THE MULTILINEAR KAKEYA INEQUALITY
|
|
- Molly Harrington
- 5 years ago
- Views:
Transcription
1 THE MULTILINEAR KAKEYA INEQUALITY 1. The dscusso from last tme, heurstcs ad memores Suppose that {T} s a Kakeya set of tubes R. Each tube has radus 1 ad legth N, ad there are N 1 tubes. Suppose that T N γ. The umber of ut cubes from the ut cube lattce that tersect T s N γ. We use the polyomal ham sadwch theorem to choose a polyomal P so that Z(P) bsects each of these ut cubes. The degree of Z(P) s N 1 γ/. What does such a polyomal tell us? Let Q(K) be ths set of cubes. Cosder oe of the tubes, T. Let l be a le parallel to the axs of T, a radomly chose parallel le the tube T. For almost every choce of l, we have l Z(P) deg(p) N 1 γ/. O the other had, the tube T cotas N cubes of Q(K), ad Z(P) bsects each of them. Let Q(T ) be ths lst of cubes. They are dsjot, so we get Averageq Q(T ),l l Z(P) q N deg(p) N. For a typcal cube q, we kow that Z(P) bsects q, ad yet Average l Z(P) q N γ/ s much smaller tha 1. Ths s oly possble f the surface Z(P) q s approxmately parallel to the tube T. We ca make a revsed pcture of the surface Z(P) the tube T. So the geometry of the surface Z(P) s coected wth the geometry of the tubes T. If we try to mtate Dvr s proof of the fte feld Nkodym or Kakeya cojectures, we are led to the followg questo. Exted each tube T a further legth N, ad let q be a ut cube the exteso. Is t true that Z(P) approxmately bsects q? Ths type of questo looks dffcult, ad t may be ulkely. The surface Z(P) s approxmately taget to the tube T sde of T, but t s hard to kow whether Z(P) wll bed sharply as soo as t leaves T ad come owhere ear to q. I spet a whle tryg to force Z(P) to ht q, ad t was pretty frustratg. I would charge dow oe of the tubes T, tryg to p Z(P) ad carry t dow to q, ad Z(P) would stay wth me for a whle ad the swg out of the way, whle I wet chargg harmlessly by... However, the structure that we observed above does say somethg terestg about Kakeya sets. We otced that for a typcal cube q T, the surface Z(P) s approxmately taget to T. But there are may dfferet tubes T j cotag q. Wth the method above, we ca argue that T j s approxmately taget to Z(P) for 1 1 γ/
2 2 THE MULTILINEAR KAKEYA INEQUALITY most of the tubes. I fact, there must be a hyperplae π(q), ad the tubes T j must usually be almost taget to π(q). Ths s a somewhat surprsg structure, called plaess. Plaess was frst dscovered by Katz, Laba, ad Tao, the paper A mproved boud o the Mkowsk dmeso of Bescovtch sets R 3. (A. of Math. (2) 152 (2000), o. 2, ) Plaess was oe of the observatos/tools that allowed 3 them to prove that a Kakeya set of tubes R (wth mld addtoal hypotheses) has volume at least N 2.5+ǫ. Later, Beett, Carbery, ad Tao proved stroger ad more geeral plaess estmates the paper O the multlear restrcto ad Kakeya cojectures Acta Math. 196 (2006), o. 2, We wll come to ther work below. If we had a hypothetcal Kakeya set of tubes, a typcal cube would le may tubes T j. Wthout ay experece, we mght guess that the dfferet tubes T j q would pot a bush of drectos that was pretty dese o the ut sphere. Suprsgly, they eed to cocetrate ear to a plae. Aother way to say ths s that they do t form a whole lot of jots. Durg the course, we met may theorems about the cdece patters of les space. Each of these questos ca be adapted to a questo about log th tubes stead of les. Usually the adapted questo s wde ope. But for the jots problem, the adapted questo has a ce aswer based o the deas we have just bee dscussg. 2. The geeralzed Looms-Whtey equalty We prove here a aalogue of the jots theorem wth log th tubes stead of perfect les. Theorem 2.1. (Beett-Carbery-Tao, Guth) Suppose that T j,a are cylders R for 1 j ad 1 a A. Each cylder has radus 1 ad fte legth. The axs of a cylder T j,a makes a agle of < (100) 1 wth the x j -axs. Let I be the pots whch le oe cylder for each value of j = 1... I equatos I := j=1( A a=1t j,a ). The the volume of I s A 1. Remarks. If the tubes T j,a are parallel to the x j -axs, the ths estmate follows from the Looms-Whtey equalty. We see that the projecto of I to ay coord ate hyperplae les A ut balls, ad the Looms-Whtey gves I A 1. The case of axs-parallel cylders s bascally equvalet to the Looms-Whtey equalty. The problem here s to see that the equalty remas true f we are allowed to tlt the tubes a few degrees. Hstory. BCT proved a ty bt weaker estmate usg mootocty formulas for the heat equato. G proved ths estmate usg the polyomal method. Ths
3 THE MULTILINEAR KAKEYA INEQUALITY 3 theorem ca be thought of as a verso of jots for early-orthogoal tubes. It mples, partcular, the jots theorem for early orthogoal les. The proof volves the dea of the drected volume of a surface. Suppose S s a smooth hypersurface R wth ormal vector N. If v s a ut vector, we defe the drected volume of S perpedcular to V by the formula V S (v) := N v dvol S. S Notce that f the taget plae of S s perpedcular to v, we have N v = 1, ad f the taget plae cotas v, we have N v = 0. For example, we cosder the drected volume of the ut crcle the drecto v = (0, 1). The drected volume of a arc of the upper sem-crcle drecto v s exactly the chage the x-coordate over the arc. Therefore, the drected volume of the whole upper sem crcle s 2, ad the drected volume of the whole crcle s 4. The computato for the crcle geeralzes as follows. Let π be the orthogoal projecto from R to v R. Lemma 2.2. V S (v) = S π 1 (y) dvol(y). v As a corollary, we ca mmedately estmate the drected volume of a degree d varety a cylder T. Lemma 2.3. (Cylder estmate) Let T be a fte cylder R of radus r. Let v be a ut vector parallel to the axs of T. Let Z(P) be the vashg set of a polyomal P. The V (v) r 1 deg(p). Z(P) T Proof. Let π be the projecto from T to the cross-secto v T. Ths crosssecto s just a (-1)-dmesoal ball of radus r. For almost every y ths ball, π 1 (y) Z(P) deg(p). By the last lemma, V Z(P) T (v) s bouded by deg(p) tmes the volume of the cross-secto, whch s r 1. Lemma 2.4. If S s a hypersurface R, ad v 1,..., v are ut vectors ad the agle from v j to the x j -axs s (100) 1, the V ol 1 S 2 j V S(v j ). Proof. At a gve pot of S wth ormal vector N, we have to prove that j N v j 1/2. If e j are the coordate vectors, the t s straghtforward to check that j N v j 1 for ay ut vector N. The vectors v j are very close to e j, ad so the error has sze j e j v j (1/100). Now we ca do the proof of the theorem. Proof. Cosder the ut cubcal lattce. Let Q 1,..., Q V be all the ut cubes the lattce whch tersect the set I. We wll prove V A 1.
4 4 THE MULTILINEAR KAKEYA INEQUALITY Let P be a o-zero polyomal so that Z(P) bsects each cube Q 1,..., Q V ad degp V 1/. Ths bsecto requres a certa amout of area, therefore: V ol 1 Z(P) Q 1. Let T j (Q ) be a tube from our lst, drecto j, whch tersects Q. Let v j, be the drecto of ths tube. The drectos v 1,,..., v, are early orthoormal, ad so VZ(P) Q (v j,) V ol 1Z(P) Q 1. j=1 For each cube, choose oe drecto j so that V Z(P) Q v j, 1, ad assg the cube Q to the tube T j (Q ). We have V cubes ad A tubes, so oe of the tubes has V/A cubes assged to t. Let T be ths tube, ad let v be ts drecto. We have V/A cubes Q obeyg the followg codtos: The cube Q tersects T. V Z(P) Q (v) 1. Let T be a wder cylder wth radus 2 ad wth the same cetral axs as T. All of the cubes Q le T. Therefore, we have V/A V Z(P) T The last equalty s by the cylder estmate. Rearragg we get V A 1. 1/ (v) V. 3. Multlear Kakeya The strogest verso of the Kakeya cojecture s the L p verso. If T are a Kakeya set of tubes of radus 1 ad legth N, the L p Kakeya cojecture says that for each ǫ > 0, χ T 1 C ǫ N ǫ N. (1) R Remarks: If we arrage the tubes a dsjot way, the left had sde s N. If we arrage them all cetered at the org, the the left had sde s (log N)N. If true, ths cojecture gves essetally sharp bouds for χ T p for every p. It mples that the uo of tubes has volume at least cǫn ǫ for ay ǫ > 0. Ths cojecture s stll wde ope. The multlear Kakeya cojecture allows us to cotrol a postve fracto of all the terms - a certa sese. Frst we rewrte the left had sde of (1).
5 THE MULTILINEAR KAKEYA INEQUALITY 5 χ T 1 = 1 χ T χ T 1 1 O the rght had sde we have a product of detcal copes of χ T 1. Now we edt the formula, keepg oly a costat fracto of the terms each copy of 1 χ T 1. Let I(j) be the subset of tubes T where the agle betwee v(t ) ad the x j axs s (100) 1. For each j, the umber of such tubes s N 1 - they form a postve fracto of all of the tubes. Theorem 3.1. (Beett-Carbery-Tao) For ay ǫ > 0, there exsts a costat C ǫ so that for ay Kakeya set of tubes, 1 χ 1 C ǫ N ǫ T N. j=1 I(j) (I ths equalty, the N ǫ factor ca actually be removed, see my paper O the edpot case of the Beett-Carbery-Tao multlear Kakeya equalty. But ths takes a lot of extra work.) Ths equalty s a geeralzato of the last theorem. We expla how they are related ad we sketch the extra steps eeded to prove the theorem. For ay tegers µ 1,..., µ 0, cosder the set of pots: The left had sde s I(µ) := {x R 2 µ j χ < 2 µ j+1 T for all j.} I(j) I(µ) 2 µ j/( 1). µ j Therefore, the theorem follows from the followg lemma: Lemma 3.2. For each µ as above, I(µ) N j 2 µ j/( 1). The lemma shows that each term the sum above has sze N, ad the umber of terms s (log N), ad so we get a boud for the total of N (log N), whch proves the theorem. If µ = 0, we have I(0) cotaed the -fold tersecto set I defed above, ad the equalty follows from the Theorem the last secto. The other values of µ are farly smlar. Let us radomly choose I (j) I(j), cludg each tube wth probablty 2 µ j. Let I be the pots lyg oe tube T, I (j) for each j. A pot of I(µ) les
6 6 THE MULTILINEAR KAKEYA INEQUALITY I wth probablty 1. Wth hgh probablty, the sze of I (j) s N 1 2 µ j. Therefore, our boud for I(µ) follows from the followg lemma. Lemma 3.3. Let T j,a a = 1...A j be cylders of radus 1 early parallel to the x j axs. The the volume of the set of pots lyg at least oe tube of each drecto 1 s j=1 A 1 j. If all the A j happe to be equal, ths lemma s exactly the theorem from the last secto. The case of uequal A j requres a extra refemet the proof. We cut each cube Q to may smaller peces, ad we choose P to bsect each smaller pece. The smaller peces are arraged to a grd, cut more fely the drectos j where A j s small ad more coarsely the drectos where A j s large. Detals the exercses... (More detals. Take a cube Q. Pck tubes T j (Q ). Chage coordates so that the vectors v(t j (Q )) become exactly orthogoal. I these coordates, Q s ot qute a cube, but cotas a slghtly smaller cube Q. Chop Q to a grd, where the j th drecto s cut subdvded to j =j A j. Choose Z(P) to bsect each of these peces....) 4. Sharp turs of algebrac varetes? So far, the polyomal method has ot led to ay progress o the Kakeya problem. There are major dffcultes applyg the methods we have see to log th tubes stead of perfect les. I the proof of fte feld Kakeya or Nkodym, we use parameter coutg to fd a polyomal that vashes some places, ad the we argue that the polyomal also must vash somewhere else. Ths step plays a key role most of the proofs we have see ths course. It s hard to see whether somethg lke ths ca work the settg of tubes. Suppose as the frst secto that K s the uo of a Kakeya set of 1 N tubes wth surprsgly small volume, ad that P s a polyomal so that Z(P) bsects each cube of the ut lattce that tersects K. Pck a tube T from the Kakeya set, ad mage extedg t to twce ts legth, ad let q be a ut cube ths exteso. Is there ay hope that Z(P) also roughly bsects q? We kow that Z(P) bsects all the cubes T, ad we ve also see that most of these cubes Z(P) s roughly parallel to T. If Z(P) keeps gog the drecto of ts taget plae, t wll come reasoably close to q (although t s stll ot clear t wll really ht q). But t s ot at all clear whether Z(P) wll cotue the drecto of ts taget plae. Perhaps Z(P) wll curve dramatcally ad go owhere ear q. It mght be helpful to uderstad better how may sharp beds there ca be a degree d algebrac surface. Here s a toy problem that gets at some of these ssues.
7 THE MULTILINEAR KAKEYA INEQUALITY 7 Let P be a polyomal two varables. Let Pos(P) := {x R 2 P(x) > 0}. For a gve degree d, how closely ca Pos(P) look lke the square [ 1, 1] 2? Recall that the Hausdorff dstace from Pos(P) to [ 1, 1] 2 s < ǫ f [ 1, 1] 2 les the ǫ- eghborhood of Pos(P) ad Pos(P) les the ǫ-eghborhood of the square. Let ǫ(d) be the fmum over all degree d polyomals P of dst Haus (Pos(P), [ 1, 1] 2 ). Ca we descrbe the order of magtude of ǫ(d)? Very lttle s kow about ths. We kow that ǫ(d) > 0 for each d. The reaso s that dst Haus (Pos(P), [ 1, 1] 2 ) vares lower sem-cotuously as P moves V (d) \ {0}. Multplyg P by a postve costat does ot chage Pos(P), ad so we ca restrct atteto to polyomals the ut sphere of V (d). By compactess the fmum s attaed. But f dst Haus (Pos(P), [ 1, 1] 2 ) were zero, we would have P = 0 o the boudary of the square. The P would vash o the le x = 1, ad (x 1) would factor out of P. Wrte P as (1 x) a P 1 (x, y), where (1 x) does ot dvde P 1. The polyomal P 1 vashes at oly ftely may pots of the le x = 1. If a s eve, the we see that P 1 eeds to vash o the sde of the square where x = 1, ad the 1 x dvdes P 1, ad we get a cotradcto. If a s odd, the we see that P 1 eeds to vash o the part of the le x = 1 where y > 1. Ths stll mples that 1 x dvdes P 1, ad we get a cotradcto. If d s eve, a ce example s the polyomal P d = 1 x d y d. The set Pos(P d ) s the ut ball the L d orm. As d, t approaches the square, whch s the ut ball the L orm. For every eve d, Pos(P d ) [ 1, 1] 2, ad P d > 0 o the square [ (1/2) 1/d, (1/2) 1/d ]. Now 1 (1/2) 1/d 2 1/d, ad so dst Haus (Pos(Pd), [ 1, 1] ) 1/d. Hece ǫ(d) 1/d. It seems plausble that P d s ear-optmal ad that ǫ(d) 1/d. The hard problem s to gve quattatve lower bouds o ǫ(d). I do t kow of ay explct lower boud the lterature. I worked o the problem, ad I had a pla for a lower boud of the form e ed... I thk the moral ssue s to gve quattatve bouds o how sharply a degree d curve ca make a certa type of tur. It s mportat to keep md the followg example. The zero set of the hyperbola xy = ǫ makes a very sharp tur ear the org, whch looks somethg lke the corer of a square. But the hyperbola has two braches, ad so stead of beg postve o approxmately oe quartat, t s postve o two opposte quartats, ad ts postve set does ot really look lke the eghborhood of a corer of a square. A algebrac curve ca make a arbtrarly sharp tur f t looks locally lke a hyperbola wth two braches, but t s harder for t to make a sharp tur wth oly oe brach. I mght have goe o too log about ths toy problem. A soluto to ths problem would ot drectly lead to ay bouds o Kakeya. Tryg to go further wth the polyomal method ad tubes, ths type of estmate seems to come up. I geeral, t
8 8 THE MULTILINEAR KAKEYA INEQUALITY mght be helpful to have more quattatve estmates about the geometry of degree d algebrac surfaces.
9 MIT OpeCourseWare 18.S997 The Polyomal Method Fall 2012 For formato about ctg these materals or our Terms of Use, vst:
Ideal multigrades with trigonometric coefficients
Ideal multgrades wth trgoometrc coeffcets Zarathustra Brady December 13, 010 1 The problem A (, k) multgrade s defed as a par of dstct sets of tegers such that (a 1,..., a ; b 1,..., b ) a j = =1 for all
More informationCHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More informationFeature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)
CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.
More informationC-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory
ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?
More informationLecture 9: Tolerant Testing
Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have
More informationL5 Polynomial / Spline Curves
L5 Polyomal / Sple Curves Cotets Coc sectos Polyomal Curves Hermte Curves Bezer Curves B-Sples No-Uform Ratoal B-Sples (NURBS) Mapulato ad Represetato of Curves Types of Curve Equatos Implct: Descrbe a
More informationCS286.2 Lecture 4: Dinur s Proof of the PCP Theorem
CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationChapter 9 Jordan Block Matrices
Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More informationThe Selection Problem - Variable Size Decrease/Conquer (Practice with algorithm analysis)
We have covered: Selecto, Iserto, Mergesort, Bubblesort, Heapsort Next: Selecto the Qucksort The Selecto Problem - Varable Sze Decrease/Coquer (Practce wth algorthm aalyss) Cosder the problem of fdg the
More information5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
More informationUNIT 2 SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
Numercal Computg -I UNIT SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS Structure Page Nos..0 Itroducto 6. Objectves 7. Ital Approxmato to a Root 7. Bsecto Method 8.. Error Aalyss 9.4 Regula Fals Method
More informationPhysics 114 Exam 2 Fall Name:
Physcs 114 Exam Fall 015 Name: For gradg purposes (do ot wrte here): Questo 1. 1... 3. 3. Problem Aswer each of the followg questos. Pots for each questo are dcated red. Uless otherwse dcated, the amout
More informationPTAS for Bin-Packing
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,
More informationLaboratory I.10 It All Adds Up
Laboratory I. It All Adds Up Goals The studet wll work wth Rema sums ad evaluate them usg Derve. The studet wll see applcatos of tegrals as accumulatos of chages. The studet wll revew curve fttg sklls.
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b
CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package
More informationExercises for Square-Congruence Modulo n ver 11
Exercses for Square-Cogruece Modulo ver Let ad ab,.. Mark True or False. a. 3S 30 b. 3S 90 c. 3S 3 d. 3S 4 e. 4S f. 5S g. 0S 55 h. 8S 57. 9S 58 j. S 76 k. 6S 304 l. 47S 5347. Fd the equvalece classes duced
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationCIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights
CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:
More informationFor combinatorial problems we might need to generate all permutations, combinations, or subsets of a set.
Addtoal Decrease ad Coquer Algorthms For combatoral problems we mght eed to geerate all permutatos, combatos, or subsets of a set. Geeratg Permutatos If we have a set f elemets: { a 1, a 2, a 3, a } the
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationAssignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class)
Assgmet 5/MATH 7/Wter 00 Due: Frday, February 9 class (!) (aswers wll be posted rght after class) As usual, there are peces of text, before the questos [], [], themselves. Recall: For the quadratc form
More information18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
More informationNaïve Bayes MIT Course Notes Cynthia Rudin
Thaks to Şeyda Ertek Credt: Ng, Mtchell Naïve Bayes MIT 5.097 Course Notes Cytha Rud The Naïve Bayes algorthm comes from a geeratve model. There s a mportat dstcto betwee geeratve ad dscrmatve models.
More informationLecture 07: Poles and Zeros
Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto
More informationClass 13,14 June 17, 19, 2015
Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral
More informationD KL (P Q) := p i ln p i q i
Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationDimensionality Reduction and Learning
CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that
More informationChapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements
Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall
More information18.657: Mathematics of Machine Learning
8.657: Mathematcs of Mache Learg Lecturer: Phlppe Rgollet Lecture 3 Scrbe: James Hrst Sep. 6, 205.5 Learg wth a fte dctoary Recall from the ed of last lecture our setup: We are workg wth a fte dctoary
More information9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d
9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,
More informationMATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More information1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.
Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please
More informationTHE COMPLETE ENUMERATION OF FINITE GROUPS OF THE FORM R 2 i ={R i R j ) k -i=i
ENUMERATON OF FNTE GROUPS OF THE FORM R ( 2 = (RfR^'u =1. 21 THE COMPLETE ENUMERATON OF FNTE GROUPS OF THE FORM R 2 ={R R j ) k -= H. S. M. COXETER*. ths paper, we vestgate the abstract group defed by
More informationLattices. Mathematical background
Lattces Mathematcal backgroud Lattces : -dmesoal Eucldea space. That s, { T x } x x = (,, ) :,. T T If x= ( x,, x), y = ( y,, y), the xy, = xy (er product of xad y) x = /2 xx, (Eucldea legth or orm of
More informationA tighter lower bound on the circuit size of the hardest Boolean functions
Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the
More informationMean is only appropriate for interval or ratio scales, not ordinal or nominal.
Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationSection l h l Stem=Tens. 8l Leaf=Ones. 8h l 03. 9h 58
Secto.. 6l 34 6h 667899 7l 44 7h Stem=Tes 8l 344 Leaf=Oes 8h 5557899 9l 3 9h 58 Ths dsplay brgs out the gap the data: There are o scores the hgh 7's. 6. a. beams cylders 9 5 8 88533 6 6 98877643 7 488
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationMultiple Choice Test. Chapter Adequacy of Models for Regression
Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to
More informationHomework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015
Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bg-o ) otato. I ths problem, you wll prove some basc facts
More informationPGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation
PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad
More informationMA/CSSE 473 Day 27. Dynamic programming
MA/CSSE 473 Day 7 Dyamc Programmg Bomal Coeffcets Warshall's algorthm (Optmal BSTs) Studet questos? Dyamc programmg Used for problems wth recursve solutos ad overlappg subproblems Typcally, we save (memoze)
More information6. Nonparametric techniques
6. Noparametrc techques Motvato Problem: how to decde o a sutable model (e.g. whch type of Gaussa) Idea: just use the orgal data (lazy learg) 2 Idea 1: each data pot represets a pece of probablty P(x)
More informationComputational Geometry
Problem efto omputatoal eometry hapter 6 Pot Locato Preprocess a plaar map S. ve a query pot p, report the face of S cotag p. oal: O()-sze data structure that eables O(log ) query tme. pplcato: Whch state
More informationPacking of graphs with small product of sizes
Joural of Combatoral Theory, Seres B 98 (008) 4 45 www.elsever.com/locate/jctb Note Packg of graphs wth small product of szes Alexadr V. Kostochka a,b,,gexyu c, a Departmet of Mathematcs, Uversty of Illos,
More informationd dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin
Learzato of the Swg Equato We wll cover sectos.5.-.6 ad begg of Secto 3.3 these otes. 1. Sgle mache-fte bus case Cosder a sgle mache coected to a fte bus, as show Fg. 1 below. E y1 V=1./_ Fg. 1 The admttace
More informationCentroids & Moments of Inertia of Beam Sections
RCH 614 Note Set 8 S017ab Cetrods & Momets of erta of Beam Sectos Notato: b C d d d Fz h c Jo L O Q Q = ame for area = ame for a (base) wdth = desgato for chael secto = ame for cetrod = calculus smbol
More informationMultiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades
STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos
More informationMedian as a Weighted Arithmetic Mean of All Sample Observations
Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of
More informationLog1 Contest Round 2 Theta Complex Numbers. 4 points each. 5 points each
01 Log1 Cotest Roud Theta Complex Numbers 1 Wrte a b Wrte a b form: 1 5 form: 1 5 4 pots each Wrte a b form: 65 4 4 Evaluate: 65 5 Determe f the followg statemet s always, sometmes, or ever true (you may
More informationGeneralized Linear Regression with Regularization
Geeralze Lear Regresso wth Regularzato Zoya Bylsk March 3, 05 BASIC REGRESSION PROBLEM Note: I the followg otes I wll make explct what s a vector a what s a scalar usg vec t or otato, to avo cofuso betwee
More informationMA 524 Homework 6 Solutions
MA 524 Homework 6 Solutos. Sce S(, s the umber of ways to partto [] to k oempty blocks, ad c(, s the umber of ways to partto to k oempty blocks ad also the arrage each block to a cycle, we must have S(,
More informationSpecial Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
More information13. Dedekind Domains. 13. Dedekind Domains 117
3. Dedekd Domas 7 3. Dedekd Domas I the last chapter we have maly studed -dmesoal regular local rgs,. e. geometrcally the local propertes of smooth pots o curves. We ow wat to patch these local results
More informationWe have already referred to a certain reaction, which takes place at high temperature after rich combustion.
ME 41 Day 13 Topcs Chemcal Equlbrum - Theory Chemcal Equlbrum Example #1 Equlbrum Costats Chemcal Equlbrum Example #2 Chemcal Equlbrum of Hot Bured Gas 1. Chemcal Equlbrum We have already referred to a
More informationCan we take the Mysticism Out of the Pearson Coefficient of Linear Correlation?
Ca we tae the Mstcsm Out of the Pearso Coeffcet of Lear Correlato? Itroducto As the ttle of ths tutoral dcates, our purpose s to egeder a clear uderstadg of the Pearso coeffcet of lear correlato studets
More informationv 1 -periodic 2-exponents of SU(2 e ) and SU(2 e + 1)
Joural of Pure ad Appled Algebra 216 (2012) 1268 1272 Cotets lsts avalable at ScVerse SceceDrect Joural of Pure ad Appled Algebra joural homepage: www.elsever.com/locate/jpaa v 1 -perodc 2-expoets of SU(2
More information1. A real number x is represented approximately by , and we are told that the relative error is 0.1 %. What is x? Note: There are two answers.
PROBLEMS A real umber s represeted appromately by 63, ad we are told that the relatve error s % What s? Note: There are two aswers Ht : Recall that % relatve error s What s the relatve error volved roudg
More informationThe conformations of linear polymers
The coformatos of lear polymers Marc R. Roussel Departmet of Chemstry ad Bochemstry Uversty of Lethbrdge February 19, 9 Polymer scece s a rch source of problems appled statstcs ad statstcal mechacs. I
More informationPart 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
More information1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i.
CS 94- Desty Matrces, vo Neuma Etropy 3/7/07 Sprg 007 Lecture 3 I ths lecture, we wll dscuss the bascs of quatum formato theory I partcular, we wll dscuss mxed quatum states, desty matrces, vo Neuma etropy
More informationf f... f 1 n n (ii) Median : It is the value of the middle-most observation(s).
CHAPTER STATISTICS Pots to Remember :. Facts or fgures, collected wth a defte pupose, are called Data.. Statstcs s the area of study dealg wth the collecto, presetato, aalyss ad terpretato of data.. The
More informationBounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy
Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationn -dimensional vectors follow naturally from the one
B. Vectors ad sets B. Vectors Ecoomsts study ecoomc pheomea by buldg hghly stylzed models. Uderstadg ad makg use of almost all such models requres a hgh comfort level wth some key mathematcal sklls. I
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More information22 Nonparametric Methods.
22 oparametrc Methods. I parametrc models oe assumes apror that the dstrbutos have a specfc form wth oe or more ukow parameters ad oe tres to fd the best or atleast reasoably effcet procedures that aswer
More informationAN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM BRUNO GRENET
AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM BRUNO GRENET Abstract. The Permaet versus Determat problem s the followg: Gve a matrx X of determates over a feld of characterstc dfferet from
More informationDATE: 21 September, 1999 TO: Jim Russell FROM: Peter Tkacik RE: Analysis of wide ply tube winding as compared to Konva Kore CC: Larry McMillan
M E M O R A N D U M DATE: 1 September, 1999 TO: Jm Russell FROM: Peter Tkack RE: Aalyss of wde ply tube wdg as compared to Kova Kore CC: Larry McMlla The goal of ths report s to aalyze the spral tube wdg
More informationUnit 9. The Tangent Bundle
Ut 9. The Taget Budle ========================================================================================== ---------- The taget sace of a submafold of R, detfcato of taget vectors wth dervatos at
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationLecture 16: Backpropogation Algorithm Neural Networks with smooth activation functions
CO-511: Learg Theory prg 2017 Lecturer: Ro Lv Lecture 16: Bacpropogato Algorthm Dsclamer: These otes have ot bee subected to the usual scruty reserved for formal publcatos. They may be dstrbuted outsde
More informationENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections
ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty
More information2. Independence and Bernoulli Trials
. Ideedece ad Beroull Trals Ideedece: Evets ad B are deedet f B B. - It s easy to show that, B deedet mles, B;, B are all deedet ars. For examle, ad so that B or B B B B B φ,.e., ad B are deedet evets.,
More informationCLASS NOTES. for. PBAF 528: Quantitative Methods II SPRING Instructor: Jean Swanson. Daniel J. Evans School of Public Affairs
CLASS NOTES for PBAF 58: Quattatve Methods II SPRING 005 Istructor: Jea Swaso Dael J. Evas School of Publc Affars Uversty of Washgto Ackowledgemet: The structor wshes to thak Rachel Klet, Assstat Professor,
More informationbest estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best
Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationInvestigating Cellular Automata
Researcher: Taylor Dupuy Advsor: Aaro Wootto Semester: Fall 4 Ivestgatg Cellular Automata A Overvew of Cellular Automata: Cellular Automata are smple computer programs that geerate rows of black ad whte
More informationLogistic regression (continued)
STAT562 page 138 Logstc regresso (cotued) Suppose we ow cosder more complex models to descrbe the relatoshp betwee a categorcal respose varable (Y) that takes o two (2) possble outcomes ad a set of p explaatory
More informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More informationSt John s College. Preliminary Examinations July 2014 Mathematics Paper 1. Examiner: G Evans Time: 3 hrs Moderator: D Grigoratos Marks: 150
St Joh s College Prelmar Eamatos Jul 04 Mathematcs Paper Eamer: G Evas Tme: 3 hrs Moderator: D Grgoratos Marks: 50 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY. Ths questo paper cossts of pages, cludg
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationENGI 3423 Simple Linear Regression Page 12-01
ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable
More informationOverview of the weighting constants and the points where we evaluate the function for The Gaussian quadrature Project two
Overvew of the weghtg costats ad the pots where we evaluate the fucto for The Gaussa quadrature Project two By Ashraf Marzouk ChE 505 Fall 005 Departmet of Mechacal Egeerg Uversty of Teessee Koxvlle, TN
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationStatistics: Unlocking the Power of Data Lock 5
STAT 0 Dr. Kar Lock Morga Exam 2 Grades: I- Class Multple Regresso SECTIONS 9.2, 0., 0.2 Multple explaatory varables (0.) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (0.2) Exam 2 Re- grades Re-
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationThe Occupancy and Coupon Collector problems
Chapter 4 The Occupacy ad Coupo Collector problems By Sarel Har-Peled, Jauary 9, 08 4 Prelmares [ Defto 4 Varace ad Stadard Devato For a radom varable X, let V E [ X [ µ X deote the varace of X, where
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More information