The On-Line Heilbronn s Triangle Problem in d Dimensions
|
|
- Donald Farmer
- 5 years ago
- Views:
Transcription
1 Discrete Comput Geom 38: DI: 0.007/s x Discrete & Computatioal Geometry 2007 Spriger Sciece+Busiess Media, Ic. The -Lie Heilbro s Triagle Problem i d Dimesios Gill Barequet ad Alia Shaikhet Departmet of Computer Sciece, The Techio Israel Istitute of Techology, Haifa 32000, Israel {barequet dalia}@cs.techio.ac.il Abstract. I this paper we show a lower boud for the o-lie versio of Heilbro s triagle problem i d dimesios. Specifically, we provide a icremetal costructio for positioig poits i the d-dimesioal uit cube, for which every simplex defied by d + of these poits has volume / d+ l d d for d 5.. Itroductio The off-lie versio of the ow famous triagle problem was posed by Heilbro [R] more tha 50 years ago. It is formulated as follows: Give poits i the uit square, what is H2 off-lie, the maximum possible area of the smallest triagle defied by some three of these poits? There is a large gap betwee the best curretly kow lower ad upper bouds o H2 off-lie, log / 2 [KPS2] ad / 8/7 ε for ay ε>0 [KPS]. Jiag et al. [JLV] showed that the expected area of the smallest triagle, whe the poits are put uiformly at radom i the uit square, is / 3. Barequet [B] geeralized the off-lie problem to d dimesios: Give poits i the d-dimesioal uit cube, what is Hd off-lie, the maximum possible volume of the smallest simplex defied by some d + of these poits? The best curretly kow lower boud o Hd off-lie is log / d [L]. ther versios, i which the dimesio of the optimized simplex is lower tha that of the cube, were ivestigated i [L2], [L3], ad [BN]. Work o this paper has bee supported i part by the Fud for the Promotio of Research at the Techio. A prelimiary versio of this paper appeared i CCN 06.
2 52 G. Barequet ad A. Shaikhet The o-lie versio of the triagle problem is harder tha the off-lie versio because the value of is ot specified i advace. I other words, the poits are positioed oe after the other i a d-dimesioal uit cube, while is icremeted by oe after every poitpositioig step. The procedure ca be stopped at ay time, ad the already-positioed poits must have the property that every subset of d + poits defies a polytope whose volume is at least some quatity Hd o-lie, where the goal is to maximize this quatity. Schmidt [S] showed that H2 o-lie = / 2. Barequet [B2] used ested packig argumets to demostrate that H3 o-lie = / 0/3 = / ad H4 o-lie = / 27/24 = / I this paper we preset a otrivial geeralizatio of the latter method to d dimesios, showig that for a fixed value of d 5 we have Hd o-lie = / d+ l d d We provide a icremetal procedure for positioig poits oe by oe i a d-dimesioal uit cube so that o subset of up to d + poits is too dese. Specifically, the distace betwee ay two poits is at least a / /d for some costat a > 0, o three poits defie a triagle whose area is less tha a 2 / 2/d for some costat a 2 > 0, ad so o. The values of the costats are tued at the ed of the costructio. It is the prove that all the d-dimesioal simplices defied by d + -tuples of the poits have volume / d+ l d d The Costructio 2.. Notatio ad Pla We use the followig otatio. Let p i, p i2,...,p iq be ay q poits i R d. The p i p i2 deotes the distace betwee two poits p i, p i2 ; p i p i2 p i3 deotes the area of the triagle p i p i2 p i3 ; p i p i2 p i3 p i4 deotes the three-dimesioal volume of the tetrahedro p i p i2 p i3 p i4 ; ad, i geeral, p i p i2 p iq deotes the volume of the q - dimesioal simplex p i p i2 p iq. We deote by C d the d-dimesioal uit cube, ad by B d r a d-dimesioal ball of radius r. The lie defied by the pair of poits p i, p i2 is deoted by l i i 2. Throughout the costructio we refer to d as a fixed costat. Therefore, we omit factors that deped solely o d, except whe they appear i powers of. We wat to costruct a set S of poits i C d such that: [] p i p i2 V = a / /d, for ay pair of distict poits p i, p i2 S ad for some costat a > 0. [2] p i p i2 p i3 V 2 = a 2 / 2/d, for ay triple of distict poits p i, p i2, p i3 S ad for some costat a 2 > 0. [3] p i p i2 p i3 p i4 V 3 = a 3 / 4d2 5d /dd d 2, for ay quadruple of distict poits p i, p i2, p i3, p i4 S ad for some costat a 3 > 0.. [q ] p i p i2 p iq V q = a q V q 2 /a q 2 dq 2+q 3/dd q+2, for ay q-tuple 4 q d + of distict poits p i, p i2,...,p iq S ad for some costat a q > 0. Note that coditio [q ] holds for q = 4, i which case it coicides with [3].
3 The -Lie Heilbro s Triagle Problem i d Dimesios 53 The goal is to costruct S icremetally. That is, assume that we have already costructed a subset S v of v poits, for v<, which satisfies coditios [] [q ] above. We wat to show that there exists a ew poit p C d that satisfies: [ ] pp i V = a / /d, for each poit p i S. [2 ] pp i p i2 V 2 = a 2 / 2/d, for ay pair of distict poits p i, p i2 S. [3 ] pp i p i2 p i3 V 3 = a 3 / 4d2 5d /dd d 2, for ay triple of distict poits p i, p i2, p i3 S.. [q ] pp i p i2 p iq V q = a q V q 2 /a q 2 dq 2+q 3/dd q+2, for ay q -tuple 4 q d + of distict poits p i, p i2,...,p iq S. We will show this by summig up the volumes of the forbidde portios of C d where oe of the iequalities [ ] [q ] is violated, ad by showig that the sum of these volumes is less tha. This implies the existece of the desired poit p, which we the add to S v to form S v+. We cotiue i this maer util the etire set S is costructed Forbidde Balls The forbidde regios where oe of the iequalities [ ] is violated are v d-dimesioal balls of radius r = a / /d. Their total volume is at most v v Br d = = Forbidde Cyliders The forbidde regios where oe of the iequalities [2 ] is violated are v 2 d-dimesioal cyliders G ij, for i < j v. The cylider G ij is cetered at l ij, its legth is at most d, ad its cross-sectio perpedicular to l ij is a d -dimesioal sphere of radius r i, j = i< j v 2a 2 2/d p i p j = i< j v 2/d p i p j see Fig.. The overall volume of the cyliders withi C d is at most Br d i, j d =. 2 p i p j d To boud this sum, we fix p i ad sum over p j. We use a d-dimesioal spherical Recall that Br d =πd/2 r d /Ɣd/2 + = r d, where Ɣ is the cotiuous geeralizatio of the factorial fuctio.
4 54 G. Barequet ad A. Shaikhet `ij 2a2 2=d jp ip j j G ij p j p i 2a2 2=d jp ip j j Fig.. A cylider i R d. packig argumet that exploits the fact that S v satisfies []. Specifically, we have j i /d p i p j d t= M t d /d a d, 2 t d where M t is the umber of poits of S v that lie i the d-dimesioal spherical shell cetered at p i with ier radius a t/ /d ad outer radius a t + / /d ; see Fig. 2. There are /d such spherical shells withi C d. Because of [], the umber of such =d 3 2 a =d p i Fig. 2. A spherical packig of balls i R d.
5 The -Lie Heilbro s Triagle Problem i d Dimesios 55 poits is M t = t d. This follows by a argumet of packig spheres of volume / withi a shell whose volume is t d /. Hece, the sum i 2 is. Summig this over all p i, we obtai a fial boud of v. Substitutig this i, we see that the total volume of the forbidde cyliders is v/ = Forbidde Prisms The forbidde regios where oe of the iequalities [3 ] is violated are v 3 d-dimesioal prisms ϕ ijk, for i < j < k v. The base area a portio of a two-dimesioal flat of ϕ ijk is at most d, ad its height is a d 2-dimesioal sphere of radius r i, j,k = 4d2 5d /dd d 2 p i p j p k The overall volume of the prisms withi C d is at most B d 2 i< j<k v r i, j,k d = i< j<k v. 4d2 5d /dd p i p j p k d 2. 3 To boud this sum, we fix p i, p j ad sum over p k. We use a d-dimesioal cylidrical packig argumet that exploits the fact that S v satisfies [] ad [2]. The cyliders are cetered at l ij ; see Fig. 3, where the lie l ij emaates from p i toward p j through the `ij =d p j 3 2 a =d p i Fig. 3. i R d. A d-dimesioal cylidrical packig a extruded d -dimesioal spherical packig of balls
6 56 G. Barequet ad A. Shaikhet dth dimesio. Specifically, we have k i, j p i p j p k N 0 2d 2/d /d d 2 a d t= 2 d 2 N t d 2/d a d 2, 4 t d 2 p i p j d 2 where N 0 is the umber of poits of S v that lie i the iermost d-dimesioal cylider of the packig cetered at l ij ad of radius a / /d, ad N t is the umber of poits of S v that lie i the cylidrical shell cetered at l ij with ier radius a t/ /d ad outer radius a t + / /d. bviously, N 0 = /d, sice the volume of the d -dimesioal cross-sectioal sphere of the iermost cylider is / d /d ad because of []. Also, we have N t = t d 2 /d. This follows by a argumet of packig spheres of volume / withi a shell whose volume is t d 2 / d /d. Hece, the quatity i 4 is 2d2 3d /dd + / p i p j d 2. Substitutig this i 3, we obtai the upper boud o the total volume of the forbidde prisms: + 2 3d2 4d /dd p i p j d 2 i< j v = v d2 4d /dd i< j v. 5 p i p j d 2 We boud the sum i the secod summad similarly to our boudig of the term i 2 i Sectio 2.3. We fix p i ad use a d-dimesioal spherical packig argumet withi spherical shells cetered at p i. Arguig as above, we obtai j i /d p i p j d 2 t= M t d 2/d a d 2 = t d 2 /d t= t d d 2/d a d 2 t d 2 =. Summig this over all p i, we obtai a fial boud of v. Substitutig this i 5, we see that the total volume of the forbidde prisms is v 2 + v =. 2 2d2 3d /dd 2.5. Geeral Forbidde Zoes I Sectios we computed the total volume of the forbidde zoes i which the respective iequalities [ ] [3 ] are violated. These zoes correspod to q = 2, 3, 4, respectively. I this sectio we aalyze the geeral case 4 < q d +. The smallest value of q for which this aalysis holds is 5, sice for q = 5 we eed coditios [4] ad [3], which are precisely the bottom of the rage i which the recursive relatio [q ] holds. The forbidde regios where oe of the iequalities [q ] is violated are v q d-dimesioal zoes ψ i i 2 i q for i < i 2 < < i q v ad 4 < q
7 The -Lie Heilbro s Triagle Problem i d Dimesios 57 d +, whose bases are portios of q 2-dimesioal flats with volume at most d q 2/2. The height of the zoe ψ i i 2 i q is a d q +2-dimesioal sphere of radius r i,...,i q = V q / p i p i2 p iq. The total volume of the zoes withi C d isat most V d q+2 Br d q+2 i d q 2/2 q =. 6,...,i q p i p i2 p iq d q+2 i < <i q v i < <i q v To boud this sum, we fix p i, p i2,...,p iq 2 ad sum over p iq. We use a packig argumet that exploits the fact that S v satisfies [] [q 2]. The packig cosists of the Cartesia product of the q 3-dimesioal flat π = π i i 2 i q 2 that passes through p i, p i2,...,p iq 2, ad spheres whose ceters belog to π ad exted to the d q + 3- dimesioal space orthogoal to π. Specifically, we have i q i,...,i q 2 p i p i2 p iq d q+2 Z 0 /d /d d q+2 V d q+2 + Z t, 7 a q 2 t= t p i p i2 p iq 2 where Z 0 is the umber of poits of S v that lie i the iermost shape of the packig cetered at the flat π ad of radius a / /d, ad Z t is the umber of poits of S v that lie i the shell cetered at π with ier radius a t/ /d ad outer radius a t + / /d. bviously, Z 0 = q 3/d, sice the volume of the iermost shape is / d q+3/d ad because of []. Also, we have Z t = t d q+2 q 3/d. This follows by a argumet of packig spheres of volume / withi a shell whose volume is t d q+2 / d q+3/d. Hece, the sum i 7 is i <...<i q 2 v q 3/d V d q+2 q 2 + V d q+2 q 2. 8 p i p i2 p iq 2 d q+2 Substitutig this i 6, we obtai the upper boud o the total volume of the forbidde zoes: V d q+2 q 3/d q + p i p i2 p iq 2 d q+2 = Combiig this with the equality d q+2 q 3/d v q 2 Vq + V q 2 V q = i < <i q 2 v a q V q 2 a q 2 dq 2+q 3/dd q+2, Vq V q 3 d q+2.
8 58 G. Barequet ad A. Shaikhet we see that the total forbidde volume is + i < <i q 2 v Vq V q 3 d q+2. 9 I order to show that the boud i 9 is, it remais to prove that the secod summad i it is. From [q ] ad [q 2] we kow that ad V q = V q 2 = a q V q 2 a q 2 dq 2+q 3/dd q+2 a q 2 V q 3 a q 3 dq 3+q 4/dd q+3, respectively. By substitutig this i the secod summad of 9, we obtai d q+2 Vq i < <i q 2 V v q 3 aq = < a q 3 aq a q 3 d q+2 Thus, it suffices to prove that i.e. kowig that v, that i < <i q 2 v 2q 3d2 + 2q 2 +3q 22d 2q 2 +2q 7/dd q+3 d q+2 v q 2 2q 3d2 + 2q 2 +3q 22d 2q 2 +2q 7/dd q q 3d2 + 2q 2 +3q 22d 2q 2 +2q 7/dd q+3 >v q 2, 2q 3d 2 + 2q 2 + 3q 22d 2q 2 + 2q 7 dd q + 3 which, after simple maipulatios, is However, it is easily verified that q 4d 2 q 4 2 d 2q 2 + 2q 7 > 0. > q 2, q 4d 2 q 4 2 d 2q 2 + 2q 7 = q 4d q + 2d > 0, usig the iequalities q 5, d q, ad d 4.
9 The -Lie Heilbro s Triagle Problem i d Dimesios Summatio We are ow ready to boud Hd o-lie, the maximum possible volume of the smallest simplex defied by some d + ofthe poits put i the d-dimesioal uit cube. I other words, we wat to boud V d from below. For this purpose we use its recursive defiitio ad write d+ a q a d V d = V a q=4 q 2 dq 2+q 3/dd q+2 2 = d+. dq 2+q 3/dd q+2+2/d q=4 Let us boud from above the power of i : d+ q=4 dq 2 + q 3 dd q + 2 q=4 + 2 d d+ q /d = d q + 2 d = d 2 t= d + /d t t d+ q=4 d q d q + 2 d 2/d + 2 d < d + /dld d 2 2/d+2/d = d + ld d ld 2.265/d +2/d < 2 d + ld d , where we use the facts which ca be verified by elemetary calculus that. k t= /t < lk for k 3 that is, d 5, ad 2. 2/d ld 2.265/d < for d 5. We see that V d > a d / d+ l d d Note that whe d teds to ifiity, some terms vaish ad d 2 t= /t l d 2 approaches the Euler Mascheroi costat, γ = Thus, for large-eough values of d,wehavev d >a d / d+ l d d It remais to show that the costats a, a 2,...,a d ca be fixed so that the total volume of the forbidde zoes is strictly less tha. To this aim ote that amog these costats, the total volume of the forbidde balls depeds solely o a, the total volume of the forbidde cyliders depeds oly o a, a 2, ad so o. This allows us to fix the values of the costats sequetially so that the total volume of ay type of forbidde zoe is strictly less tha /d. Specifically, we fix a to make the total volume of the forbidde balls less tha /d. The, havig a already fixed, we fix a 2 to make the total volume of the forbidde cyliders less tha /d, ad so o. The grad total of the volume of the forbidde zoes is, thus, less tha d/d =. See [B2] for the implemetatio of this techique for d = 3, 4. Note that the values of a, a 2,...,a d deped oly o d ad ot o. This allows the iterative procedure to cotiue ad ifiitum, while the forbidde zoes of each iteratio properly cotai the respective forbidde zoes of the precedig iteratio.
10 60 G. Barequet ad A. Shaikhet This completes the proof of the mai theorem: Theorem. H o-lie d = / d+ l d d for d Coclusio I this paper we show, by usig ested packig argumets, that Hd o-lie = / d+ l d d for all d 5. For large-eough values of d we have Hd o-lie = / d+ l d d This compares favorably with the bestkow lower boud [L] i the off-lie case Hd off-lie = log / d. Ackowledgmet The authors thak Micha Sharir for helpful discussios o the triagle problem ad o ested packig argumets. Refereces [B] G. Barequet, A lower boud for Heilbro s triagle problem i d dimesios, SIAM J. Discrete Math., 4 200, [B2] G. Barequet, The o-lie Heilbro s triagle problem, Discrete Math., , 7 4. [BN] G. Barequet ad J. Naor, Large k-d simplices i the d-dimesioal uit cube, Far East J. Appl. Math., , [JLV] T. Jiag, M. Li, ad P. Vitáyi, The average-case area of Heilbro-type triagles, Radom Structures Algorithms, , [KPS] J. Komlós, J. Pitz, ad E. Szemerédi, Heilbro s triagle problem, J. Lodo Math. Soc. 2, 24 98, [KPS2] J. Komlós, J. Pitz, ad E. Szemerédi, A lower boud for Heilbro s problem, J. Lodo Math. Soc. 2, , [L] H. Lefma, Heilbro s problem i higher dimesio, Combiatorica, , [L2] H. Lefma, Large triagles i the d-dimesioal uit-cube, Proc. 0th A. It. Computig ad Combiatorics Cof., Jeju Islad, South Korea, pp , Lecture Notes i Computer Sciece, 306, Spriger-Verlag, Berli, August [L3] H. Lefma, Distributios of poits i d dimesios ad large k-poit simplices, Proc. th A. It. Computig ad Combiatorics Cof., Kumig, Chia, pp , Lecture Notes i Computer Sciece, 3595, Spriger-Verlag, Berli, August [R] K.F. Roth, a problem of Heilbro, Proc. Lodo Math. Soc., 26 95, [S] W.M. Schmidt, a problem of Heilbro, J. Lodo Math. Soc. 2, 4 97, Received September 6, 2006, ad i revised form Jauary 9, lie publicatio May 8, 2007.
Several properties of new ellipsoids
Appl. Math. Mech. -Egl. Ed. 008 9(7):967 973 DOI 10.1007/s10483-008-0716-y c Shaghai Uiversity ad Spriger-Verlag 008 Applied Mathematics ad Mechaics (Eglish Editio) Several properties of ew ellipsoids
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationReal Variables II Homework Set #5
Real Variables II Homework Set #5 Name: Due Friday /0 by 4pm (at GOS-4) Istructios: () Attach this page to the frot of your homework assigmet you tur i (or write each problem before your solutio). () Please
More informationROSE WONG. f(1) f(n) where L the average value of f(n). In this paper, we will examine averages of several different arithmetic functions.
AVERAGE VALUES OF ARITHMETIC FUNCTIONS ROSE WONG Abstract. I this paper, we will preset problems ivolvig average values of arithmetic fuctios. The arithmetic fuctios we discuss are: (1)the umber of represetatios
More informationA Lower Bound on the Density of Sphere Packings via Graph Theory. Michael Krivelevich, Simon Litsyn, and Alexander Vardy.
IMRN Iteratioal Mathematics Research Notices 004, No. 43 A Lower Boud o the Desity of Sphere Packigs via Graph Theory Michael Krivelevich, Simo Litsy, ad Alexader Vardy 1 Itroductio A sphere packig P i
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationON HEILBRONN S PROBLEM IN HIGHER DIMENSION
COMBINATORICA Bolyai Society Spriger-Verlag Combiatorica 23 (4 (2003 669 680 ON HEILBRONN S PROBLEM IN HIGHER DIMENSION HANNO LEFMANN Received April 26, 2000 Heilbro cojectured that give arbitrary poits
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationFew remarks on Ramsey-Turán-type problems Benny Sudakov Λ Abstract Let H be a fixed forbidden graph and let f be a function of n. Denote by RT n; H; f
Few remarks o Ramsey-Turá-type problems Bey Sudakov Abstract Let H be a fixed forbidde graph ad let f be a fuctio of. Deote by ; H; f () the maximum umber of edges a graph G o vertices ca have without
More informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationLecture 2. The Lovász Local Lemma
Staford Uiversity Sprig 208 Math 233A: No-costructive methods i combiatorics Istructor: Ja Vodrák Lecture date: Jauary 0, 208 Origial scribe: Apoorva Khare Lecture 2. The Lovász Local Lemma 2. Itroductio
More informationBIRKHOFF ERGODIC THEOREM
BIRKHOFF ERGODIC THEOREM Abstract. We will give a proof of the poitwise ergodic theorem, which was first proved by Birkhoff. May improvemets have bee made sice Birkhoff s orgial proof. The versio we give
More informationRelations between the continuous and the discrete Lotka power function
Relatios betwee the cotiuous ad the discrete Lotka power fuctio by L. Egghe Limburgs Uiversitair Cetrum (LUC), Uiversitaire Campus, B-3590 Diepebeek, Belgium ad Uiversiteit Atwerpe (UA), Campus Drie Eike,
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationIT is well known that Brouwer s fixed point theorem can
IAENG Iteratioal Joural of Applied Mathematics, 4:, IJAM_4 0 Costructive Proof of Brouwer s Fixed Poit Theorem for Sequetially Locally No-costat ad Uiformly Sequetially Cotiuous Fuctios Yasuhito Taaka,
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationMarcinkiwiecz-Zygmund Type Inequalities for all Arcs of the Circle
Marcikiwiecz-ygmud Type Iequalities for all Arcs of the Circle C.K. Kobidarajah ad D. S. Lubisky Mathematics Departmet, Easter Uiversity, Chekalady, Sri Laka; Mathematics Departmet, Georgia Istitute of
More informationA note on log-concave random graphs
A ote o log-cocave radom graphs Ala Frieze ad Tomasz Tocz Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh PA53, USA Jue, 08 Abstract We establish a threshold for the coectivity of
More informationREGRESSION WITH QUADRATIC LOSS
REGRESSION WITH QUADRATIC LOSS MAXIM RAGINSKY Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X, Y ), where, as before, X is a R d
More informationThe random version of Dvoretzky s theorem in l n
The radom versio of Dvoretzky s theorem i l Gideo Schechtma Abstract We show that with high probability a sectio of the l ball of dimesio k cε log c > 0 a uiversal costat) is ε close to a multiple of the
More informationROTATION-EQUIVALENCE CLASSES OF BINARY VECTORS. 1. Introduction
t m Mathematical Publicatios DOI: 10.1515/tmmp-2016-0033 Tatra Mt. Math. Publ. 67 (2016, 93 98 ROTATION-EQUIVALENCE CLASSES OF BINARY VECTORS Otokar Grošek Viliam Hromada ABSTRACT. I this paper we study
More informationThe multiplicative structure of finite field and a construction of LRC
IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationLarge holes in quasi-random graphs
Large holes i quasi-radom graphs Joaa Polcy Departmet of Discrete Mathematics Adam Mickiewicz Uiversity Pozań, Polad joaska@amuedupl Submitted: Nov 23, 2006; Accepted: Apr 10, 2008; Published: Apr 18,
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationA Simple Proof of the Shallow Packing Lemma
A Simple Proof of the Shallow Packig Lemma Nabil Mustafa To cite this versio: Nabil Mustafa. A Simple Proof of the Shallow Packig Lemma. Discrete ad Computatioal Geometry, Spriger Verlag, 06, 55 (3), pp.739-743.
More informationA Quantitative Lusin Theorem for Functions in BV
A Quatitative Lusi Theorem for Fuctios i BV Adrás Telcs, Vicezo Vespri November 19, 013 Abstract We exted to the BV case a measure theoretic lemma previously proved by DiBeedetto, Giaazza ad Vespri ([1])
More informationThe Boolean Ring of Intervals
MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,
More informationSome remarks for codes and lattices over imaginary quadratic
Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationResolvent Estrada Index of Cycles and Paths
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 8, 1 (216), 1-1. Resolvet Estrada Idex of Cycles ad Paths Bo Deg, Shouzhog Wag, Iva Gutma Abstract:
More informationif > 6 is sucietly large). Nevertheless, Pichasi has show that the umber of radial poits of a o-colliear set P of poits i the plae that lie i a halfpl
Radial Poits i the Plae Jaos Pach y Micha Sharir z Jauary 6, 00 Abstract A radial poit for a ite set P i the plae is a poit q 6 P with the property that each lie coectig q to a poit of P passes through
More informationCATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01
CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ
More informationMetric Space Properties
Metric Space Properties Math 40 Fial Project Preseted by: Michael Brow, Alex Cordova, ad Alyssa Sachez We have already poited out ad will recogize throughout this book the importace of compact sets. All
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationA Recurrence Formula for Packing Hyper-Spheres
A Recurrece Formula for Packig Hyper-Spheres DokeyFt. Itroductio We cosider packig of -D hyper-spheres of uit diameter aroud a similar sphere. The kissig spheres ad the kerel sphere form cells of equilateral
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationRegression with quadratic loss
Regressio with quadratic loss Maxim Ragisky October 13, 2015 Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X,Y, where, as before,
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationCANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS
CANTOR SETS WHICH ARE MINIMAL FOR QUASISYMMETRIC MAPS HRANT A. HAKOBYAN Abstract. We show that middle iterval Cator sets of Hausdorff dimesio are miimal for quasisymmetric maps of a lie. Combiig this with
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More informationDisjoint Systems. Abstract
Disjoit Systems Noga Alo ad Bey Sudaov Departmet of Mathematics Raymod ad Beverly Sacler Faculty of Exact Scieces Tel Aviv Uiversity, Tel Aviv, Israel Abstract A disjoit system of type (,,, ) is a collectio
More informationCOMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION (x 1) + x = n = n.
COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where
More informationA constructive analysis of convex-valued demand correspondence for weakly uniformly rotund and monotonic preference
MPRA Muich Persoal RePEc Archive A costructive aalysis of covex-valued demad correspodece for weakly uiformly rotud ad mootoic preferece Yasuhito Taaka ad Atsuhiro Satoh. May 04 Olie at http://mpra.ub.ui-mueche.de/55889/
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 12
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract I this lecture we derive risk bouds for kerel methods. We will start by showig that Soft Margi kerel SVM correspods to miimizig
More informationLecture 7: Density Estimation: k-nearest Neighbor and Basis Approach
STAT 425: Itroductio to Noparametric Statistics Witer 28 Lecture 7: Desity Estimatio: k-nearest Neighbor ad Basis Approach Istructor: Ye-Chi Che Referece: Sectio 8.4 of All of Noparametric Statistics.
More informationAn Extremal Property of the Regular Simplex
Covex Geometric Aalysis MSRI Publicatios Volume 34, 1998 A Extremal Property of the Regular Simplex MICHAEL SCHMUCKENSCHLÄGER Abstract. If C is a covex body i R such that the ellipsoid of miimal volume
More informationLONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES
J Lodo Math Soc (2 50, (1994, 465 476 LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES Jerzy Wojciechowski Abstract I [5] Abbott ad Katchalski ask if there exists a costat c >
More informationMinimal surface area position of a convex body is not always an M-position
Miimal surface area positio of a covex body is ot always a M-positio Christos Saroglou Abstract Milma proved that there exists a absolute costat C > 0 such that, for every covex body i R there exists a
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationA Simplified Binet Formula for k-generalized Fibonacci Numbers
A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationIndependence number of graphs with a prescribed number of cliques
Idepedece umber of graphs with a prescribed umber of cliques Tom Bohma Dhruv Mubayi Abstract We cosider the followig problem posed by Erdős i 1962. Suppose that G is a -vertex graph where the umber of
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationOn a Smarandache problem concerning the prime gaps
O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationIntroduction to Computational Biology Homework 2 Solution
Itroductio to Computatioal Biology Homework 2 Solutio Problem 1: Cocave gap pealty fuctio Let γ be a gap pealty fuctio defied over o-egative itegers. The fuctio γ is called sub-additive iff it satisfies
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationLecture #20. n ( x p i )1/p = max
COMPSCI 632: Approximatio Algorithms November 8, 2017 Lecturer: Debmalya Paigrahi Lecture #20 Scribe: Yua Deg 1 Overview Today, we cotiue to discuss about metric embeddigs techique. Specifically, we apply
More informationPRACTICE FINAL/STUDY GUIDE SOLUTIONS
Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)
More informationLecture 6 Simple alternatives and the Neyman-Pearson lemma
STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull
More informationA NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS
Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationChapter IV Integration Theory
Chapter IV Itegratio Theory Lectures 32-33 1. Costructio of the itegral I this sectio we costruct the abstract itegral. As a matter of termiology, we defie a measure space as beig a triple (, A, µ), where
More informationProc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS
Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of
More informationLecture 12: Subadditive Ergodic Theorem
Statistics 205b: Probability Theory Sprig 2003 Lecture 2: Subadditive Ergodic Theorem Lecturer: Jim Pitma Scribe: Soghwai Oh 2. The Subadditive Ergodic Theorem This theorem is due
More informationOnline hypergraph matching: hiring teams of secretaries
Olie hypergraph matchig: hirig teams of secretaries Rafael M. Frogillo Advisor: Robert Kleiberg May 29, 2008 Itroductio The goal of this paper is to fid a competitive algorithm for the followig problem.
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationP. Z. Chinn Department of Mathematics, Humboldt State University, Arcata, CA
RISES, LEVELS, DROPS AND + SIGNS IN COMPOSITIONS: EXTENSIONS OF A PAPER BY ALLADI AND HOGGATT S. Heubach Departmet of Mathematics, Califoria State Uiversity Los Ageles 55 State Uiversity Drive, Los Ageles,
More informationSAMPLING LIPSCHITZ CONTINUOUS DENSITIES. 1. Introduction
SAMPLING LIPSCHITZ CONTINUOUS DENSITIES OLIVIER BINETTE Abstract. A simple ad efficiet algorithm for geeratig radom variates from the class of Lipschitz cotiuous desities is described. A MatLab implemetatio
More informationON THE FUZZY METRIC SPACES
The Joural of Mathematics ad Computer Sciece Available olie at http://www.tjmcs.com The Joural of Mathematics ad Computer Sciece Vol.2 No.3 2) 475-482 ON THE FUZZY METRIC SPACES Received: July 2, Revised:
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationCS 330 Discussion - Probability
CS 330 Discussio - Probability March 24 2017 1 Fudametals of Probability 11 Radom Variables ad Evets A radom variable X is oe whose value is o-determiistic For example, suppose we flip a coi ad set X =
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationA RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a
More informationLecture 27. Capacity of additive Gaussian noise channel and the sphere packing bound
Lecture 7 Ageda for the lecture Gaussia chael with average power costraits Capacity of additive Gaussia oise chael ad the sphere packig boud 7. Additive Gaussia oise chael Up to this poit, we have bee
More information16 Riemann Sums and Integrals
16 Riema Sums ad Itegrals Defiitio: A partitio P of a closed iterval [a, b], (b >a)isasetof 1 distict poits x i (a, b) togetherwitha = x 0 ad b = x, together with the covetio that i>j x i >x j. Defiitio:
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationAMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.
J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.
More informationA collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More informationFundamental Theorem of Algebra. Yvonne Lai March 2010
Fudametal Theorem of Algebra Yvoe Lai March 010 We prove the Fudametal Theorem of Algebra: Fudametal Theorem of Algebra. Let f be a o-costat polyomial with real coefficiets. The f has at least oe complex
More information