea: we generate samples frm ur Bayes net, then cmpute prbabilities using (weighted)
More informationDistributions, spatial statistics and a Bayesian perspective
Distributins, spatial statistics and a Bayesian perspective Dug Nychka Natinal Center fr Atmspheric Research Distributins and densities Cnditinal distributins and Bayes Thm Bivariate nrmal Spatial statistics
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More information1 The limitations of Hartree Fock approximation
Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
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 informationSAMPLE ASSESSMENT TASKS MATHEMATICS SPECIALIST ATAR YEAR 11
SAMPLE ASSESSMENT TASKS MATHEMATICS SPECIALIST ATAR YEAR Cpyright Schl Curriculum and Standards Authrity, 08 This dcument apart frm any third party cpyright material cntained in it may be freely cpied,
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 informationGAUSS' LAW E. A. surface
Prf. Dr. I. M. A. Nasser GAUSS' LAW 08.11.017 GAUSS' LAW Intrductin: The electric field f a given charge distributin can in principle be calculated using Culmb's law. The examples discussed in electric
More informationThe special theory of relativity
The special thery f relatiity The preliminaries f special thery f relatiity The Galilean thery f relatiity states that it is impssible t find the abslute reference system with mechanical eperiments. In
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 informationAPPROXIMATING EMBEDDINGS OF POLYHEDRA IN CODIMENSION THREE(J)
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vlume 170, August 1972 APPROXIMATING EMBEDDINGS OF POLYHEDRA IN CODIMENSION THREE(J) BY J. L. BRYANT ABSTRACT. Let F be a p-dimensinal plyhedrn and let
More informationREADING STATECHART DIAGRAMS
READING STATECHART DIAGRAMS Figure 4.48 A Statechart diagram with events The diagram in Figure 4.48 shws all states that the bject plane can be in during the curse f its life. Furthermre, it shws the pssible
More informationSource Coding and Compression
Surce Cding and Cmpressin Heik Schwarz Cntact: Dr.-Ing. Heik Schwarz heik.schwarz@hhi.fraunhfer.de Heik Schwarz Surce Cding and Cmpressin September 22, 2013 1 / 60 PartI: Surce Cding Fundamentals Heik
More informationThermodynamics and Equilibrium
Thermdynamics and Equilibrium Thermdynamics Thermdynamics is the study f the relatinship between heat and ther frms f energy in a chemical r physical prcess. We intrduced the thermdynamic prperty f enthalpy,
More informationPERSISTENCE DEFINITIONS AND THEIR CONNECTIONS
prceedings f the american mathematical sciety Vlume 109, Number 4, August 1990 PERSISTENCE DEFINITIONS AND THEIR CONNECTIONS H. I. FREEDMAN AND P. MOSON (Cmmunicated by Kenneth R. Meyer) Abstract. We give
More information20 Faraday s Law and Maxwell s Extension to Ampere s Law
Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet
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 informationChapter 9 Vector Differential Calculus, Grad, Div, Curl
Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2-Space and 3-Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields
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 informationDECOMPOSITION OF SYMPLECTIC STRUCTURES
Transnatinal Jurnal f Mathematical Analysis and Applicatins Vl., Issue, 03, Pages 7-44 ISSN 347-9086 Published Online n Nvember 30, 03 03 Jyti Academic Press http://jytiacademicpress.net DECOMPOSITION
More informationLecture 13: Electrochemical Equilibria
3.012 Fundamentals f Materials Science Fall 2005 Lecture 13: 10.21.05 Electrchemical Equilibria Tday: LAST TIME...2 An example calculatin...3 THE ELECTROCHEMICAL POTENTIAL...4 Electrstatic energy cntributins
More information1996 Engineering Systems Design and Analysis Conference, Montpellier, France, July 1-4, 1996, Vol. 7, pp
THE POWER AND LIMIT OF NEURAL NETWORKS T. Y. Lin Department f Mathematics and Cmputer Science San Jse State University San Jse, Califrnia 959-003 tylin@cs.ssu.edu and Bereley Initiative in Sft Cmputing*
More informationIntroduction: A Generalized approach for computing the trajectories associated with the Newtonian N Body Problem
A Generalized apprach fr cmputing the trajectries assciated with the Newtnian N Bdy Prblem AbuBar Mehmd, Syed Umer Abbas Shah and Ghulam Shabbir Faculty f Engineering Sciences, GIK Institute f Engineering
More informationSection I5: Feedback in Operational Amplifiers
Sectin I5: eedback in Operatinal mplifiers s discussed earlier, practical p-amps hae a high gain under dc (zer frequency) cnditins and the gain decreases as frequency increases. This frequency dependence
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 informationENGI 4430 Parametric Vector Functions Page 2-01
ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr
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 informationYou need to be able to define the following terms and answer basic questions about them:
CS440/ECE448 Sectin Q Fall 2017 Midterm Review Yu need t be able t define the fllwing terms and answer basic questins abut them: Intr t AI, agents and envirnments Pssible definitins f AI, prs and cns f
More informationInterference is when two (or more) sets of waves meet and combine to produce a new pattern.
Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme
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 informationWhen a substance heats up (absorbs heat) it is an endothermic reaction with a (+)q
Chemistry Ntes Lecture 15 [st] 3/6/09 IMPORTANT NOTES: -( We finished using the lecture slides frm lecture 14) -In class the challenge prblem was passed ut, it is due Tuesday at :00 P.M. SHARP, :01 is
More informationMore Tutorial at
Answer each questin in the space prvided; use back f page if extra space is needed. Answer questins s the grader can READILY understand yur wrk; nly wrk n the exam sheet will be cnsidered. Write answers,
More informationA Novel Isolated Buck-Boost Converter
vel slated uck-st Cnverter S-Sek Kim *,WOO-J JG,JOOG-HO SOG, Ok-K Kang, and Hee-Jn Kim ept. f Electrical Eng., Seul atinal University f Technlgy, Krea Schl f Electrical and Cmputer Eng., Hanyang University,
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 informationDead-beat controller design
J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Dead-beat cntrller design In sampled data cntrl systems the cntrller is realised by an intelligent device, typically by a PLC (Prgrammable
More informationinitially lcated away frm the data set never win the cmpetitin, resulting in a nnptimal nal cdebk, [2] [3] [4] and [5]. Khnen's Self Organizing Featur
Cdewrd Distributin fr Frequency Sensitive Cmpetitive Learning with One Dimensinal Input Data Aristides S. Galanpuls and Stanley C. Ahalt Department f Electrical Engineering The Ohi State University Abstract
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 informationTree Structured Classifier
Tree Structured Classifier Reference: Classificatin and Regressin Trees by L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stne, Chapman & Hall, 98. A Medical Eample (CART): Predict high risk patients
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 informationInstructions: Show all work for complete credit. Work in symbols first, plugging in numbers and performing calculations last. / 26.
CM ROSE-HULMAN INSTITUTE OF TECHNOLOGY Name Circle sectin: 01 [4 th Lui] 02 [5 th Lui] 03 [4 th Thm] 04 [5 th Thm] 05 [4 th Mech] ME301 Applicatins f Thermdynamics Exam 1 Sep 29, 2017 Rules: Clsed bk/ntes
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 informationSurface and Contact Stress
Surface and Cntact Stress The cncept f the frce is fundamental t mechanics and many imprtant prblems can be cast in terms f frces nly, fr example the prblems cnsidered in Chapter. Hwever, mre sphisticated
More information