Homology groups of disks with holes
|
|
- Kory Clark
- 6 years ago
- Views:
Transcription
1 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. Let S j dente the bundary f E j, and let F = D n j IntE j. Then the fllwing hld: (i) The hmlgy grups H q (F ) are zer if q 0, n 1, and H 0 (F ) = Z. (ii) In the remaining dimensin we have H n 1 (F ) = Z k, and the inclusin induced mappings H n 1 (S j ) H n 1 (F ) send generatrs f the dmains int a set f free generatrs fr the cdmain. (iii) If we define standard generatrs fr H n 1 (S j ) by taking the images f the standard generatr fr H n 1 (S n 1 ) under the cannial hmemrphims S n 1 S j, then the image f the geeratr fr H n 1 (S n 1 ) in H n 1 (F ) is equal t the sums f the standard free generatrs fr H n 1 (F ). Gemetrically, F is a disk with k hles; a picture f ne example is included n the last page f this dcument. The standard hmemrphisms S n 1 S j arise frm the hmemrphisms frm R n t itself which send v R n t (r j v) + p j, where r j is the radius f E j. Prf f the therem The first step is t replace F by an pen set. CLAIM 1. The clsed set F is a strng defrmatin retract f R n {p 1,, p k }. Prf f Claim 1. Observe that R n {p 1,, p k } is the unin f F with the punctured clsed disks E j {p j }, and the intersectin is j S j. Therefre it is enugh t shw that fr each j the sphere S j is a strng defrmatin retract f E j {p j }. We can cnstruct the retractins E j {p j } S j by the standard frmula ρ j (v) = p j + r j v p j (v p j) fr pushing pints int the bundary radially, and if ϕ j : S j E j {p j } is the inclusin mapping then ϕ ρ j is hmtpic t the identity by a straight line hmtpy. CLAIM 2. D n. Prf f Claim 2. disk D n R n. If D R n is a clsed metric disk, then (R n ) D is hmemrphic t the interir f The first step is t reduce the prf t the familiar case where D is the unit If the result is true when D = D n, then it is als true fr every clsed disk E centered at the rigin, fr the hmemrphism M r : R n R n defined by v r v (where v > 0) sends D n t the disk f radius r, and fr general reasns it extends t a hmemrphism f the ne pint cmpactificatin (R n ). If r is the radius f E, then this hmemrphism sends D n t E and hence als sends the cmplement f D n hmemrphically t the cmplement f E. Next, if the result is true fr every clsed disk E centered at the rigin, then it is true fr all clsed metric disks D, fr if p is the center f D, then T (v) = v + p is a hmemrphism f R n t itself and hence extends t a hmemrphism frm (R n ) t itself. If we chse E t have the 1
2 same radius as D, then this hmemrphism maps D t E and hence als maps the cmplement f D t the cmplement f E. Finally, we have t shw the result is true fr D n. Let G R n be the set f all vectrs v such that v 1. We claim that G is hmemrphic t D n {0}; an explicit hmemrphism is given by sending v t v 2 v, s that the image f v pints in the same directin but has length v 1. If we extend this hmemrphism t ne pint cmpactificatins and nte that the ne pint cmpactificatin f D n {0} is hmemrphic t D n, we btain a hmemrphism frm G t D n such that the unit sphere is sent t itself. Taking cmplements f the unit sphere, we see that G S n 1 = (R n ) D n is hmemrphic t the interir f D n. STEP 3. Cmputatin f H (R n {p 1,, p k }) fr n 2. Let U j dente the interir f the disk E j. Then by excisin we have H (R n, R n {p 1,, p k }) = H ( j U j, j U j {p j }) = j H (U j, U j {p j }) where the secnd ismrphism hlds because the hmlgy f a space splits int the direct sum f the hmlgy grups f its arc cmpnents. These relative grups are Z in dimensin n and zer therwise, and since n 2 the result in this case fllws because the lng exact hmlgy sequence f (R n, R n {p 1,, p k }) yields ismrphisms frm H q+1 (R n, R n {p 1,, p k }) t H q (R n {p 1,, p k }) if q > 0 and frm H 0 f the latter t H 0 (R n ) = Z. Befre prceeding t the final step, we shall discuss the cnstructin f cannical generatrs in mre detail. Fr ur purpses it will suffice t begin by taking a cannical generatr fr H n 1 (bbr n, R n = {0}) = Z; there are ways f chsing such generatrs fr all values f n cannically, but we shall nt try t explain hw this can be dne. Cnsider the fllwing cmmutative diagram, in which p Int D n and we identify S n with the ne pint cmpactificatin f R n : H q 1 (S n {p}) H q 1 (S n D n ) H q 1 (S n {0}) H q (S n, S n {p}) H q (S n, S n D n ) H q (S n, S n {0}) H q (R n, R n {p}) H q (R n, R n D n ) H q (R n, R n {0}) By Claim 2 we knw that S n D n is cntractible, and we als knw that S n {p} is cntractible fr all p in the interir f D n by the symmetry prperties f S n and the fact that S n {v} = R n if v is the pint at infinity. Therefre the hmmrphisms in the first rw f the diagram are ismrphisms. Next, the vertical arrws frm the secnd rw t the first are the bundary hmmrphisms in lng exact sequences f pairs, and therefre a Five Lemma argument shws that the hmmrphisms in the secnd rw are als ismrphisms. Finally, the vertical arrws frm the third rw t the secnd are excisin ismrphisms, and therefre the hmmrphisms in the third rw are als ismrphisms. We can then use the third rw t define a cannical generatr fr H n (R n, R n {p}) by taking the class crrespnding t the chsen generatr fr H n (R n, R n {0}). STEP 4. Cmputatin f the image f H n (R n, R n {0}) = Z in H n (R n, R n {p 1,, p k }) = Z k fr n 2. 2
3 By the splitting result mentined earlier, it suffices t cnsider the maps H n (R n, R n {0}) H n (R n, R n {p j }) fr each j, and the preceding discussin shws that these maps are ismrphisms which preserve cannical generatrs. Therefre the image f H n (R n, R n {0}) = Z in H n (R n, R n {p 1,, p k }) = Z k is merely the sum f the cannical free generatrs r the cdmain. STEP 5. Cmputatin f the image f H n 1 (S n 1 ) = Z in H n 1 (F ) = Z k fr n 2. Fr each j such that 1 j k, let F j = D n U j, where U j is the small pen disk centered at p j. We shall begin by analyzing a cmmutative diagram which is related t the previus ne: H n (R n, R n {p 1,, p k }) = = j H n (R n, R n {p j }) = H n 1 (R n {p 1,, p k }) j H n 1 (R n {p j }) = = H n 1 (F ) j H n 1 (F j ) The arrw in the first rw is an ismrphism by excisin, the arrws frm the first t secnd rws are ismrphisms by the lng exact hmlgy sequences fr the pairs (the adjacent terms in each case are psitive dimensinal hmlgy grups f R n ), and the arrws frm the third t secnd rws are ismrphisms by Step 1. By the direct sum decmpsitins n the right, it suffices t analyze the image in hmlgy when we are nly remving the pint p j r the pen disk U j centered at p j, where 1 j k. At this pint we need t be careful abut chsing the right signs fr ur free generatrs f hmlgy grups, especially in view f the applicatin we have in mind. If Σ is a sphere f radius r centered at p R n, we take the hmemrphism S n 1 Σ cnstructed in Step 2: First stretch r shrink the sphere S n 1 f radius 1 centered at 0 t a cncentric sphere f radius r, and then map this t the crrespnding sphere centered at p via the translatin v v +p. Then by the cmments in the preceding paragraph we shall have prved the prpsitin if we can shw the fllwing: Let S 1 R n be the sphere f radius a centered at p, suppse that the disk it bunds is cntained in the interir f the disk f radius b centered at sme pint q, and let S 2 dente the bundary sphere f that disk. Let f 1 : S n 1 S 1 and f : S n 1 S 2 be the hmemrphisms given as abve. Chse a generatr ω f H n 1 (S n 1 ). Then the images f f 1 (ω) and f 2 (ω) in R n {p} are equal. T prve this, let S 3 be the sphere f radius b centered at p, and let j 1 and j 3 dente the inclusins int R n = {p}. Then j 3 f 3 j 1 f 1 by the radial stretching hmtpy sending (x, t) t (1 t) f 1 (x) + t f 3 (x). Therefre f 1 (ω) = f 3 (ω). By definitin we als have f 2 (x) = f 3 (x) + q p; if j 2 is th inclusin f S 2 in R n {p}, it will suffice t prve that j 3 f 2 j 2 f 2, and this will fllw if the image f the straight line hmtpy H(x, t) = f 3 (x) + (1 t)(q p) is cntained in R n {p}. Since q p = d and a is the radius f the disk bunded by S 1, the cnditin that ne sphere is cntained in the interir f the pen disk bunded by the ther means that d + a < b (Prf: If w is chsen s that w p is a negative multiple f q p and w p = a, then w S 1, s that w is als in the pen disk bunded by S 2 and therefre b > w q = w p + q p = a + d). We need t shw that x = b implies that H(x, t) p, r equivalently that x = b implies that H(x, t) p > 0. But we have H(x, t) p = f 3 (x) p + (1 t) (q p) 3
4 which means that if 0 t 1 (hence als 0 1 t 1) then H(x, t) p f 3 (x) p (1 t) q p > b (1 t) d which is what we wanted t prve. b d > a > 0 A degree frmula We shall use the therem t prve an abstract, multidimensinal versin f a result which plays a key rle in cmplex analysis when n = 2. COROLLARY. Let F be as in the therem, and suppse that we are given a cntinuus mapping g : F R n {0}, and fr each j let E j be the subdisk in D n whse interir is remved t frm F. Let h j : S n 1 F be the cmpsite f the standard hmemrphism S n 1 Bdy E j with the inclusin f Bdy E j in F. Then we have a summatin frmula deg (g S n 1 ) = j deg (g h j ). Sectin 4.5 f Ahlfrs, Cmplex Analysis (Third Editin), describes implicatns f this result fr cmplex functin thery. Prf f the crllary. Let ω be the standard generatr f H n 1 (S n 1 ) described in Step 5, and let h 0 : S n 1 F be the inclusin mapping. Then by the therem we have h 0 (ω) = j h j (ω) and if we apply g t bth sides we btain a similar identity with h j replaced by g h j = (g h j ) fr all j. By the definitin f degree we knw that the image f ω under the latter map is equal t the degree f g h j times ω if j > 0, and if h = 0 then the image f ω is equal t the degree f g S n 1 times ω. 4
5 Drawing f a disk with hles The fllwing is a picture f a typical set F satisfying the cnditins in the main therem. Nte that the hles may be irregularly distributed thrughut the disk and that the radii f the hles may als differ. Als, in general the center f the circle might nt be ne f the deleted center pints. In the 2 dimensinal case, the main therem implies that uter circle with a cunterclckwise parametrizatin is hmlgus t the sum f the inner circles with cunterclckwise parametrizatins (in ther wrds, the tw 1 dimensinal cnfiguratins determine the same hmlgy class).
SOLUTIONS 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationVersion 1 lastupdate:11/27/149:47:29am Preliminar verison prone to errors and subjected to changes. The version number says all!
7. Del Characters Versin 1 lastupdate:11/27/149:47:29am Preliminar verisn prne t errrs and subjected t changes. The versin number says all! Characters f finite abelian grups Let A be a finite abelian grup.
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 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 information6.3: Volumes by Cylindrical Shells
6.3: Vlumes by Cylindrical Shells Nt all vlume prblems can be addressed using cylinders. Fr example: Find the vlume f the slid btained by rtating abut the y-axis the regin bunded by y = 2x x B and y =
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 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 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 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 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 informationMath 302 Learning Objectives
Multivariable Calculus (Part I) 13.1 Vectrs in Three-Dimensinal Space Math 302 Learning Objectives Plt pints in three-dimensinal space. Find the distance between tw pints in three-dimensinal space. Write
More informationQ1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2.
Phys10 Final-133 Zer Versin Crdinatr: A.A.Naqvi Wednesday, August 13, 014 Page: 1 Q1. A string, f length 0.75 m and fixed at bth ends, is vibrating in its fundamental mde. The maximum transverse speed
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 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 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 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 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 informationPhys102 Second Major-102 Zero Version Coordinator: Al-Shukri Thursday, May 05, 2011 Page: 1
Crdinatr: Al-Shukri Thursday, May 05, 2011 Page: 1 1. Particles A and B are electrically neutral and are separated by 5.0 μm. If 5.0 x 10 6 electrns are transferred frm particle A t particle B, the magnitude
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 informationThe Electromagnetic Form of the Dirac Electron Theory
0 The Electrmagnetic Frm f the Dirac Electrn Thery Aleander G. Kyriaks Saint-Petersburg State Institute f Technlgy, St. Petersburg, Russia* In the present paper it is shwn that the Dirac electrn thery
More informationM-theory from the superpoint
M-thery frm the superpint Jhn Huerta http://math.ucr.edu/~huerta CAMGSD Institut Superir Técnic Iberian Strings 2017 Lisbn 16 19 January Prlgue Figure : R 0 1 Prlgue Figure : R 0 1 R 0 1 has a single dd
More informationHigher Mathematics Booklet CONTENTS
Higher Mathematics Bklet CONTENTS Frmula List Item Pages The Straight Line Hmewrk The Straight Line Hmewrk Functins Hmewrk 3 Functins Hmewrk 4 Recurrence Relatins Hmewrk 5 Differentiatin Hmewrk 6 Differentiatin
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 information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
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 informationV. Balakrishnan and S. Boyd. (To Appear in Systems and Control Letters, 1992) Abstract
On Cmputing the WrstCase Peak Gain f Linear Systems V Balakrishnan and S Byd (T Appear in Systems and Cntrl Letters, 99) Abstract Based n the bunds due t Dyle and Byd, we present simple upper and lwer
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 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 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 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 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 informationI. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is
Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes,
More informationTHE LIFE OF AN OBJECT IT SYSTEMS
THE LIFE OF AN OBJECT IT SYSTEMS Persns, bjects, r cncepts frm the real wrld, which we mdel as bjects in the IT system, have "lives". Actually, they have tw lives; the riginal in the real wrld has a life,
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 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 informationPart a: Writing the nodal equations and solving for v o gives the magnitude and phase response: tan ( 0.25 )
+ - Hmewrk 0 Slutin ) In the circuit belw: a. Find the magnitude and phase respnse. b. What kind f filter is it? c. At what frequency is the respnse 0.707 if the generatr has a ltage f? d. What is the
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
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 informationPure adaptive search for finite global optimization*
Mathematical Prgramming 69 (1995) 443-448 Pure adaptive search fr finite glbal ptimizatin* Z.B. Zabinskya.*, G.R. Wd b, M.A. Steel c, W.P. Baritmpa c a Industrial Engineering Prgram, FU-20. University
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 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 information1. What is the difference between complementary and supplementary angles?
Name 1 Date Angles Intrductin t Angles Part 1 Independent Practice 1. What is the difference between cmplementary and supplementary angles? 2. Suppse m TOK = 49. Part A: What is the measure f the angle
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 informationCHM112 Lab Graphing with Excel Grading Rubric
Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline
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 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 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 informationExperiment #3. Graphing with Excel
Experiment #3. Graphing with Excel Study the "Graphing with Excel" instructins that have been prvided. Additinal help with learning t use Excel can be fund n several web sites, including http://www.ncsu.edu/labwrite/res/gt/gt-
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 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 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 informationKinematic transformation of mechanical behavior Neville Hogan
inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized
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 informationQUASICOMPONENTS AND SHAPE THEORY
Vlume 13, 1988 Pages 73 82 http://tplgy.auburn.edu/tp/ QUASICOMPONENTS AND SHAPE THEORY by Jerzy Dydak and Manuel Alns Mrón Tplgy Prceedings Web: http://tplgy.auburn.edu/tp/ Mail: Tplgy Prceedings Department
More informationTrigonometric Ratios Unit 5 Tentative TEST date
1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin
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 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 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 informationAIP Logic Chapter 4 Notes
AIP Lgic Chapter 4 Ntes Sectin 4.1 Sectin 4.2 Sectin 4.3 Sectin 4.4 Sectin 4.5 Sectin 4.6 Sectin 4.7 4.1 The Cmpnents f Categrical Prpsitins There are fur types f categrical prpsitins. Prpsitin Letter
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 information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationSMARANDACHE GROUPOIDS
SMARANDACHE GROUPOIDS W. B. Vsnth Kndsmy Deprtment f Mthemtics Indin Institute f Technlgy Mdrs Chenni - 6 6 Indi. E-mil: vsntk@md.vsnl.net.in Astrct: In this pper we study the cncept f Smrndche Grupids
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 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 informationStandard Title: Frequency Response and Frequency Bias Setting. Andrew Dressel Holly Hawkins Maureen Long Scott Miller
Template fr Quality Review f NERC Reliability Standard BAL-003-1 Frequency Respnse and Frequency Bias Setting Basic Infrmatin: Prject number: 2007-12 Standard number: BAL-003-1 Prject title: Frequency
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 informationChapter 8: The Binomial and Geometric Distributions
Sectin 8.1: The Binmial Distributins Chapter 8: The Binmial and Gemetric Distributins A randm variable X is called a BINOMIAL RANDOM VARIABLE if it meets ALL the fllwing cnditins: 1) 2) 3) 4) The MOST
More informationGetting Involved O. Responsibilities of a Member. People Are Depending On You. Participation Is Important. Think It Through
f Getting Invlved O Literature Circles can be fun. It is exciting t be part f a grup that shares smething. S get invlved, read, think, and talk abut bks! Respnsibilities f a Member Remember a Literature
More information4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression
4th Indian Institute f Astrphysics - PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw
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 informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More information