Convex Bodies with Minimal Mean Width
|
|
- Brett Page
- 6 years ago
- Views:
Transcription
1 Covex Boies with Miimal Mea With A.A. Giaopoulos 1, V.D. Milma 2, a M. Ruelso 3 1 Departmet of Mathematics, Uiversity of Crete, Iraklio, Greece 2 School of Mathematical Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel 3 Departmet of Mathematics, Uiversity of Missouri, Columbia, MO 65211, USA 1 Itrouctio Let K be a covex boy i R, a {T K T SL(} be the family of its positios. I [GM] it was show that for may atural fuctioals of the form T f(t K, T SL(, the solutio T 0 of the problem mi{f(t K T SL(} is isotropic with respect to a appropriate measure epeig o f. The purpose of this ote is to provie applicatios of this poit of view i the case of the mea with fuctioal T w(t K uer various costraits. Recall that the with of K i the irectio of u S 1 is efie by w(k, u = h K (u + h K ( u, where h K (y = max x K x, y is the support fuctio of K. The with fuctio w(k, is traslatio ivariat, therefore we may assume that o it(k. The mea with of K is give by w(k = w(k, uσ(u = 2 h K (uσ(u, S 1 S 1 where σ is the rotatioally ivariat probability measure o the uit sphere S 1. We say that K has miimal mea with if w(t K w(k for every T SL(. The followig isotropic characterizatio of the miimal mea with positio was prove i [GM]: Fact. A covex boy K i R has miimal mea with if a oly if h K (u u, θ 2 σ(u = w(k S 2 1 for every θ S 1. Moreover, if U SL( a UK has miimal mea with, we must have U O(. Research of the seco ame author partially supporte by the Israel Sciece Fouatio foue by the Acaemy of Scieces a Humaities. Research of the thir ame author was supporte i part by NSF Grat DMS
2 2 A.A. Giaopoulos et al. Our first result is a applicatio of this fact to a reverse Urysoh iequality problem: The classical Urysoh iequality states that w(k ( K /ω 1/ where ω is the volume of the Eucliea uit ball D, with equality if a oly if K is a ball. A atural questio is to ask for which boies K a := max mi w(t K K =1 T SL( is attaie, a what is the precise orer of growth of a as. Examples such as the regular simplex or the cross-polytope show that a c log( + 1. O the other ha, it is kow that every symmetric covex boy K i R has a image T K with T K = 1 for which w(t K c 1 log[(xk, l 2 + 1], where X K = (R, K a eotes the Baach-Mazur istace. This statemet follows from a iequality of Pisier [Pi], combie with work of Lewis [L], Figiel a Tomczak-Jaegerma [FT]. Joh s theorem [J] implies that mi w(t K c 2 log( + 1, T SL( for every symmetric covex boy K with K = 1, a a simple argumet base o the ifferece boy a the Rogers-Shephar iequality [RS] shows that the same hols true without the symmetry assumptio. Therefore, c log( + 1 a c 3 log( + 1. Here, we shall give a precise estimate for the miimal mea with of zoois (this is the class of symmetric covex boies which ca be approximate by Mikowski sums of lie segmets i the Hausorff sese: Theorem A. Let Z be a zooi i R with volume Z = 1. The, where Q = [ 1/2, 1/2]. mi w(t Z w(q = 2ω 1, T SL( ω For our seco applicatio, we cosier the class of origi symmetric covex boies i R. Every symmetric boy K iuces a orm x K = mi{λ 0 : x λk} o R, a we write X K for the orme space (R, K. The polar boy of K is efie by x K = max y K x, y = h K (x, a will be eote by K. Wheever we write (1/a x x K b x, we assume that a, b are the smallest positive umbers for which this iequality hols true for every x R. We cosier the average M(K = x K σ(x S 1
3 Covex Boies with Miimal Mea With 3 of the orm K o S 1, a efie M (K = M(K. Thus, M (K is half the mea with of K. We will say that K has miimal M if M(K M(T K for every T SL(. Equivaletly, if K has miimal mea with. Our purpose is to show that if K has miimal M, the the volume raius of K is boue by a fuctio of b a M. Actually, it is of the orer of b/m. The precise formulatio is as follows: Theorem B. Let K be a symmetric covex boy i R with miimal M, such that (1/a x x K b x, x R. The, ( 1/ b K M c b ( 2b (1/bD M log, M where c > 0 is a absolute costat. Our last result cocers optimizatio of the with fuctioal uer a ifferet coitio. We say that a -imesioal symmetric covex boy K is i the Gauss-Joh positio if the miimum of the fuctioal E g T K uer the costrait T K D is attaie for T = I. That is, K has miimal mea with uer the coitio T K D (it miimizes M uer the coitio a(t K 1. We ca cosier this optimizatio problem oly for positive self-ajoit operators T. Sice the orm of T shoul be boue to guaratee that T K D a the orm of T 1 shoul be boue as well, there exists T for which the miimum is attaie. Deote by γ the staar Gaussia measure i R. The, we have the followig ecompositio. Theorem C. Let K be i the Gauss-Joh positio. The there exist: m (+1/2, cotact poits x 1,..., x m K S 1 a umbers c 1,..., c m > 0 such that m c i = 1 a ( m (x x I x K γ(x = x K γ(x c i x i x i. R R The Gauss-Joh positio is ot equivalet to the classical Joh positio. Examples show that, whe K is i the Gauss-Joh positio, the istace betwee D a the Joh ellipsoi may be of orer / log. 2 Reverse Urysoh Iequality for Zoois The proof of Theorem A will make use of a characterizatio of the miimal surface positio, which was give by Petty [Pe] (see also [GP]: Recall that the area measure σ K of a covex boy K is efie o S 1 a correspos
4 4 A.A. Giaopoulos et al. to the usual surface measure o K via the Gauss map: For every Borel V S 1, we have σ K (V = ν ({x b(k : the outer ormal to K at x is i V }, where ν is the ( 1-imesioal surface measure o K. If A(K is the surface area of K, we obviously have A(K = σ K (S 1. We say that K has miimal surface area if A(K A(T K for every T SL(. With these efiitios, we have: 2.1 Theorem. A covex boy K i R has miimal surface area if a oly if u, θ 2 σ K (u = A(K S 1 for every θ S 1. Moreover, if U SL( a UK has miimal surface area, we must have U O(. Recall also the efiitio of the projectio boy ΠK of K: it is the symmetric covex boy whose support fuctio is efie by h ΠK (θ = P θ (K where P θ (K is the orthogoal projectio of K oto θ, θ S 1. It is kow that Z is a zooi i R if a oly if there exists a covex boy K i R such that Z = ΠK. By the formula for the area of projectios, this ca be writte i the form h Z (x = 1 x, u σ K (u. 2 S 1 The, the characterizatio of the miimal mea with positio a Theorem 2.1 imply the followig: 2.2 Lemma. Let Z = ΠK be a zooi. The, Z has miimal mea with if a oly if K has miimal surface area. Proof. The proof (moulo the characterizatio of the miimal mea with positio may be fou i [Pe]: By Cauchy s surface area formula, A(K = ω h Z (θσ(θ. ω 1 S 1 If f 2 is a spherical harmoic of egree 2, the Fuk-Hecke formula shows that S 1 f 2 (u u, τ σ(u = c f 2 (τ for all u, τ S 1, where c is a costat epeig oly o the imesio. Therefore, f 2 (uh Z (uσ(u = 1 f 2 (u u, τ σ(uσ K (τ S 2 1 S 1 S 1 = c f 2 (τσ K (τ. 2 S 1
5 Covex Boies with Miimal Mea With 5 Sice u u, θ 2 is homogeeous of egree 2, this implies h Z (u u, θ 2 σ(u = c u, θ 2 σ K (u S 2 1 S 1 for every θ S 1. The characterizatios of the miimal mea with a the miimal surface area positios make it clear that Z has miimal mea with if a oly if K has miimal surface area. Our ext lemma is a well-kow fact, prove by K. Ball [B]: 2.3 Lemma Let {u j } j m be uit vectors i R a {c j } j m be positive umbers satisfyig m I = c j u j u j. j=1 If Z = m j=1 α j[ u j, u j ] for some α j > 0, the Z 2 m ( αj j=1 c j cj. We apply this result to the projectio boy of a covex boy with miimal surface area. 2.4 Lemma If K has miimal surface area, the A(K ΠK 1/. Proof. We may assume that K is a polytope with facets F j a ormals u j, j = 1,..., m. The, Theorem 2.1 is equivalet to the statemet I = m c j u j u j j=1 where c j = F j /A(K (see [GP]. O the other ha, ΠK = A(K 2 m c j [ u j, u j ]. j=1 We ow apply Lemma 2.3 for Z = ΠK, with α j = A(K 2 c j: m ( A(K ΠK 2. 2 j=1 Remark. I the previous argumet, equality ca hol oly if (u j j m is a orthoormal basis of R (see [Ba]. This meas that if K is a polytope the equality i Lemma 2.3 ca hol oly if K is a cube.
6 6 A.A. Giaopoulos et al. Proof of Theorem A. Let Z be a zooi with miimal mea with a volume Z = 1. By Lemma 2.2, Z is the projectio boy ΠK of some covex boy K with miimal surface area. We have w(z = 2 h Z (uσ(u = 2 S 1 P u (K σ(u = 2ω 1 S ω 1 A(K. By Lemma 2.4, the area of K is boue by Z 1/ =. We have equality whe K is a cube, a this correspos to the case Z = Q. Therefore, w(z w(q = 2ω 1 ω. Remark. Urysoh s iequality a Theorem A show that if Z is a zooi with Z = 1, the where α, β 1 as. 2 2 α mi πe w(t Z β, T SL( π 3 Volume Ratio of Symmetric Covex Boies with Miimal M For the proof of Theorem B we will ee the followig fact which was prove i [GM]: 3.1 Theorem. Let K be a symmetric covex boy i R with miimal M. The, for every λ (0, 1 there exists a [(1 λ]-imesioal subspace E of R such that b r(λ x x K b x, x E, (3.1 b where r(λ c log( 2b Mλ 1/2 Mλ, a c > 0 is a absolute costat. Actually, the proof of Theorem 3.1 shows that the statemet hols true for a raom [(1 λ]-imesioal subspace E of R. Oe ca assume that for every k we have the result with probability greater tha c log (this formulatio is correct whe 0, where 0 N is absolute. This assumptio o the measure of subspaces satisfyig (3.1 implies that there is a icreasig sequece of subspaces E 1 E 2... E k0, where k 0 = [ c log 2 ] a ime k = k, so that (3.1 hols for each E k with r = r(k/. We will also ee the followig
7 Covex Boies with Miimal Mea With Lemma. Let K be a symmetric covex boy i R, such that (1/a x x K b x. If E is a k-imesioal subspace of R, the ( K D Ca( where C > 0 is a absolute costat. k 1/2 k K E, D k Proof. Let E be a k-imesioal subspace of R. Replacig K by (1/aK, we may assume that a = 1, so K D. Usig Bru s theorem we see that K = K (E + y y P E (K K E This shows that P E (K K E D k. K D D k D k D K E D k Proof of Theorem B. We first observe that ( C k ab C log. ( k/2 K E. D k Iee, let e S 1 be such that e K = b a let γ be a staar ormal variable. The cb = E γe E g K. Similarly, ca E g K. Multiplyig these iequalities, we obtai ab CE g K E g K C log. The last iequality follows from the fact that M is miimal for K, a Pisier s iequality [Pi]. Assume ow that b = 1. Let t = [log ]. For s = 1, 2,..., t put k s = [(1 1/s] a let E s = E ks be a subspace from our flag. The by Theorem 3.1 we have (1/a s x x K x o E s, where a s r(( k s / c M s1/2 log ( 2s M s c M log ( 2 =: s c(m. M Now, Lemma 3.2 shows that K ( D Ca kt K Et t (C log 2 / log K E t D kt D kt C K E t D kt
8 8 A.A. Giaopoulos et al. a K E s+1 D ks+1 (Ca s s ks+1 ks K E s D ks ( Cc(Ms 2 k s+1 k s K E s, D ks for all s = 1, 2,..., t. Sice a 1 c(m, we have K E 1 / D k1 c(m k1. Hece, multiplyig the iequalities above, we get K D C (Cc(M kt k1 c(m k1 By the efiitio of k s, ( t s 2(ks+1 ks exp c therefore s=2 t s=2 t s 2(ks+1 ks. s=2 ( 1/ K C 1 c(m. (1/bD log s s 2 e c, The left ha sie iequality is a immeiate cosequece of Höler s iequality: ( 1/ ( K = b (1/bD ( b S 1 x K σ(x 1/ S 1 x K 1 = b M. 4 Gauss-Joh Positio We prove Theorem C. Cosier the followig optimizatio problem: F (T = T 1 x K γ(x mi (4.1 R uer the costrait H x (T = T x for x K. Assume that the boy K is i the Gauss-Joh positio, amely the miimum i (4.1 is attaie for T = I. Let W be the set of the cotact poits of K: W = K S 1. First we apply a argumet of Joh [J] to show that we
9 Covex Boies with Miimal Mea With 9 ca cosier oly fiitely may costraits. Sice the paper [J] is ot easily available, we shall sketch the argumet. Let T be a self-ajoit operator a let T s = I + st. We shall prove that if for every x W, the Iee, assume that s H x(t s s=0 < 0 s F (T s s=0 0. a = sup x W s H x(t s s=0 < 0. Let W ε be a ε-eighborhoo of W : W ε = {x K ist(x, W < ε}. There exists a ε > 0 such that s H x(t s s=0 < a 2 for every x W ε. So, there exists s 0 > 0 such that for ay 0 < s < s 0 a ay x W ε H x (T s < H x (I 0. O the other ha, if x K \ W ε the Here, a H x (T s H x (T s H x (I + H x (I. H x (T s H x (I = T s x 2 x 2 T (1 + T s sup H x (I = sup x 2 1 < 0 x K\W ε x K\W ε sice K \ W ε is compact. Thus for a sufficietly small s we have H x (T s < 0 for all x K. So, sice I is the solutio of the miimizatio problem (4.1, s F (T s s=0 0. Sice s H x(t s s=0 = H x (I, T a s F (T s s=0 = F (I, T, this meas that the vector F (I caot be separate from the set { H x (I x W } by a hyperplae. By Carathéoory s theorem, there exist M (+1/2 cotact poits x 1... x M W a umbers λ 1... λ M > 0 such that M M F (I = λ i H xi (I = λ i x i x i. (4.2
10 10 A.A. Giaopoulos et al. Now we have to calculate F (I. We have F (T = (2π /2 so, 2 /2 x T 1 x K e x R = (2π /2 T x et T x K e 2 /2 x, R F (I = ((2π /2 x K e x 2 /2 x I R (2π /2 x K e x 2 /2 x xx R = (I x x x K γ(x. R Combiig it with (4.2 we obtai R (I x x x K γ(x + Takig the trace, we get ( Tr (I x x x K γ(x R = x K γ(x x 2 x K γ(x R R = r e 2 /2 r ω K m(ω 0 S 1 0 = x K γ(x. R M λ i x i x i = 0. r +2 e 2 /2 r ω K m(ω S 1 Fially, puttig λ i = c i R x K γ(x, we obtai the ecompositio ( M (I x x x K γ(x = x K γ(x c i x i x i, R R where M c i = 1. This completes the proof of Theorem C. We procee to compare D with the Joh ellipsoi i the Gauss-Joh positio: Propositio. Let K be a symmetric covex boy i R which is i the Gauss- Joh positio. The, (i (2/π 1/2 1 D K D ;
11 Covex Boies with Miimal Mea With 11 (ii It may happe that c log D is ot cotaie i K. Proof. (i Let T 0 be a operator which puts K ito the maximal volume positio. The mi F (T F (T 0. From the other sie, if there exists y S 1 such that y K < (2/π 1/2 1 the x K γ(x x, y/ y K γ(x (2/π 1/2 1/ y K >. R R (ii Let K = B1 1 + [ e, e ]. Let T be a positive self-ajoit operator such that T K is i the Gauss-Joh positio. We first prove that T is a iagoal operator. Let G O( be the group geerate by the operators U i = I 2e i e i, i = 1,..., a let m be the uiform measure o G. Notice that U i K = K for every i. The, x T K γ(x = Ux T U(K γ(xm(u R G R ( U 1 T 1 U m(u x γ(x. R G Put We claim that W = Iee, sice for ay i a G U 1 T 1 Um(U = iag(t 1. W ( iag(t 1. e i, iag(t 1 e i = e i, (T 1 e i e i, ( iag(t 1 ei = e i, T e i 1, the claim follows from the fact that for ay θ S 1 θ, T 1 θ θ, T θ 1. Let S = iag(t. Sice W S 1, we have F (T = x T K γ(x R W x K γ(x R S 1 x K γ(x = F (S. R [Notice that sice T K D, ( SK = U 1 T U m(u (K D, G so the restrictios of the optimizatio problem (4.1 are satisfie.]
12 12 A.A. Giaopoulos et al. Let ow G O( be the group geerate by the operators U ij = I e i e i e j e j + e i e j + e j e i for i, j = 1,..., 1, i j. Arguig the same way we ca show that there exist a, b > 0 such that F (S F (T 0, where ( 1 T 0 = a e i e i + be e. Sice the vertices of T 0 K are cotact poits, We have x T0K = max a 2 + b 2 = 1. ( x i, b 1 x. a 1 1 Deote x 1 = 1 x i a let t = t(x = (b/a x 1. The, ψ(b = x T0Kγ(x R ( 1 t = a 1 x 1 e x2 /2 x + 2 b 1 x e x2 /2 x γ(x 2π 2π = R 1 R 1 t ( 2 a x 1Φ(t + 2 b 1 e t2 /2 γ(x, 2π where Φ(t = (1/ 2π t /2 u. We have to show that b c log 0 e u2. We may assume that b c/. Puttig a = (1 b 2 1/2 a ifferetiatig, we get after some calculatios ( b ψ(b = 2a 3 b x 1 Φ(t 2 b 2 e t2 /2 γ(x. 2π R 1 Sice b c/ a x 1 C with probability at least 1/2, we have Φ(t > c with probability 1/2, for some absolute costat c > 0. So, which is positive whe b c log /. Remark. The ual problem f(t = uer the costrait b ψ(b c Cb 2 exp( c 2 b 2, R sup x, y γ(x max y T K h x (T = T x for x K is very ifferet. The examples suggest that the matrix T for which the maximum is attaie may be sigular. t
13 Covex Boies with Miimal Mea With 13 Refereces [B] [Ba] [BM] [FT] [GM] [GP] [J] [L] [MS] [Pe] [Pi] [RS] Ball K.M. (1991 Shaows of covex boies. Tras. Amer. Math. Soc. 327: Barthe F. (1998 O a reverse form of the Brascamp-Lieb iequality. Ivet. Math. 134: Bourgai J., Milma V.D. (1987 New volume ratio properties for covex symmetric boies i R. Ivet. Math. 88: Figiel T., Tomczak-Jaegerma N. (1979 Projectios oto Hilbertia subspaces of Baach spaces. Israel J. Math. 33: Giaopoulos A.A., Milma V.D. Extremal problems a isotropic positios of covex boies. Israel J. Math., to appear Giaopoulos A.A., Papaimitrakis M. (1999 Isotropic surface area measures. Mathematika 46:1 13 Joh F. (1948 Extremum problems with iequalities as subsiiary coitios. Courat Aiversary Volume, Itersciece, New York, Lewis D.R. (1979 Ellipsois efie by Baach ieal orms. Mathematika 26:18 29 Milma V.D., Schechtma G. (1986 Asymptotic theory of fiite imesioal orme spaces. Lecture Notes i Mathematics, 1200, Spriger, Berli Petty C.M. (1961 Surface area of a covex boy uer affie trasformatios. Proc. Amer. Math. Soc. 12: Pisier G. (1982 Holomorphic semi-groups a the geometry of Baach spaces. A. of Math. 115: Rogers C.A., Shephar G. (1957 The ifferece boy of a covex boy. Arch. Math. 8:
An 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 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 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 informationChapter 2 Transformations and Expectations
Chapter Trasformatios a Epectatios Chapter Distributios of Fuctios of a Raom Variable Problem: Let be a raom variable with cf F ( ) If we efie ay fuctio of, say g( ) g( ) is also a raom variable whose
More informationA remark on p-summing norms of operators
A remark o p-summig orms of operators Artem Zvavitch Abstract. I this paper we improve a result of W. B. Johso ad G. Schechtma by provig that the p-summig orm of ay operator with -dimesioal domai ca be
More informationSparsification using Regular and Weighted. Graphs
Sparsificatio usig Regular a Weighte 1 Graphs Aly El Gamal ECE Departmet a Cooriate Sciece Laboratory Uiversity of Illiois at Urbaa-Champaig Abstract We review the state of the art results o spectral approximatio
More informationSeveral 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 informationDefinition 2 (Eigenvalue Expansion). We say a d-regular graph is a λ eigenvalue expander if
Expaer Graphs Graph Theory (Fall 011) Rutgers Uiversity Swastik Kopparty Throughout these otes G is a -regular graph 1 The Spectrum Let A G be the ajacecy matrix of G Let λ 1 λ λ be the eigevalues of A
More informationLecture 6 Testing Nonlinear Restrictions 1. The previous lectures prepare us for the tests of nonlinear restrictions of the form:
Eco 75 Lecture 6 Testig Noliear Restrictios The previous lectures prepare us for the tests of oliear restrictios of the form: H 0 : h( 0 ) = 0 versus H : h( 0 ) 6= 0: () I this lecture, we cosier Wal,
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 informationA REMARK ON A PROBLEM OF KLEE
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property
More informationJOHN S DECOMPOSITION OF THE IDENTITY IN THE NON-CONVEX CASE. Jesus Bastero* and Miguel Romance**
JOHN S DECOMPOSITION OF THE IDENTITY IN THE NON-CONVEX CASE Jesus Bastero* ad Miguel Romace** Abstract We prove a extesio of the classical Joh s Theorem, that characterices the ellipsoid of maximal volume
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 informationOFF-DIAGONAL MULTILINEAR INTERPOLATION BETWEEN ADJOINT OPERATORS
OFF-DIAGONAL MULTILINEAR INTERPOLATION BETWEEN ADJOINT OPERATORS LOUKAS GRAFAKOS AND RICHARD G. LYNCH 2 Abstract. We exted a theorem by Grafakos ad Tao [5] o multiliear iterpolatio betwee adjoit operators
More informationarxiv: v4 [math.co] 5 May 2011
A PROBLEM OF ENUMERATION OF TWO-COLOR BRACELETS WITH SEVERAL VARIATIONS arxiv:07101370v4 [mathco] 5 May 011 VLADIMIR SHEVELEV Abstract We cosier the problem of eumeratio of icogruet two-color bracelets
More informationOn triangular billiards
O triagular billiars Abstract We prove a cojecture of Keyo a Smillie cocerig the oexistece of acute ratioal-agle triagles with the lattice property. MSC-iex: 58F99, 11N25 Keywors: Polygoal billiars, Veech
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More information6.451 Principles of Digital Communication II Wednesday, March 9, 2005 MIT, Spring 2005 Handout #12. Problem Set 5 Solutions
6.51 Priciples of Digital Commuicatio II Weesay, March 9, 2005 MIT, Sprig 2005 Haout #12 Problem Set 5 Solutios Problem 5.1 (Eucliea ivisio algorithm). (a) For the set F[x] of polyomials over ay fiel F,
More informationAnalytic Number Theory Solutions
Aalytic Number Theory Solutios Sea Li Corell Uiversity sl6@corell.eu Ja. 03 Itrouctio This ocumet is a work-i-progress solutio maual for Tom Apostol s Itrouctio to Aalytic Number Theory. The solutios were
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More information1 Review and Overview
CS229T/STATS231: Statistical Learig Theory Lecturer: Tegyu Ma Lecture #12 Scribe: Garrett Thomas, Pega Liu October 31, 2018 1 Review a Overview Recall the GAN setup: we have iepeet samples x 1,..., x raw
More informationBoundaries and the James theorem
Boudaries ad the James theorem L. Vesely 1. Itroductio The followig theorem is importat ad well kow. All spaces cosidered here are real ormed or Baach spaces. Give a ormed space X, we deote by B X ad S
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 information1 = 2 d x. n x n (mod d) d n
HW2, Problem 3*: Use Dirichlet hyperbola metho to show that τ 2 + = 3 log + O. This ote presets the ifferet ieas suggeste by the stuets Daiel Klocker, Jürge Steiiger, Stefaia Ebli a Valerie Roiter for
More informationLinear Support Vector Machines
Liear Support Vector Machies David S. Roseberg The Support Vector Machie For a liear support vector machie (SVM), we use the hypothesis space of affie fuctios F = { f(x) = w T x + b w R d, b R } ad evaluate
More informationThe structure of Fourier series
The structure of Fourier series Valery P Dmitriyev Lomoosov Uiversity, Russia Date: February 3, 2011) Fourier series is costructe basig o the iea to moel the elemetary oscillatio 1, +1) by the expoetial
More informationON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX0000-0 ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2
More informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More information(average number of points per unit length). Note that Equation (9B1) does not depend on the
EE603 Class Notes 9/25/203 Joh Stesby Appeix 9-B: Raom Poisso Poits As iscusse i Chapter, let (t,t 2 ) eote the umber of Poisso raom poits i the iterval (t, t 2 ]. The quatity (t, t 2 ) is a o-egative-iteger-value
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 informationA note on equiangular tight frames
A ote o equiagular tight frames Thomas Strohmer Departmet of Mathematics, Uiversity of Califoria, Davis, CA 9566, USA Abstract We settle a cojecture of Joseph Rees about the existece a costructio of certai
More informationThe Brunn-Minkowski Theorem and Influences of Boolean Variables
Lecture 7 The Bru-Mikowski Theorem ad Iflueces of Boolea Variables Friday 5, 005 Lecturer: Nati Liial Notes: Mukud Narasimha Theorem 7.1 Bru-Mikowski). If A, B R satisfy some mild assumptios i particular,
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 informationIntroduction to Optimization Techniques. How to Solve Equations
Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually
More informationLecture 23 Rearrangement Inequality
Lecture 23 Rearragemet Iequality Holde Lee 6/4/ The Iequalities We start with a example Suppose there are four boxes cotaiig $0, $20, $50 ad $00 bills, respectively You may take 2 bills from oe box, 3
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
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 informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationALGEBRA HW 7 CLAY SHONKWILER
ALGEBRA HW 7 CLAY SHONKWILER Prove, or isprove a salvage: If K is a fiel, a f(x) K[x] has o roots, the K[x]/(f(x)) is a fiel. Couter-example: Cosier the fiel K = Q a the polyomial f(x) = x 4 + 3x 2 + 2.
More informationTheorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.
Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is
More informationApplication to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
More informationON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS * M. JA]IMOVI], I. KRNI] 1.
Yugoslav Joural of Operatios Research 1 (00), Number 1, 49-60 ON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS M. JA]IMOVI], I. KRNI] Departmet of Mathematics
More information6.3.3 Parameter Estimation
130 CHAPTER 6. ARMA MODELS 6.3.3 Parameter Estimatio I this sectio we will iscuss methos of parameter estimatio for ARMAp,q assumig that the orers p a q are kow. Metho of Momets I this metho we equate
More informationLecture 7: October 18, 2017
Iformatio ad Codig Theory Autum 207 Lecturer: Madhur Tulsiai Lecture 7: October 8, 207 Biary hypothesis testig I this lecture, we apply the tools developed i the past few lectures to uderstad the problem
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationBETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
More information6. Kalman filter implementation for linear algebraic equations. Karhunen-Loeve decomposition
6. Kalma filter implemetatio for liear algebraic equatios. Karhue-Loeve decompositio 6.1. Solvable liear algebraic systems. Probabilistic iterpretatio. Let A be a quadratic matrix (ot obligatory osigular.
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 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 informationSymmetrization and isotropic constants of convex bodies
Symmetrizatio ad isotropic costats of covex bodies J. Bourgai B. lartag V. Milma IAS, Priceto el Aviv el Aviv November 18, 23 Abstract We ivestigate the effect of a Steier type symmetrizatio o the isotropic
More informationLecture 19. sup y 1,..., yn B d n
STAT 06A: Polyomials of adom Variables Lecture date: Nov Lecture 19 Grothedieck s Iequality Scribe: Be Hough The scribes are based o a guest lecture by ya O Doell. I this lecture we prove Grothedieck s
More informationarxiv: v1 [math.mg] 29 Nov 2018
AN EXTREMAL PROBLEM OF REGULAR SIMPLICES THE HIGHER-DIMENSIONAL CASE ÁKOS GHORVÁTH arxiv:99v [mathmg] 9 Nov Abstract The ew result of this paper coected with the followig problem: Cosider a supportig hyperplae
More informationRemarks on the Geometry of Coordinate Projections in R n
Remarks o the Geometry of Coordiate Projectios i R S. Medelso R. Vershyi Abstract We study geometric properties of coordiate projectios. Amog other results, we show that if a body K R has a almost extremal
More informationLECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK)
LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) Everythig marked by is ot required by the course syllabus I this lecture, all vector spaces is over the real umber R. All vectors i R is viewed as a colum
More informationNotes 19 : Martingale CLT
Notes 9 : Martigale CLT Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: [Bil95, Chapter 35], [Roc, Chapter 3]. Sice we have ot ecoutered weak covergece i some time, we first recall
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 informationSupplemental Material: Proofs
Proof to Theorem Supplemetal Material: Proofs Proof. Let be the miimal umber of traiig items to esure a uique solutio θ. First cosider the case. It happes if ad oly if θ ad Rak(A) d, which is a special
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationA NEW AFFINE INVARIANT FOR POLYTOPES AND SCHNEIDER S PROJECTION PROBLEM
A NEW AFFINE INVARIANT FOR POLYTOPES AND SCHNEIDER S PROJECTION PROBLEM Erwi Lutwak, Deae Yag, ad Gaoyog Zhag Departmet of Mathematics Polytechic Uiversity Brookly, NY 1101 Abstract. New affie ivariat
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm
More informationANSWERS TO MIDTERM EXAM # 2
MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18
More informationIntroduction to Optimization Techniques
Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors
More informationA Characterization of Compact Operators by Orthogonality
Australia Joural of Basic ad Applied Scieces, 5(6): 253-257, 211 ISSN 1991-8178 A Characterizatio of Compact Operators by Orthogoality Abdorreza Paahi, Mohamad Reza Farmai ad Azam Noorafa Zaai Departmet
More informationarxiv: v1 [math.mg] 16 Dec 2017
arxiv:171.05949v1 [math.g] 16 ec 017 ESTIATES FOR OENTS OF GENERAL EASURES ON CONVEX BOIES SERGEY BOBOV, BO AZ LARTAG, AN ALEXANER OLOBSY Abstract. For p 1, N, ad a origi-symmetric covex body i R, let
More informationON THE EXTENDED AND ALLAN SPECTRA AND TOPOLOGICAL RADII. Hugo Arizmendi-Peimbert, Angel Carrillo-Hoyo, and Jairo Roa-Fajardo
Opuscula Mathematica Vol. 32 No. 2 2012 http://dx.doi.org/10.7494/opmath.2012.32.2.227 ON THE EXTENDED AND ALLAN SPECTRA AND TOPOLOGICAL RADII Hugo Arizmedi-Peimbert, Agel Carrillo-Hoyo, ad Jairo Roa-Fajardo
More informationGeneralized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces
Turkish Joural of Aalysis a Number Theory, 08, Vol. 6, No., 43-48 Available olie at http://pubs.sciepub.com/tjat/6// Sciece a Eucatio Publishig DOI:0.69/tjat-6-- Geeralize Dyamic Process for Geeralize
More informationReview Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =
Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:
More information5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.
40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a self-adjoit matrix Eigevalues of self-adjoit matrices are easy to calculate. This sectio shows how this is doe usig
More informationarxiv: v1 [math.pr] 13 Oct 2011
A tail iequality for quadratic forms of subgaussia radom vectors Daiel Hsu, Sham M. Kakade,, ad Tog Zhag 3 arxiv:0.84v math.pr] 3 Oct 0 Microsoft Research New Eglad Departmet of Statistics, Wharto School,
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 information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationAPPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS
APPLICATION OF YOUNG S INEQUALITY TO VOLUMES OF CONVEX SETS 1. Itroductio Let C be a bouded, covex subset of. Thus, by defiitio, with every two poits i the set, the lie segmet coectig these two poits is
More informationThe value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng
The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture 9: Pricipal Compoet Aalysis The text i black outlies mai ideas to retai from the lecture. The text i blue give a deeper uderstadig of how we derive or get
More informationVECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS
Dedicated to Professor Philippe G. Ciarlet o his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst sectio of this paper deals with the properties
More informationAN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS. Robert DEVILLE and Catherine FINET
2001 vol. XIV, um. 1, 95-104 ISSN 1139-1138 AN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS Robert DEVILLE ad Catherie FINET Abstract This article is devoted to a extesio of Simos iequality. As a cosequece,
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationProbabilistic and Average Linear Widths in L -Norm with Respect to r-fold Wiener Measure
joural of approximatio theory 84, 3140 (1996) Article No. 0003 Probabilistic ad Average Liear Widths i L -Norm with Respect to r-fold Wieer Measure V. E. Maiorov Departmet of Mathematics, Techio, Haifa,
More informationarxiv: v3 [math.mg] 28 Aug 2017
ESTIMATING VOLUME AND SURFACE AREA OF A CONVEX BODY VIA ITS PROJECTIONS OR SECTIONS ALEXANDER KOLDOBSKY, CHRISTOS SAROGLOU, AND ARTEM ZVAVITCH arxiv:6.0892v3 [math.mg] 28 Aug 207 Dedicated to the memory
More informationOn equivalent strictly G-convex renormings of Banach spaces
Cet. Eur. J. Math. 8(5) 200 87-877 DOI: 0.2478/s533-00-0050-3 Cetral Europea Joural of Mathematics O equivalet strictly G-covex reormigs of Baach spaces Research Article Nataliia V. Boyko Departmet of
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationIntroductory Analysis I Fall 2014 Homework #7 Solutions
Itroductory Aalysis I Fall 214 Homework #7 Solutios Note: There were a couple of typos/omissios i the formulatio of this homework. Some of them were, I believe, quite obvious. The fact that the statemet
More informationSelf-normalized deviation inequalities with application to t-statistic
Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric
More informationOn the Stability of Multivariate Trigonometric Systems*
Joural of Mathematical Aalysis a Applicatios 35, 5967 999 Article ID jmaa.999.6386, available olie at http:www.iealibrary.com o O the Stability of Multivariate Trigoometric Systems* Wechag Su a Xigwei
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationThe Wasserstein distances
The Wasserstei distaces March 20, 2011 This documet presets the proof of the mai results we proved o Wasserstei distaces themselves (ad ot o curves i the Wasserstei space). I particular, triagle iequality
More informationAlgorithms in The Real World Fall 2002 Homework Assignment 2 Solutions
Algorithms i The Real Worl Fall 00 Homewor Assigmet Solutios Problem. Suppose that a bipartite graph with oes o the left a oes o the right is costructe by coectig each oe o the left to raomly-selecte oes
More informationA non-reflexive Banach space with all contractions mean ergodic
A o-reflexive Baach space with all cotractios mea ergodic Vladimir P. Fof, Michael Li Be-Gurio Uiversity Przemyslaw Wojtaszczyk Uiversity of Warsaw May 4, 2009 Dedicated to the memory of Aryeh Dvoretzky
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 information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationOn matchings in hypergraphs
O matchigs i hypergraphs Peter Frakl Tokyo, Japa peter.frakl@gmail.com Tomasz Luczak Adam Mickiewicz Uiversity Faculty of Mathematics ad CS Pozań, Polad ad Emory Uiversity Departmet of Mathematics ad CS
More information<, if ε > 0 2nloglogn. =, if ε < 0.
GLASNIK MATEMATIČKI Vol. 52(72)(207), 35 360 THE DAVIS-GUT LAW FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED BANACH SPACE VALUED RANDOM ELEMENTS Pigya Che, Migyag Zhag ad Adrew Rosalsky Jia Uversity, P.
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More information