Covering an ellipsoid with equal balls


 Randolf Caldwell
 10 months ago
 Views:
Transcription
1 Journal of Combinatorial Theory, Series A 113 (2006) Covering an ellipsoid with equal balls Ilya Dumer College of Engineering, University of California, Riverside, CA 92521, USA Received 1 October 2005 Available online 13 June 2006 Abstract The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, our goal is to find the minimum number of unit balls needed to cover this ellipsoid. A tight asymptotic bound on the logarithm of this number is obtained Elsevier Inc. All rights reserved. Keywords: Ellipsoid; Euclidean space; Spherical covering; Unit ball 1. Introduction 1.1. Ellipsoids and coverings Consider the ball Bε n(y) of radius ε centered at some point y = (y 1,...,y n ) in an n dimensional Euclidean space R n : { Bε n def (y) = x R n n (x i y i ) 2 ε }. 2 For any subset A R n, a subset M ε (A) R n is called its εcovering if A is contained in the union of the balls of radius ε centered at points y M ε (A): A Bε n (y). y M ε (A) address: /$ see front matter 2006 Elsevier Inc. All rights reserved. doi: /j.jcta
2 1668 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) The socalled εentropy [1] ℵ ε (A) of a set A is defined as ℵ ε (A) = log min M ε (A), M ε (A) where minimum is taken over all coverings and the logarithm is base e. Below we study the εentropy of an arbitrary ellipsoid { } def = x R n n x 2 i 1, (1) E n a a 2 i where a = (a 1,...,a n ) is a vector with n positive symbols. Without loss of generality, we assume that 0 <a 1 a 2 a n. By linear transformation of R n, we can always replace a subset A and its covering M ε (A) using the unit balls B1 n (y) on the rescaled subset A/ε. More generally, we can use different scaling factors b i for different axis x i. Thus, the following three problems are equivalent: (1) covering an ellipsoid Ea n with unit balls; (2) covering an ellipsoid Ea n with balls of radius ε; (3) covering an ellipsoid Ea n with (smaller) ellipsoids { } Eb n def (y) = x R n n (x i y i ) 2 1. b 2 i Due to this equivalence, we will consider coverings with unit balls and remove the subscript ε from our notation. Our main goal is to find the asymptotic (unit) entropy ℵ(Ea n ) as a function of n and a. Here we consider the subsets of ellipsoids such that ℵ(Ea n) Coverings of the balls Optimal coverings have been long studied for an Euclidean ball Bρ n = Bn ρ (0). Various bounds on its minimum covering size are obtained in papers [2,3]. In particular, it follows from these papers that for any n 1 and ρ 1, ℵ ( Bρ) n n log ρ + c log(n + 1). (2) Here and in the sequel, c and c i denote some universal constants. We also mention the Few Coxeter Rogers lower bound ℵ ( Bρ) n n log ρ + c0 if ρ>n. For more details, we refer to the monographs [4,5], and survey [6], which give a detailed account of the subject along with an extensive bibliography. Coverings of other sets have also been studied for general convex bodies (see [7] and references therein). 2. Prior and present results Note that ℵ(E 1 a ) = log a for n = 1. Thus, we assume that n 2. Given some θ (0, 1/2), we decompose the set of positions N ={1,...,n} into the three consecutive subsets:
3 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) J θ,0 = { j:0<aj 2 1 θ} ; J θ,1 = { j:1 θ<aj 2 1} ; J ={j: a j > 1}. Let μ = μ θ = J θ,1 + J, m= J. (3) We also assume that m 1, since otherwise Ea n is covered with the single ball Bn. Given an ellipsoid Ea n, consider the geometric mean of the m largest halfaxes and the sum of their logarithms ρ def = a 1/m j, (4) j J K def = n j=n m+1 log a j = m log ρ. (5) Note that the ball Bρ m has the same volume as the mdimensional subellipsoid { def = x R m x 2 } j 1 E m a a 2 j J j spanned over the m largest axes of the original ellipsoid Ea n. We begin with a lower bound. Theorem 1 (Generalized packing bound [7]). The entropy of any ellipsoid Ea n satisfies inequality ℵ ( Ea n ) K. (6) The following theorem is valid for all ellipsoids and directly follows from the more general results [7] for convex bodies. It is also a reformulation of Theorem 3 of [8]. Theorem 2. For any θ (0, 1/2), the entropy of an ellipsoid Ea n satisfies inequality ℵ ( Ea n ) K + μθ log(3/θ). (7) From now on, consider the sets of ellipsoids Ea n with K, so that subellipsoids Em a have growing size. According to Theorem 2, asymptotic equality ℵ ( Ea n ) = K + o(k) (8) holds if there exists θ (0, 1/2) such that K/μ θ. (9) Note, however, that condition (9) is very restrictive and holds only for the sets of expanding ellipsoids, such that ρ. In particular, it fails on a ball Bρ n of any given radius ρ>1. The following asymptotic bound of [8] removes this drawback. Theorem 3. Asymptotic equality (8) holds for the ellipsoids Ea n provided that ( ) log a n K log ρ = o. log n (10)
4 1670 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Note that condition (10) implies that K log n, (11) in which case the volume of the largest subellipsoid Ea m exceeds any polynomial in n. Our main goal is to obtain asymptotic equality (8) for a broader class of ellipsoids. Firstly, we shall refine condition (10) so that the largest coefficient a n will depend on m and ρ only. Secondly, we show that all positions of the subset J θ,0 are insignificant for the entropy ℵ(Ea n ), which allows us to replace parameter n in (11) with parameter μ θ. Finally, we show that asymptotic bounds only slightly depend on θ and can be extended to the case of θ 0. These results are summarized as follows. Theorem 4. Consider the set of ellipsoids Ea n that satisfy restriction ( ) log a n K log ρ = o (12) log m for any m 2. Then for any parameter θ (0, 1/2), any ellipsoid Ea n has entropy ℵ ( Ea n ) K + o(k) + c1 log(μ θ + 1) + c 2 log(1/θ), K, (13) where c 1 and c 2 are universal constants. By taking θ = e K/log K, we obtain the following corollary. Corollary 5. Consider the set of ellipsoids Ea n that satisfy condition (12). Let m = m (K) = { a j : aj 2 1 e K/log K}. Then ellipsoids Ea n satisfy asymptotic equality (8) if K. log m (14) Note that new conditions (12) and (14) loosen former conditions (10) and (11). In particular, restriction (12) holds whenever the longest halfaxis a n is a polynomial ρ s of increasing degree s, as long as s = o(m log ρ/log m). Also, (14) admits any ellipsoid whose size grows faster than a polynomial in m. Finally, this number m includes only those m halfaxes a j, which either exceed 1 or are arbitrarily close to 1 (within an exponentially declining margin e K/o(K) for any function o(k) ). On the other hand, the following lemma shows that there exist ellipsoids, whose entropy (13) is dominated by the term log μ θ. Lemma 6. Consider an ellipsoid E n a with halfaxes a 1 = =a n 1 = 1 and any a n > 1. No n unit balls can cover E n a. 3. Proofs Proof of Theorem 4. The proof includes three main steps. In the first step, any ellipsoid Ea n will be enclosed into a finite number of subsets DR n, each of which is a direct product of the balls (of lesser dimensions). In the second step, we design a covering for each DR n. In the third step, we obtain a universal upper bound on the entropy ℵ(Ea n ) and optimize its asymptotic parameters.
5 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Step 1. Divide the set N of n positions into some number t + 1 of consecutive intervals J 0 = J θ,0, J 1 = J θ,1, J 2,...,J t, where t m + 1. Below we use notation J i =[n i + 1,n i+1 ] for any interval J i and denote its length s i = n i+1 n i. Here n 0 = 0 and n t+1 = n. For any θ, m, and μ = μ θ, we also use the parameter z = 2(μ + 1) 2 /θ 2 and define t + 1 approximation grids P i, where P 0 : pl 0 = e lθ/2, l = 0,..., θ 2 log z, P 1 : pl 1 = e l(log z)/μ, l = 0,...,μ, P i : p l = e l(log z)/m, l = 0,...,m, i = 2,...,t. Thus, all t + 1 grids have the same range [1/z, 1], and the last t 1 grids are identical. Without loss of generality, we assume that θ 2 log z is an integer. These grids are used as follows. For any point x Ea n, we take any interval J i and consider the subvector x Ji = (x j j J i ), i = 0,...,t, of length s i. Then we define the vector R = R(x) = (r 0,...,r t ) with symbols (15) r i = j J i x 2 j /a2 j, i = 0,...,t. By the definition of an ellipsoid E n a, r i 1. i=0 Each symbol r i is then rounded off to the two closest (but not necessarily different) points r i and r i on the grid P i taken as follows: { ri = r i = 1/z if r i 1/z, (17) r i r i r i, r i, r i P i if r i > 1/z. Finally, for any vector R, we use its approximation R = ( r 0,..., r t ). These vectors R form the set {R} of size N P 0 P 1 P i t 1 = 2(m + 1) t 1 (μ + 1)(log z)/θ. (18) Now, for any i, define the ball { B s i } ρ i = x Ji xj 2 ρ2 i, ρi 2 = a2 n i+1 r i, j J i of dimension s i and radius ρ i. Then we consider the direct products DR n D n R def = t i=0 { B s i ρ i = x R n j J i of all t + 1 balls: (16) xj 2 } an 2 r i,i= 0,...,t. (19) i+1
6 1672 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Lemma 7. The original ellipsoid E n a is contained in the union of the sets Dn R : E n a {R} D n R. (20) Proof. For any point x Ea n, consider the corresponding vectors R(x) and R. Recall that a ni+1 = max{a j,j J i }. Then xj 2 /a2 n i+1 xj 2 /a2 j = r i r i, j J i j J i and x Ji B s i ρ i for all i. Thus, by considering all possible vectors R, we find a subset DR n that covers any point x Ea n, and (20) holds. Step 2. Given any vector R, our next goal is to cover each subset DR n defined in (19) with unit balls. In doing so, we cover each ball B s i ρ i,i= 0,...,t,with the balls B s i e i of some radius e i. Then the direct product DR n of the balls is completely covered by the direct product of their coverings. Given any vector R, we choose the covering radii { ei 2 def a 2 n1 r 0 if i = 0, = ε i = r i (1 2μ 1 ) if i 1. (21) The following lemma shows that the direct product of the covering balls is contained in the unit ball: t i=0 B s i e i B n. Lemma 8. For any vector R, vector E = (ε 0,...,ε t ) satisfies restrictions ε i 1. i=0 (22) Proof. First, consider the intervals J i,i= 0,...,t,on which r i 1/z. Since t μ, ε i 1/z t + 1 θ 2 z μ + 1. (23) i: r i 1/z i: r i 1/z Next we proceed with r i > 1/z. For r 0 > 1/z, we use the fact that a 2 n 1 1 θ in definition (21). Also, e θ/2 1 + θ for θ [0, 1]. Then ε 0 r 0 (1 θ) r 0 (1 + θ)(1 θ)= r 0 ( 1 θ 2 ). (24) Also, definition (21) shows that ( ε i r i 1 1 2μ 1 i t r i >1/z Thus, (22) follows from (23) (25): ) ( (1 r 0 ) 1 1 ). (25) 2μ
7 i=0 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) ( ε i r 0 1 θ 2 ) ( + (1 r 0 ) 1 1 ) + θ 2 2μ μ + 1 { max 1 θ 2, 1 1 } + θ μ μ + 1 Step 3. Our goal is to estimate the entropy ℵ ( ( Ea n ) ) ℵ DR n max ℵ( D n ) R + log N. (26) R R Consider any set DR n. According to (21), the first ball Bs 0 ρ 0 is entirely covered by the ball of the same radius e 0. Also, for the remaining t balls B s i ρ i, the universal bound (2) gives the following estimates: ℵ ( DR n ) ( ) = ℵ ei B s i ρ i s i log ρ i + c log(s i + 1). (27) e i Note that log(1 α) 2α for any α [0, 1/2]. Thus for all i 1, definitions (15), (17), and (21) give inequalities log r i log r i + log r i log z ε i r i ε i m + 1 μ, log r 1 = log r 1 + log r 1 log z ε 1 r 1 ε 1 μ + 1 μ. These two inequalities are used as follows. For i = 1, the interval J 1 has length s 1 = μ m. Then s 1 log ρ 1 = s 1 e 1 2 log a2 n 1 r 1 s 1 ε 1 2 log r 1 log z + 1 ε (28) Similarly, for any i 2, we obtain a uniform estimates for all vectors R: s i log ρ i s i e i 2 log a2 n i+1 r i ( log z s i log a ni+1 + ε i m + 1 ) s i μ 2 s i log a ni+1 + log z (29) Now note that log(s i + 1) log(μ + 1) + (t 1) log(m + 1). (30) Thus, estimates (26) (30) give the universal bound ℵ ( Ea n ) s i log a ni+1 + c log(s i + 1) + log z + log N + 1 s i log a ni+1 + c 1 log(μ + 1) + c 2 log θ + C(t 1) log(m + 1). (31)
8 1674 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) For m = 1, we have t 2, and bound (13) readily follows from (31). For m 2 and K, let η = η(k) be a positive function such that ( ) lim η = 0, log a n ηk K log ρ = o. (32) log m Obviously, our original condition (12) can be replaced with (32) if function η approaches 0 slowly enough. We then choose the intervals J i of length m log m s i s =, i = 2,...,t 1, ηk s t s. Then ηk t 1 = m/s, log m (t 1) log(m + 1) 2ηK + log(m + 1). (33) Finally, note that restriction (32) can be rewritten as ( ηk 2 ) log a n = o. m log m Then the first term in (31) gives n s i log a ni+1 log a j + (s i 1) log a n i+1 a n K + (s 1) log a ni +1 a n m+1 j=n m = K + o(k). The latter bound combined with (31) and (33) gives our main estimate (13), and the proof of Theorem 4 is completed. Proof of Lemma 6. Assume that n unit balls cover Ea n. Then the centers of the balls belong to some hyperplane H in R n. First, suppose that H does not contain the longest axis x n. Consider the orthogonal line OA H from the origin O that crosses the surface of Ea n at some point A. Since a n > 1, this point A is located at the distance d(a,h) > 1 and is not covered by any unit ball. Secondly, let H contain the axis x n. We take any positive parameter λ< a n 1 (34) a n + 1 and consider L n + 1 points O i = ( 0,...,0,λ i), i = 1,...,L, on the axis x n. Next, we consider the L lines O i A i H, all orthogonal to H. Here the points A i belong to the surface of Ea n. For each A i, let D i H be the center of its covering ball B(D i ). By the definitions of the ellipsoid Ea n and the unit ball B(D i), the Euclidean distance d(x,y) satisfies the following: d 2 (O, O i )/an 2 + d2 (O i,a i ) = 1, d 2 (O i,d i ) + d 2 (O i,a i ) 1.
9 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Thus, d(o i,d i ) d(o,o i )/a n = λ i /a n. Then for any two indices i<j, d(o i,d i ) + d(o j,d j ) λi + λ j λ i 1 + λ < 2λi a n a n a n + 1, in accordance with restriction (34). On the other hand, d(o i,o j ) = λ i λ j λ i (1 λ) > Now we see that 2λi a n + 1. d(o i,d i ) + d(o j,d j )<d(o i,o j ), (35) and no two points A i and A j can have the same center D = D i = D j, by contradiction to (35). Thus, no L 1 separate balls with centers on the same hyperplane H can cover all L points A i (despite the fact that these points have vanishing distance if n or a n 1). This contradiction shows that the centers D i may not belong to the same hyperplane and there are more than n centers needed for complete covering. Note that similar arguments can also be extended to the case when some coefficients a i are less than 1. In particular, Lemma 6 can be verified for any a n > 1 and any parameter c (1,a n ) if we take an ellipsoid En a, where a1 2 = =a2 n 1 1 λ2n+1 (c 1) 2an 2, λ= a n c a n + c Concluding remarks Replacing an original ellipsoid with direct products of the balls was first used in [8]. Present design differs in the following aspects. Firstly, exponentially declining steps are now used instead of the uniform quantization of [8]. Secondly, different approximation grids are applied to different positions. Finally, for each vector R, our radii e i are specified directly. Instead of this, nonconvex optimization was performed in [8] to find the worst vectors R, which give the highest contribution to the entire entropy. Our design includes two parts. In the first, approximation, part, we increase and round off the original quantities r i employed to cover the axes of an ellipsoid. To minimize the overhead caused by this expansion, the approximation grids {P i } have to be stretched to a very low level 1/z. These grids should also have sufficient density (to yield small approximation errors) and sufficiently small size (to avoid prohibitively many subsets D n R ). Exponentially declining levels { r i} resolved these problems. In the second, covering, part, we have to compensate for the increase in radius e 0 employed on the first interval J θ,0. This problem is addressed by using a small multiplying step e θ/2 in the first grid P 0. To obtain sufficiently small approximation errors r i /ε i on the remaining intervals, we define the radii e i through the second set of approximation levels { r i }. In turn, this double approximation allowed us to meet the restrictions of Theorem 4 and Corollary 5. Acknowledgment The author thanks V. Prelov for helpful suggestions.
10 1676 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) References [1] A.N. Kolmogorov, V.M. Tikhomirov, εentropy and εcapacity, Uspekhi Mat. Nauk 14 (1959) [2] C.A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963) [3] J.L. VergerGaugry, Covering a ball with smaller equal balls in R n, Discrete Comput. Geom. 33 (2005) [4] K. Böröczky Jr., Finite Packing and Covering, Cambridge Tracts in Math., vol. 154, Cambridge Univ. Press, Cambridge, [5] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, SpringerVerlag, New York, [6] G. Fejes Tóth, New results in the theory of packing and covering, in: P.M. Gruber, J.M. Wills (Eds.), Convexity and Its Applications, 1983, pp [7] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, Cambridge Univ. Press, Cambridge, [8] I. Dumer, V. Prelov, M. Pinsker, On the coverings of an ellipsoid in the Euclidean space, IEEE Trans. Inform. Theory 50 (10) (2004)
Covering a sphere with caps: Rogers bound revisited
Covering a sphere with caps: Rogers bound revisited Ilya Dumer Abstract We consider coverings of a sphere S n r of radius r with the balls of radius one in an ndimensional Euclidean space R n. Our goal
More informationarxiv:math/ v1 [math.mg] 31 May 2006
Covering spheres with spheres arxiv:math/060600v1 [math.mg] 31 May 006 Ilya Dumer College of Engineering, University of California at Riverside, Riverside, CA 951, USA dumer@ee.ucr.edu Abstract Given a
More informationCovering Spheres with Spheres
Discrete Comput Geom 2007) 38: 665 679 DOI 10.1007/s0045400790007 Covering Spheres with Spheres Ilya Dumer Received: 1 June 2006 / Revised: 14 July 2007 / Published online: 2 September 2007 Springer
More information4488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 10, OCTOBER /$ IEEE
4488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 10, OCTOBER 2008 List Decoding of Biorthogonal Codes the Hadamard Transform With Linear Complexity Ilya Dumer, Fellow, IEEE, Grigory Kabatiansky,
More informationCovering the Plane with Translates of a Triangle
Discrete Comput Geom (2010) 43: 167 178 DOI 10.1007/s0045400992031 Covering the Plane with Translates of a Triangle Janusz Januszewski Received: 20 December 2007 / Revised: 22 May 2009 / Accepted: 10
More informationSoftDecision Decoding Using Punctured Codes
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 47, NO 1, JANUARY 2001 59 SoftDecision Decoding Using Punctured Codes Ilya Dumer, Member, IEEE Abstract Let a ary linear ( )code be used over a memoryless
More informationTHIS paper is aimed at designing efficient decoding algorithms
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 7, NOVEMBER 1999 2333 SortandMatch Algorithm for SoftDecision Decoding Ilya Dumer, Member, IEEE Abstract Let a qary linear (n; k)code C be used
More informationON FREE PLANES IN LATTICE BALL PACKINGS ABSTRACT. 1. Introduction
ON FREE PLANES IN LATTICE BALL PACKINGS MARTIN HENK, GÜNTER M ZIEGLER, AND CHUANMING ZONG ABSTRACT This note, by studying relations between the length of the shortest lattice vectors and the covering minima
More informationLARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011
LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS S. G. Bobkov and F. L. Nazarov September 25, 20 Abstract We study large deviations of linear functionals on an isotropic
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 19912 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationLATTICE POINT COVERINGS
LATTICE POINT COVERINGS MARTIN HENK AND GEORGE A. TSINTSIFAS Abstract. We give a simple proof of a necessary and sufficient condition under which any congruent copy of a given ellipsoid contains an integral
More informationA Combinatorial Bound on the List Size
1 A Combinatorial Bound on the List Size Yuval Cassuto and Jehoshua Bruck California Institute of Technology Electrical Engineering Department MC 13693 Pasadena, CA 9115, U.S.A. Email: {ycassuto,bruck}@paradise.caltech.edu
More informationUniversal convex coverings
Bull. London Math. Soc. 41 (2009) 987 992 C 2009 London Mathematical Society doi:10.1112/blms/bdp076 Universal convex coverings Roland Bacher Abstract In every dimension d 1, we establish the existence
More informationEXACT INTERPOLATION, SPURIOUS POLES, AND UNIFORM CONVERGENCE OF MULTIPOINT PADÉ APPROXIMANTS
EXACT INTERPOLATION, SPURIOUS POLES, AND UNIFORM CONVERGENCE OF MULTIPOINT PADÉ APPROXIMANTS D S LUBINSKY A We introduce the concept of an exact interpolation index n associated with a function f and open
More informationA strongly polynomial algorithm for linear systems having a binary solution
A strongly polynomial algorithm for linear systems having a binary solution Sergei Chubanov Institute of Information Systems at the University of Siegen, Germany email: sergei.chubanov@unisiegen.de 7th
More informationThe 123 Theorem and its extensions
The 123 Theorem and its extensions Noga Alon and Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract It is shown
More informationBoundedly complete weakcauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weakcauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationSPHERE PACKINGS CONSTRUCTED FROM BCH AND JUSTESEN CODES
SPHERE PACKINGS CONSTRUCTED FROM BCH AND JUSTESEN CODES N. J. A. SLOANE Abstract. BoseChaudhuriHocquenghem and Justesen codes are used to pack equa spheres in Mdimensional Euclidean space with density
More informationUC Riverside UC Riverside Previously Published Works
UC Riverside UC Riverside Previously Published Works Title Softdecision decoding of ReedMuller codes: A simplied algorithm Permalink https://escholarship.org/uc/item/5v71z6zr Journal IEEE Transactions
More informationCS 6820 Fall 2014 Lectures, October 320, 2014
Analysis of Algorithms Linear Programming Notes CS 6820 Fall 2014 Lectures, October 320, 2014 1 Linear programming The linear programming (LP) problem is the following optimization problem. We are given
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationMeans of unitaries, conjugations, and the Friedrichs operator
J. Math. Anal. Appl. 335 (2007) 941 947 www.elsevier.com/locate/jmaa Means of unitaries, conjugations, and the Friedrichs operator Stephan Ramon Garcia Department of Mathematics, Pomona College, Claremont,
More informationAnother LowTechnology Estimate in Convex Geometry
Convex Geometric Analysis MSRI Publications Volume 34, 1998 Another LowTechnology Estimate in Convex Geometry GREG KUPERBERG Abstract. We give a short argument that for some C > 0, every n dimensional
More informationBOUNDS FOR SOLID ANGLES OF LATTICES OF RANK THREE
BOUNDS FOR SOLID ANGLES OF LATTICES OF RANK THREE LENNY FUKSHANSKY AND SINAI ROBINS Abstract. We find sharp absolute consts C and C with the following property: every wellrounded lattice of rank 3 in
More informationJeongHyun Kang Department of Mathematics, University of West Georgia, Carrollton, GA
#A33 INTEGERS 10 (2010), 379392 DISTANCE GRAPHS FROM P ADIC NORMS JeongHyun Kang Department of Mathematics, University of West Georgia, Carrollton, GA 30118 jkang@westga.edu Hiren Maharaj Department
More informationsatisfying ( i ; j ) = ij Here ij = if i = j and 0 otherwise The idea to use lattices is the following Suppose we are given a lattice L and a point ~x
Dual Vectors and Lower Bounds for the Nearest Lattice Point Problem Johan Hastad* MIT Abstract: We prove that given a point ~z outside a given lattice L then there is a dual vector which gives a fairly
More informationarxiv: v1 [cs.cg] 16 May 2011
Collinearities in Kinetic Point Sets arxiv:1105.3078v1 [cs.cg] 16 May 2011 Ben D. Lund George B. Purdy Justin W. Smith Csaba D. Tóth August 24, 2018 Abstract Let P be a set of n points in the plane, each
More informationRolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1
Journal of Mathematical Analysis and Applications 265, 322 33 (2002) doi:0.006/jmaa.200.7708, available online at http://www.idealibrary.com on Rolle s Theorem for Polynomials of Degree Four in a Hilbert
More informationA Note on the Distribution of the Distance from a Lattice
Discrete Comput Geom (009) 4: 6 76 DOI 0007/s00454008935 A Note on the Distribution of the Distance from a Lattice Ishay Haviv Vadim Lyubashevsky Oded Regev Received: 30 March 007 / Revised: 30 May
More informationSphere Packings. Ji Hoon Chun. Thursday, July 25, 2013
Sphere Packings Ji Hoon Chun Thursday, July 5, 03 Abstract The density of a (point) lattice sphere packing in n dimensions is the volume of a sphere in R n divided by the volume of a fundamental region
More informationHelly's Theorem and its Equivalences via Convex Analysis
Portland State University PDXScholar University Honors Theses University Honors College 2014 Helly's Theorem and its Equivalences via Convex Analysis Adam Robinson Portland State University Let us know
More informationCombinatorial Optimization
Combinatorial Optimization 20172018 1 Maximum matching on bipartite graphs Given a graph G = (V, E), find a maximum cardinal matching. 1.1 Direct algorithms Theorem 1.1 (Petersen, 1891) A matching M is
More informationAn example of a convex body without symmetric projections.
An example of a convex body without symmetric projections. E. D. Gluskin A. E. Litvak N. TomczakJaegermann Abstract Many crucial results of the asymptotic theory of symmetric convex bodies were extended
More informationOn bisectors in Minkowski normed space.
On bisectors in Minkowski normed space. Á.G.Horváth Department of Geometry, Technical University of Budapest, H1521 Budapest, Hungary November 6, 1997 Abstract In this paper we discuss the concept of
More informationMeasuring Ellipsoids 1
Measuring Ellipsoids 1 Igor Rivin Temple University 2 What is an ellipsoid? E = {x E n Ax = 1}, where A is a nonsingular linear transformation of E n. Remark that Ax = Ax, Ax = x, A t Ax. The matrix Q
More informationCombinatorial Generalizations of Jung s Theorem
Discrete Comput Geom (013) 49:478 484 DOI 10.1007/s0045401394913 Combinatorial Generalizations of Jung s Theorem Arseniy V. Akopyan Received: 15 October 011 / Revised: 11 February 013 / Accepted: 11
More informationA. Iosevich and I. Laba January 9, Introduction
KDISTANCE SETS, FALCONER CONJECTURE, AND DISCRETE ANALOGS A. Iosevich and I. Laba January 9, 004 Abstract. In this paper we prove a series of results on the size of distance sets corresponding to sets
More informationRademacher Averages and Phase Transitions in Glivenko Cantelli Classes
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 1, JANUARY 2002 251 Rademacher Averages Phase Transitions in Glivenko Cantelli Classes Shahar Mendelson Abstract We introduce a new parameter which
More informationA proof of the existence of good nested lattices
A proof of the existence of good nested lattices Dinesh Krithivasan and S. Sandeep Pradhan July 24, 2007 1 Introduction We show the existence of a sequence of nested lattices (Λ (n) 1, Λ(n) ) with Λ (n)
More informationSquares in products with terms in an arithmetic progression
ACTA ARITHMETICA LXXXVI. (998) Squares in products with terms in an arithmetic progression by N. Saradha (Mumbai). Introduction. Let d, k 2, l 2, n, y be integers with gcd(n, d) =. Erdős [4] and Rigge
More informationHarmonic Polynomials and DirichletType Problems. 1. Derivatives of x 2 n
Harmonic Polynomials and DirichletType Problems Sheldon Axler and Wade Ramey 30 May 1995 Abstract. We take a new approach to harmonic polynomials via differentiation. Surprisingly powerful results about
More informationCHAPTER 2. CONFORMAL MAPPINGS 58
CHAPTER 2. CONFORMAL MAPPINGS 58 We prove that a strong form of converse of the above statement also holds. Please note we could apply the Theorem 1.11.3 to prove the theorem. But we prefer to apply the
More information1: Introduction to Lattices
CSE 206A: Lattice Algorithms and Applications Winter 2012 Instructor: Daniele Micciancio 1: Introduction to Lattices UCSD CSE Lattices are regular arrangements of points in Euclidean space. The simplest
More informationSzemerédiTrotter theorem and applications
SzemerédiTrotter theorem and applications M. Rudnev December 6, 2004 The theorem Abstract These notes cover the material of two Applied postgraduate lectures in Bristol, 2004. SzemerédiTrotter theorem
More informationResearch Article Translative Packing of Unit Squares into Squares
International Mathematics and Mathematical Sciences Volume 01, Article ID 61301, 7 pages doi:10.1155/01/61301 Research Article Translative Packing of Unit Squares into Squares Janusz Januszewski Institute
More informationMATH 31BH Homework 1 Solutions
MATH 3BH Homework Solutions January 0, 04 Problem.5. (a) (x, y)plane in R 3 is closed and not open. To see that this plane is not open, notice that any ball around the origin (0, 0, 0) will contain points
More informationA Lower Estimate for Entropy Numbers
Journal of Approximation Theory 110, 120124 (2001) doi:10.1006jath.2000.3554, available online at http:www.idealibrary.com on A Lower Estimate for Entropy Numbers Thomas Ku hn Fakulta t fu r Mathematik
More informationThe Knaster problem and the geometry of highdimensional cubes
The Knaster problem and the geometry of highdimensional cubes B. S. Kashin (Moscow) S. J. Szarek (Paris & Cleveland) Abstract We study questions of the following type: Given positive semidefinite matrix
More informationDivide and Conquer. Arash Rafiey. 27 October, 2016
27 October, 2016 Divide the problem into a number of subproblems Divide the problem into a number of subproblems Conquer the subproblems by solving them recursively or if they are small, there must be
More informationAPPROXIMATE ISOMETRIES ON FINITEDIMENSIONAL NORMED SPACES
APPROXIMATE ISOMETRIES ON FINITEDIMENSIONAL NORMED SPACES S. J. DILWORTH Abstract. Every εisometry u between real normed spaces of the same finite dimension which maps the origin to the origin may by
More informationMinimizing Mean Flowtime and Makespan on MasterSlave Systems
Minimizing Mean Flowtime and Makespan on MasterSlave Systems Joseph YT. Leung,1 and Hairong Zhao 2 Department of Computer Science New Jersey Institute of Technology Newark, NJ 07102, USA Abstract The
More informationA NUMBERTHEORETIC CONJECTURE AND ITS IMPLICATION FOR SET THEORY. 1. Motivation
Acta Math. Univ. Comenianae Vol. LXXIV, 2(2005), pp. 243 254 243 A NUMBERTHEORETIC CONJECTURE AND ITS IMPLICATION FOR SET THEORY L. HALBEISEN Abstract. For any set S let seq (S) denote the cardinality
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationCSE 206A: Lattice Algorithms and Applications Spring Basis Reduction. Instructor: Daniele Micciancio
CSE 206A: Lattice Algorithms and Applications Spring 2014 Basis Reduction Instructor: Daniele Micciancio UCSD CSE No efficient algorithm is known to find the shortest vector in a lattice (in arbitrary
More informationInterpolation on lines by ridge functions
Available online at www.sciencedirect.com ScienceDirect Journal of Approximation Theory 175 (2013) 91 113 www.elsevier.com/locate/jat Full length article Interpolation on lines by ridge functions V.E.
More informationOn the constant in the reverse BrunnMinkowski inequality for pconvex balls.
On the constant in the reverse BrunnMinkowski inequality for pconvex balls. A.E. Litvak Abstract This note is devoted to the study of the dependence on p of the constant in the reverse BrunnMinkowski
More informationSpectrum for compact operators on Banach spaces
Submitted to Journal of the Mathematical Society of Japan Spectrum for compact operators on Banach spaces By Luis Barreira, Davor Dragičević Claudia Valls (Received Nov. 22, 203) (Revised Jan. 23, 204)
More informationAuerbach bases and minimal volume sufficient enlargements
Auerbach bases and minimal volume sufficient enlargements M. I. Ostrovskii January, 2009 Abstract. Let B Y denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in
More informationSELECTIVELY BALANCING UNIT VECTORS AART BLOKHUIS AND HAO CHEN
SELECTIVELY BALANCING UNIT VECTORS AART BLOKHUIS AND HAO CHEN Abstract. A set U of unit vectors is selectively balancing if one can find two disjoint subsets U + and U, not both empty, such that the Euclidean
More informationc 2000 Society for Industrial and Applied Mathematics
SIAM J. OPIM. Vol. 10, No. 3, pp. 750 778 c 2000 Society for Industrial and Applied Mathematics CONES OF MARICES AND SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES MASAKAZU KOJIMA AND LEVEN UNÇEL Abstract.
More informationGAPS BETWEEN FRACTIONAL PARTS, AND ADDITIVE COMBINATORICS. Antal Balog, Andrew Granville and Jozsef Solymosi
GAPS BETWEEN FRACTIONAL PARTS, AND ADDITIVE COMBINATORICS Antal Balog, Andrew Granville and Jozsef Solymosi Abstract. We give bounds on the number of distinct differences N a a as a varies over all elements
More informationFINITE CONNECTED HSPACES ARE CONTRACTIBLE
FINITE CONNECTED HSPACES ARE CONTRACTIBLE ISAAC FRIEND Abstract. The nonhausdorff suspension of the onesphere S 1 of complex numbers fails to model the group s continuous multiplication. Moreover, finite
More informationINDUSTRIAL MATHEMATICS INSTITUTE. B.S. Kashin and V.N. Temlyakov. IMI Preprint Series. Department of Mathematics University of South Carolina
INDUSTRIAL MATHEMATICS INSTITUTE 2007:08 A remark on compressed sensing B.S. Kashin and V.N. Temlyakov IMI Preprint Series Department of Mathematics University of South Carolina A remark on compressed
More informationSubsequences of frames
Subsequences of frames R. Vershynin February 13, 1999 Abstract Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is ndimensional then this subsequence has
More informationA note on transitive topological Markov chains of given entropy and period
A note on transitive topological Markov chains of given entropy and period Jérôme Buzzi and Sylvie Ruette November, 205 Abstract We show that, for every positive real number h and every positive integer
More informationSIMPLE AND POSITIVE ROOTS
SIMPLE AND POSITIVE ROOTS JUHA VALKAMA MASSACHUSETTS INSTITUTE OF TECHNOLOGY Let V be a Euclidean space, i.e. a real finite dimensional linear space with a symmetric positive definite inner product,. We
More informationDeviation Measures and Normals of Convex Bodies
Beiträge zur Algebra und Geometrie Contributions to Algebra Geometry Volume 45 (2004), No. 1, 155167. Deviation Measures Normals of Convex Bodies Dedicated to Professor August Florian on the occasion
More information1 Directional Derivatives and Differentiability
Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=
More informationBounds for the Eventual Positivity of Difference Functions of Partitions into Prime Powers
3 47 6 3 Journal of Integer Seuences, Vol. (7), rticle 7..3 ounds for the Eventual Positivity of Difference Functions of Partitions into Prime Powers Roger Woodford Department of Mathematics University
More informationAn algorithm for the satisfiability problem of formulas in conjunctive normal form
Journal of Algorithms 54 (2005) 40 44 www.elsevier.com/locate/jalgor An algorithm for the satisfiability problem of formulas in conjunctive normal form Rainer Schuler Abt. Theoretische Informatik, Universität
More informationON SUCCESSIVE RADII AND psums OF CONVEX BODIES
ON SUCCESSIVE RADII AND psums OF CONVEX BODIES BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE Abstract. We study the behavior of the so called successive inner and outer radii with respect to the psums
More informationTilburg University. Twodimensional maximin Latin hypercube designs van Dam, Edwin. Published in: Discrete Applied Mathematics
Tilburg University Twodimensional maximin Latin hypercube designs van Dam, Edwin Published in: Discrete Applied Mathematics Document version: Peer reviewed version Publication date: 2008 Link to publication
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationLinear distortion of Hausdorff dimension and Cantor s function
Collect. Math. 57, 2 (2006), 93 20 c 2006 Universitat de Barcelona Linear distortion of Hausdorff dimension and Cantor s function O. Dovgoshey and V. Ryazanov Institute of Applied Mathematics and Mechanics,
More informationCHAPTER 9. Embedding theorems
CHAPTER 9 Embedding theorems In this chapter we will describe a general method for attacking embedding problems. We will establish several results but, as the main final result, we state here the following:
More informationarxiv: v1 [math.oc] 21 Mar 2015
Convex KKM maps, monotone operators and Minty variational inequalities arxiv:1503.06363v1 [math.oc] 21 Mar 2015 Marc Lassonde Université des Antilles, 97159 Pointe à Pitre, France Email: marc.lassonde@univag.fr
More informationTHE information capacity is one of the most important
256 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 1, JANUARY 1998 Capacity of TwoLayer Feedforward Neural Networks with Binary Weights Chuanyi Ji, Member, IEEE, Demetri Psaltis, Senior Member,
More informationDaniel M. Oberlin Department of Mathematics, Florida State University. January 2005
PACKING SPHERES AND FRACTAL STRICHARTZ ESTIMATES IN R d FOR d 3 Daniel M. Oberlin Department of Mathematics, Florida State University January 005 Fix a dimension d and for x R d and r > 0, let Sx, r) stand
More informationA Banach space with a symmetric basis which is of weak cotype 2 but not of cotype 2
A Banach space with a symmetric basis which is of weak cotype but not of cotype Peter G. Casazza Niels J. Nielsen Abstract We prove that the symmetric convexified Tsirelson space is of weak cotype but
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne PearsonAddison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 PearsonAddison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationConflictFree Colorings of Rectangles Ranges
ConflictFree Colorings of Rectangles Ranges Khaled Elbassioni Nabil H. Mustafa MaxPlanckInstitut für Informatik, Saarbrücken, Germany felbassio, nmustafag@mpisb.mpg.de Abstract. Given the range space
More informationMaximum unionfree subfamilies
Maximum unionfree subfamilies Jacob Fox Choongbum Lee Benny Sudakov Abstract An old problem of Moser asks: how large of a unionfree subfamily does every family of m sets have? A family of sets is called
More informationON ωindependence AND THE KUNENSHELAH PROPERTY. 1. Introduction.
ON ωindependence AND THE KUNENSHELAH PROPERTY A. S. GRANERO, M. JIMÉNEZSEVILLA AND J. P. MORENO Abstract. We prove that spaces with an uncountable ωindependent family fail the KunenShelah property.
More informationA NOTE ON LINEAR FUNCTIONAL NORMS
A NOTE ON LINEAR FUNCTIONAL NORMS YIFEI PAN AND MEI WANG Abstract. For a vector u in a normed linear space, HahnBanach Theorem provides the existence of a linear functional f, f(u) = u such that f = 1.
More informationSungWook Park*, Hyuk Han**, and Se Won Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No. 1, June 2003 CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS SungWook Park*, Hyuk Han**,
More informationA LOCALIZATION PROPERTY AT THE BOUNDARY FOR MONGEAMPERE EQUATION
A LOCALIZATION PROPERTY AT THE BOUNDARY FOR MONGEAMPERE EQUATION O. SAVIN. Introduction In this paper we study the geometry of the sections for solutions to the Monge Ampere equation det D 2 u = f, u
More informationOn the Length of Lemniscates
On the Length of Lemniscates Alexandre Eremenko & Walter Hayman For a monic polynomial p of degree d, we write E(p) := {z : p(z) =1}. A conjecture of Erdős, Herzog and Piranian [4], repeated by Erdős in
More informationMULTIPLICITIES OF MONOMIAL IDEALS
MULTIPLICITIES OF MONOMIAL IDEALS JÜRGEN HERZOG AND HEMA SRINIVASAN Introduction Let S = K[x 1 x n ] be a polynomial ring over a field K with standard grading, I S a graded ideal. The multiplicity of S/I
More informationSolving a linear equation in a set of integers II
ACTA ARITHMETICA LXXII.4 (1995) Solving a linear equation in a set of integers II by Imre Z. Ruzsa (Budapest) 1. Introduction. We continue the study of linear equations started in Part I of this paper.
More informationOn the optimal order of worst case complexity of direct search
On the optimal order of worst case complexity of direct search M. Dodangeh L. N. Vicente Z. Zhang May 14, 2015 Abstract The worst case complexity of directsearch methods has been recently analyzed when
More informationOn John type ellipsoids
On John type ellipsoids B. Klartag Tel Aviv University Abstract Given an arbitrary convex symmetric body K R n, we construct a natural and nontrivial continuous map u K which associates ellipsoids to
More informationMath General Topology Fall 2012 Homework 6 Solutions
Math 535  General Topology Fall 202 Homework 6 Solutions Problem. Let F be the field R or C of real or complex numbers. Let n and denote by F[x, x 2,..., x n ] the set of all polynomials in n variables
More informationOn the Power of Robust Solutions in TwoStage Stochastic and Adaptive Optimization Problems
MATHEMATICS OF OPERATIONS RESEARCH Vol. 35, No., May 010, pp. 84 305 issn 0364765X eissn 1565471 10 350 084 informs doi 10.187/moor.1090.0440 010 INFORMS On the Power of Robust Solutions in TwoStage
More informationGeometry and topology of continuous best and near best approximations
Journal of Approximation Theory 105: 252 262, Geometry and topology of continuous best and near best approximations Paul C. Kainen Dept. of Mathematics Georgetown University Washington, D.C. 20057 Věra
More informationRelationships between upper exhausters and the basic subdifferential in variational analysis
J. Math. Anal. Appl. 334 (2007) 261 272 www.elsevier.com/locate/jmaa Relationships between upper exhausters and the basic subdifferential in variational analysis Vera Roshchina City University of Hong
More informationPolynomiality of Linear Programming
Chapter 10 Polynomiality of Linear Programming In the previous section, we presented the Simplex Method. This method turns out to be very efficient for solving linear programmes in practice. While it is
More informationMEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW
MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow
More informationA Nonlinear Lower Bound on Linear Search Tree Programs for Solving Knapsack Problems*
JOURNAL OF COMPUTER AND SYSTEM SCIENCES 13, 6973 (1976) A Nonlinear Lower Bound on Linear Search Tree Programs for Solving Knapsack Problems* DAVID DOBKIN Department of Computer Science, Yale University,
More information