Quantization dimension of probability measures supported on Cantor-like sets
|
|
- Millicent Park
- 6 years ago
- Views:
Transcription
1 J. Math. Anal. Appl. 338 (2008) Note Quantization dimension of probability measures supported on Cantor-like sets Sanguo Zhu Department of Mathematics, Huazhong University of Science and Technology, Wuhan , PR China Received 13 October 2006 Available online 22 May 2007 Submitted by M. Laczkovich Abstract Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied Elsevier Inc. All rights reserved. Keywords: Quantization dimension; Cantor-like sets; Bounded distortion; Hereditary condition 1. Introduction The quantization problem consists in studying the L r -error induced by the approximation of a given probability measure with discrete probability measures of finite supports. This problem originated in information theory and some engineering technology. Its history goes back to the 1940s (cf. [1,6,12,13]). Graf and Luschgy studied this problem systematically and gave a general mathematical treatment of it (cf. [3]). Two important objects in the quantization theory are the quantization coefficient and the quantization dimension. Let μ be a Borel probability measure on R d and let 0 <r<.thenth quantization error of μ of order r is defined by { } V n,r (μ) = inf min x a α a r dμ(x): α R d, card(α) n. (1) If the infimum in (1) is attained at some α R d with card(α) n, we call α an n-optimal set of μ of order r. The collection of all the n-optimal sets of order r is denoted by C n,r (μ). The upper and lower quantization dimension of μ of order r are defined by D r (μ) := lim sup r log n log V n,r (μ) ; D r(μ) := lim inf r log n log V n,r (μ). address: sgzhu2006@163.com X/$ see front matter 2007 Elsevier Inc. All rights reserved. doi: /j.jmaa
2 S. Zhu / J. Math. Anal. Appl. 338 (2008) If D r (μ), D r (μ) coincide, we call the common value the quantization dimension of μ of order r and denote it by D r (μ). In recent years, the quantization problem for self-similar distributions have been extensively studied. Let {f 1,...,f N } be an iterated function system of contractive similitudes on R d with contraction ratios c 1,...,c N. The corresponding self-similar set refers to the unique non-empty compact set E satisfying E = N i=1 f i (E). The self-similar measure associated with {f 1,...,f N } and a given probability vector (p 1,...,p N ) is the unique Borel probability measure satisfying μ = N i=1 p i μ fi 1. We say that {f 1,...,f N } satisfies the strong separation condition if f i (E), 1 i N, are pairwise disjoint. We say that {f 1,...,f N } satisfies the open set condition if there exists a non-empty open set U such that f i (U) U for all i = 1,...,N and f i (U) f j (U) = for any pair i, j with 1 i j N. Under the open set condition, Graf and Luschgy (cf. [2,4]) proved that the quantization dimension of μ exists and equals s r which is the solution of the following equation: N ( pi ci r ) sr sr +r = 1. i=1 The above result was extended by Lindsay and Mauldin to the F -conformal measures associated with finitely many conformal maps (cf. [10]). Pötzelberger showed that, if the strong separation condition is satisfied and the corresponding vector (log(p 1 c1 r),...,log(p N cn r )) is non-arithmetic, the quantization coefficient of a self-similar measure exists (cf. [11]). By using different methods, Graf and Luschgy extended this result to the cases where only the open set condition is satisfied (cf. [5, Theorem 4.1]). In this paper, we study the quantization dimensions of arbitrary probability measures μ supported on certain Cantorlike sets under a hereditary condition. We will establish a relationship between the quantization dimension of μ and its distribution on cylinders. This generalizes Graf and Luschgy s result on the quantization dimension of self-similar distributions. As an application, we determine the quantization dimension of the product measures supported on the Cantor-like sets provided that a hereditary condition is satisfied. The paper is organized as follows. In the next section, we give some definitions and notations. In Section 3, we define some separation conditions and a hereditary condition and then state our main theorem. In Section 4, we first establish some lemmas, on the basis of these lemmas, we give the proof of our main result. The last section is devoted to the application to product measures. 2. Definitions and notations Let (n k ) be a sequence of integers with n k 2 for all k 1. Let Ξ 0 denote the set containing only the empty word. We define n Ω k := {1, 2,...,n k }, Ξ n := Ω k, Ξ := Ω k, Ξ := Ξ k. k=1 For σ = (σ (1),...,σ(n)) Ξ n, we call the number n the length of σ and denote it by σ. For any σ Ξ Ξ with σ n, we write σ n := ( σ(1),...,σ(n) ). If σ,τ Ξ and σ τ, σ = τ σ, we call σ a predecessor of τ and denote this by σ τ. The empty word is a predecessor of any finite or infinite word. We say σ,τ are incomparable if we have neither σ τ nor τ σ. A finite set Γ Ξ is called a finite anti-chain if any two words σ, τ in Γ are incomparable. A finite anti-chain Γ is called maximal if any word σ Ξ has a predecessor in Γ.Forn 2, σ = (σ (1),..., σ (n)) Ξ n and i Ω n+1, we define σ := σ n 1, σ i = ( σ(1),...,σ(n),i ). Let f kj, 1 j n k,k 1, be contractive similitudes on R d of contraction ratios 0 <c kj < 1. We assume that {f kj } satisfies the open set condition: there exists a bounded non-empty open set V such that for k 1 and 1 j n k we have f kj (V ) V and f ki (V ) f kj (V ) = for 1 i j n k. Set k=1 f σ := f 1σ(1) f nσ (n), c σ := c 1σ(1) c nσ (n), σ Ξ n. k=1
3 744 S. Zhu / J. Math. Anal. Appl. 338 (2008) Let V denote the closure of V in R d. We call the non-empty compact set E = f σ (V), k 1 σ Ξ k the Cantor-like set determined by {f kj }.Forσ Ξ k, we call the set E σ := f σ (V) a cylinder set of order k; forthe empty word φ, wetakee φ = V. The Hausdorff and packing dimension of this type of sets have been discussed in [7,8]. 3. Statement of the main theorem We will need the following three conditions, the first two of which is on the separation property of the Cantor-like set and the third is on the mass distribution of a given probability measure. (a) Bounded distortion (BD): we say that {f kj } satisfies the bounded distortion property if c kj c>0 for all 1 j n k and all k 1. (b) Extra Strong Separation Condition (ESSC): for k 0 and σ Ξ k, we define Λ(σ ) := {τ Ξ k+1 : σ τ}; we say that {f kj } satisfies the extra strong separation condition if there exists a constant β>0 such that for any σ Ξ Ξ 0,wehave min { dist(e τ,e ρ ): τ,ρ Λ(σ ) } β max { E τ : τ Λ(σ ) }, (2) where A denotes the diameter of a set A and E σ = f σ (V). (c) Hereditary Condition (HC): Let μ be a probability measure supported on the Cantor-like set E; we say that μ satisfies the hereditary condition if (I) for any σ Ξ Ξ 0 with μ(e σ )>0, there exist at least two distinct words τ,ρ Λ(σ ) with μ(e τ ), μ(e ρ )>0; (II) there exists a constant 0 <p<1 which is independent of σ such that for all τ Λ(σ ) with μ(e τ )>0, we have μ(e τ ) pμ(e σ ). Before we state our main theorem, we need to give some more definitions and notations. For σ Ξ, we define := μ(e σ )c r σ. Set l := min { μ(e i )c r 1i : μ(e i)>0 }, where E i = f 1i (V),1 i n 1. For each n 1, we define Γ n := {σ Ξ : h ( σ ) ln } >h(σ). (3) The set Γ n is crucial in the calculation of the quantization dimension. We remark that the definition of Γ n is motivated by Graf and Luschgy s work on the quantization for self-similar distributions (cf. [4]). For each n N, according to the definition of l, thesetγ n is non-empty; and for any σ Γ n,bythe(hc),wehave l/n h ( σ ) = μ(e σ )cσ r μ(e σ ) (1 p) σ 2. This means σ 2 + log(l/n)/ log(1 p) =: K(n) and Γ n is a finite set. Moreover, for each n, Γ n is a finite antichain, but Γ n may fail to be a finite maximal anti-chain since, in general, μ(e σ ) = 0 is possible. Let d n,r, n 1 and s, s be defined by ( ) dn,r dn,r +r = 1; σ Γ n s := lim sup d n,r, s := lim inf d n,r. (4) By considering the continuous function g(t) := σ Γ n () t, one easily see that the numbers d n,r, n 1areall well defined. We are now in the position to state our main result. Theorem 1. Let E be the Cantor-like set determined by {f kj } which satisfies the conditions (BD) and (ESSC). Let μ be a probability measure supported on E satisfying the condition (HC). Then we have D r (μ) = s, D r (μ) = s, where s and s are as defined in (4).
4 S. Zhu / J. Math. Anal. Appl. 338 (2008) Main results Throughout this section, f kj,1 j n k, k 1 are contractive similitudes with contraction ratios c kj satisfying the conditions (BD) and (ESSC); E denotes the Cantor-like set determined by {f kj } and μ is an arbitrary probability measure supported on E satisfying the condition (HC). Let [x] denote the largest integer less than or equal to x. We begin with the following simple lemma which is an immediate consequence of the definitions. Lemma 2. (See [14, Lemma 6].) Let l,ζ,ξ > 0.Forφ(n):= [ζ(n/l) ξ ], we have D r (μ) = lim sup log V φ(n),r (μ), Proof. By the definition, it is easy to see that D r (μ) = lim inf r log([n/2]) = lim inf log V [n/2],r (μ) D r(μ) = lim inf Now let (n i ) be an arbitrary subsequence of N. Weset [ 1/ξ ] [ 1/ξ ] ln i ln i W i := ζ 1/ξ, Z i := ζ 1/ξ + 1. Then for large i, wehave n i 2 ξ+1 φ(w i) n i, log V φ(n),r (μ). r log n log V [n/2],r (μ). (5) n i 2 φ(z i) 2 ξ+1 n i. (6) By (5), (6) and the definition of the quantization dimension, one easily gets D r (μ) lim sup log V φ(n),r (μ), The reverse inequalities are clear. D r(μ) lim inf log V φ(n),r (μ). Let Γ n be as defined in (3). For σ Γ n, it could happen that h(σ ) l/n but = 0. This will bring us much inconvenience when considering the quantization dimension. For this reason, we need to pick out those words σ Γ n with positive μ-measure, i.e., for each n 1, we set Γ n := { σ Γ n : μ(e σ )>0 }. Let (A) ɛ denote the ɛ-neighborhood of a set A. Forα C m,r (μ) and, we define α σ := α (E σ ) β Eσ /8. The following lemma will be crucial in the proof of the main theorem. It is a generalization of [14, Lemma 9]. Lemma 3. There exists a constant L 1 such that for any m card( Γ n ), α C m,r (μ) and all we have card(α σ ) L. Proof. Let c>0 be as in the condition (BD). We set C = pc r and let M be a constant with M r > 2/C. Then for, by (HC) and (BD), we have = μ(e σ )cσ r pcr μ(e σ )cσ r = Ch ( σ ). (7) By the condition (ESSC), for any distinct words σ,τ Ξ,wehave (E σ ) β Eσ /4 (E τ ) β Eτ /4 =. By estimating the volumes, we know that there exist two constants L 1,L 2 1 which are independent of σ, τ such that (E σ ) β Eσ /4 can be covered by L 1 closed balls with radii β E σ /(8M) and E τ can be covered by L 2 closed balls with radii β E τ /(8M). Indeed, we may take
5 746 S. Zhu / J. Math. Anal. Appl. 338 (2008) L 1 := [( 16Mβ 1 + 8M + 1 ) d], L2 := [( 16Mβ ) d]. Set L := L 1 + L 2. Suppose card(α σ )>Lfor some. Then there exists some τ Γ n such that card(α τ ) = 0 since card(α) card( Γ n ). We assume that b 1,...,b L α σ.letq 1,...,q L1 be the centers of the L 1 closed balls with radii β E σ /(8M) which cover (E σ ) β Eσ /4. Lete 1,...,e L2 be the centers of the L 2 closed balls with radii β E τ /(8M) which cover E τ. Set γ := ( α \{b 1,...,b L } ) {q 1,...,q L1,e 1,...,e L2 }. Then using the condition (HC) we have min x a α a r dμ(x) βr E τ r 8 r μ(e τ ) Cβr V r 8 r h ( τ ) Cβr V r l 8 r. n E τ On the other hand, by the definition of γ,wehave min x c γ c r dμ(x) βr E σ r 8 r M r μ(e σ ) + βr E τ r 8 r M r μ(e τ ) = βr V r ( ) Cβ r V r l + h(τ) < 8 r M r 8 r. n E σ E τ Combining the above inequalities, we have min x a α a r dμ(x) > min x c γ c r dμ(x). (8) E σ E τ E σ E τ For any x E \ (E σ E τ ) and for any a α σ, we denote by B(a,β E σ /8) the closed ball centered at a and of radius β E σ /8. Let y be the intersection of the surface of B(a,β E σ /8) and the line determined by x and a. Then we have y (E σ ) β Eσ /4. Recall that (E σ ) β Eσ /4 is covered by L 1 balls of radii β E σ /(8M) centered at q j,1 j L 1. Hence there exists some point q i such that y B(q i,β E σ /(8M)). By the triangular inequality, we have x q i x y +β E σ /(8M) < x y +β E σ /8 = x a. From the above inequalities, we deduce min x a min x q i. a α σ 1 i L 1 Observing the difference between α and γ, this implies min x a min a α c γ for all x E \ (E σ E τ ). (9) Hence (8) and (9) yields V m,r (μ) = min a α a r dμ(x) > min c γ c r dμ(x) = min c γ c r dμ(x). E σ E σ This contradicts the optimality of α. The lemma follows. Lemma 4. Let L 1 be an integer and α an arbitrary subset of R d with cardinality L. Then there exists a constant D>0 such that for any σ Ξ with μ(e σ )>0, we have min x a α a r dμ(x) D. (10) E σ Proof. We assume that σ Ξ k. Choose j 1 such that 2 j > L. Set Λ j (σ ) := {τ Ξ k+j : σ τ}. By the condition (HC), there exist τ (i) Λ j (σ ), 1 i L + 1 such that
6 S. Zhu / J. Math. Anal. Appl. 338 (2008) μ(e τ (i))>0, 1 i L + 1. Suppose that for some a α there exist 1 i 1 i 2 L + 1 such that dist(a, E τ (i h ))< β 2 min{ E τ,τ Λ j (σ ) }, h= 1, 2. (11) Then by the triangular inequality we have dist(e τ (i 1 ),E τ (i 2 ))<βmin { E τ,τ Λ j (σ ) }. This contradicts the condition (ESSC). Therefore, for each a α there is at most one cylinder set E τ, τ Λ j (σ ) such that (11) holds. On the other hand, we have card(α) < L + 1; thus there exists some E τ (i), 1 i L + 1 such that μ(e τ (i))>0, min dist(a, E a α τ (i)) β 2 min{ E τ,τ Λ j (σ ) }. (12) Using (12) and the conditions (ESSC) and (HC), we deduce E σ min x a α a r dμ(x) E τ (i) min x a α a r dμ(x) μ(e τ (i))2 r β r( min { E τ,τ Λ j (σ ) }) r p j μ(e σ )2 r β r c jr E σ r =: D, where D = 2 r p j β r c jr V r. This completes the proof of the lemma. Proof of Theorem 1. Let S>d r. Then d n,r <Sfor large n. Letc>0 be as in the condition (BD) and let C = pc r. By (3) and (4), we have 1 = ( ) dn,r r+dn,r C dn,r r+dn,r ( ( )) dn,r h σ r+dn,r C dn,r r+dn,r ( ( h σ )) ( ) S r+s S l r+s C card( Γ n ), n Hence we have card( Γ n ) C 1 (n/l) r+s S.Letφ(n):= [C 1 (n/l) r+s S ]. For each, we choose an arbitrary point of E σ and denote by α the set of these points. Note that φ(n) card( Γ n ). We deduce V φ(n),r (μ) E σ r min x a α a r dμ(x) μ(e σ ) E σ r C 1 (n/l) σ Γ n where C r = C 1 V r l r+s. Thus by Lemma 2, we have D r (μ) = lim sup S. log V φ(n),r (μ) S r+s l/n V r =: C r n S+r r, By the arbitrariness of S, wehaved r (μ) d r. One can show D r (μ) d r in a similar manner by considering subsequences. Next we show the reverse inequalities. Let s<d r. Then s<d n,r for large n. By (3) and (4) we have 1 = ( ) dn,r r+dn,r ( ) s ( ) s r+s l r+s card( Γ n ). n Hence card( Γ n ) (n/l) s r+s.letα C [(n/l) s r+s ],r (μ). For each,letw 1,...,w L2 be the centers of the L 2 closed balls with radii β E σ /(8M) which cover E σ and define α σ := α σ {w 1,...,w L2 }. Thus for and all x E σ,wehave min x a min x a. a α a α σ By Lemma 3, card( α σ ) L + L 2 =: L. By Lemma 4 and (7), we deduce
7 748 S. Zhu / J. Math. Anal. Appl. 338 (2008) V [(n/l) s r+s ],r (μ) = E σ min x a α a r dμ(x) E σ DC(n/l) r+s s l/n =: C1 r n r+s r, where C1 r = DClr/(s+r). Thus by Lemma 2, for φ(n):= [(n/l) r+s s ], wehave D r (μ) = lim inf s. log V φ(n),r (μ) min x a r dμ(x) D a α σ By the arbitrariness of s, we finally get D r (μ) d r. The inequality D r (μ) d r follows similarly by considering subsequences. Remark 5. As we mentioned above, our definition of the crucial set Γ n is motivated by Graf and Luschgy s work for self-similar measures. We note that Graf and Luschgy s method to get the lower bound of the quantization dimension is based on Hölder s inequality with exponent less than one and is dependent on the existence of the quantization dimension. As a result, their method is not applicable here. 5. Application to product measures In this section, we use Theorem 1 to determine the quantization dimension of the product measures supported on a Cantor-like set E which is determined by contractive similitudes f kj s of contraction ratios c kj,1 j n k, k 1. Let p kj,1 j n k, k 1 be positive real numbers satisfying inf min p kj > 0, k 1 1 j n k n k p kj = 1, k 1. (13) j=1 By Kolmogorov consistency theorem, there exists a unique probability measure μ supported on E such that μ(e σ ) = p 1σ(1) p nσ (n), σ = ( σ(1),...,σ(n) ) Ξ n. We call this measure the product measure on E associated with {p kj }. A special product measure is the uniform probability measure μ on E which assigns a weight (n 1 n k ) 1 to each of the cylinder sets of order k, i.e., μ(e σ ) = (n 1 n k ) 1, σ Ξ k. (14) For σ Ξ n,wesetp σ = p 1σ(1) p nσ (n).letd n,r, d r,d r be defined by ( pσ cσ r ) dn,r dn,r +r = 1, σ Γ n d r := lim inf d n,r, where Γ n is as defined in (3). Then we have d r := lim sup d n,r, Theorem 6. Let E be the Cantor-like set determined by {f kj } which satisfies the (BD) and (ESSC). Let μ be the product measure associated with {p kj } satisfying (13). Then D r (μ) = d r, D r (μ) = d r. In particular, if c kj c k for all k 1 and μ is the uniform probability measure on E, then D r (μ) = lim sup k log(n 1 n k ) log(c 1 c k ), D r(μ) = lim inf k log(n 1 n k ) log(c 1 c k ). Proof. Since (13) implies the (HC), the first part of the theorem follows immediately from Theorem 1. In the following we show the remaining part. Let μ be the uniform probability measure on E. We write s k,r := log(n 1 n k ) log(c 1 c k ), ξ := lim sup s k,r, k ξ := lim inf k s k,r. Let l = n 1 1 cr 1. For each n 1, there exists some k 1 such that (n 1 n k+1 ) 1 (c 1 c k+1 ) r <l/n (n 1 n k ) 1 (c 1 c k ) r. (15)
8 S. Zhu / J. Math. Anal. Appl. 338 (2008) By (14), for σ Ξ k, = (n 1 n k ) 1 (c 1 c k ) r. Thus by (15), d n,r = s k+1,r for the above k. By Theorem 1, this implies D r (μ) ξ, D r (μ) ξ. On the other hand, for any k 2, there exists some n 1, such that l/(n + 1) (n 1 n k ) 1 (c 1 c k ) r <l/n. For this n,wehave (n 1 n k 1 ) 1 (c 1 c k 1 ) r n k ck r l/(n + 1) l/n. This implies Ξ k = Γ n and s k,r = d n,r for the above n. It follows that D r (μ) ξ,d r (μ) ξ. This completes the proof of the theorem which generalizes [9, Theorem 1.6(2)]. In the following, we give an example to illustrate Theorem 6. Example 7. Let f kj, p kj, j = 1, 2, k 1, be defined as follows: { x6 { if k = 1, x if k = 1, f k1 (x) = x f if k>1, k2 (x) = x if k>1, k k 3 { { 1+2 k 1 (1 + 2 r ) 1 if k = 1, (1 + 2 r ) 1 if k = 1, p k1 = p k2 = 1 (1 + 2 (1+2 k)r ) 1 if k>1, (1 + 2 (1+2 k)r ) 1 if k>1. Let E be the Cantor-like set on R determined by {f kj } and let μ be the product measure supported on E determined by {p kj }.Wehave D r (μ) = r log 2 r log 3 + log(1 + 2 r =: s. ) log 2 In fact, by our construction, we have p 11 c r 11 = p 12c r 12, p k1c r k1 = p k2c r k2 = ( p 11 c r 11) Ak, k 1, where A k = (1 + 2 r )(1 + 2 (1+2 k )r ) kr. Thus for each n 1, we have Γ n = Ξ k for some k 1. For any σ Ξ k, we have = μ(e σ )c r σ = p 1σ(1) p kσ(k) c r 1σ(1) cr kσ(k) = ( p 11 c r 11 Let d n,r be the solution of the following equation: ( (1 2 k + 2 r ) ) k 1 dn,r dn,r +r k 3 kr A i = 1. i=1 k 1 ) k Using the fact that A k 1 (k ), we deduce lim n r d n,r = s. By Theorem 6, D r (μ) exists and equals s. Acknowledgment I thank the referee for some helpful comments and nice suggestions. References [1] J.A. Bucklew, G.L. Wise, Multidimensional asymptotic quantization with rth power distortion measures, IEEE Trans. Inform. Theory 28 (1982) [2] S. Graf, H. Luschgy, The quantization of the Cantor distribution, Math. Nachr. 183 (1997) [3] S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Math., vol. 1730, Springer-Verlag, [4] S. Graf, H. Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr. 241 (2002) i=1 A i.
9 750 S. Zhu / J. Math. Anal. Appl. 338 (2008) [5] S. Graf, H. Luschgy, The point density measure in the quantization of self-similar probabilities, Math. Proc. Cambridge Philos. Soc. 138 (2005) [6] R. Gray, D. Neuhoff, Quantization, IEEE Trans. Inform. Theory 44 (1998) [7] S. Hua, On the dimension of generalized self-similar sets, Acta Math. Appl. Sin. 17 (4) (1994) [8] S. Hua, W. Li, Packing dimension of generalized Moran sets, Progr. Natur. Sci. 6 (2) (1996) [9] M. Kesseböhmer, S. Zhu, Stability of the quantization dimension and quantization for homogeneous Cantor measures, Math. Nachr. 280 (8) (2007) [10] L.J. Lindsay, R.D. Mauldin, Quantization dimension for conformal function system, Nonlinearity 15 (1) (2002) [11] K. Pötzelberger, The quantization error of self-similar distributions, Math. Proc. Cambridge Philos. Soc. 137 (2004) [12] P.L. Zador, Asymptotic quantization error of continuous signals and the quantization dimension, IEEE Trans. Inform. Theory 28 (1982) [13] P.L. Zador, Development and evaluation of procedures for quantizing multivariate distributions, PhD thesis, Stanford University, [14] S. Zhu, Quantization dimension for condensation systems, Math. Z., in press.
The Hausdorff measure of a class of Sierpinski carpets
J. Math. Anal. Appl. 305 (005) 11 19 www.elsevier.com/locate/jmaa The Hausdorff measure of a class of Sierpinski carpets Yahan Xiong, Ji Zhou Department of Mathematics, Sichuan Normal University, Chengdu
More informationOptimal quantization for uniform distributions on Cantor-like sets
oname manuscript o. will be inserted by the editor Optimal quantization for uniform distributions on Cantor-like sets Wolfgang Kreitmeier Received: date / Accepted: date Abstract In this paper, the problem
More informationarxiv: v8 [stat.co] 28 Jan 2018
OPTIMAL QUANTIZATION FOR NONUNIFORM CANTOR DISTRIBUTIONS LAKSHMI ROYCHOWDHURY arxiv:151.00379v8 [stat.co] 8 Jan 018 Abstract. Let P be a Borel probability measure on R such that P = 1 4 P S 1 1 + 3 4 P
More informationBounds of Hausdorff measure of the Sierpinski gasket
J Math Anal Appl 0 007 06 04 wwwelseviercom/locate/jmaa Bounds of Hausdorff measure of the Sierpinski gasket Baoguo Jia School of Mathematics and Scientific Computer, Zhongshan University, Guangzhou 5075,
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationarxiv: v1 [cs.it] 8 May 2016
NONHOMOGENEOUS DISTRIBUTIONS AND OPTIMAL QUANTIZERS FOR SIERPIŃSKI CARPETS MRINAL KANTI ROYCHOWDHURY arxiv:605.08v [cs.it] 8 May 06 Abstract. The purpose of quantization of a probability distribution is
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationA NOTE ON CORRELATION AND LOCAL DIMENSIONS
A NOTE ON CORRELATION AND LOCAL DIMENSIONS JIAOJIAO YANG, ANTTI KÄENMÄKI, AND MIN WU Abstract Under very mild assumptions, we give formulas for the correlation and local dimensions of measures on the limit
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 informationUNIFORMLY PERFECT ANALYTIC AND CONFORMAL ATTRACTOR SETS. 1. Introduction and results
UNIFORMLY PERFECT ANALYTIC AND CONFORMAL ATTRACTOR SETS RICH STANKEWITZ Abstract. Conditions are given which imply that analytic iterated function systems (IFS s) in the complex plane C have uniformly
More informationSimultaneous Accumulation Points to Sets of d-tuples
ISSN 1749-3889 print, 1749-3897 online International Journal of Nonlinear Science Vol.92010 No.2,pp.224-228 Simultaneous Accumulation Points to Sets of d-tuples Zhaoxin Yin, Meifeng Dai Nonlinear Scientific
More informationNECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES
NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)
More informationQUASI-LIPSCHITZ EQUIVALENCE OF SUBSETS OF AHLFORS DAVID REGULAR SETS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 39, 2014, 759 769 QUASI-LIPSCHITZ EQUIVALENCE OF SUBSETS OF AHLFORS DAVID REGULAR SETS Qiuli Guo, Hao Li and Qin Wang Zhejiang Wanli University,
More informationA BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras.
A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM STEVO TODORCEVIC Abstract. We describe a Borel poset satisfying the σ-finite chain condition but failing to satisfy the σ-bounded chain condition. MSC 2000:
More informationHYPERBOLIC DERIVATIVES AND GENERALIZED SCHWARZ-PICK ESTIMATES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 11, Pages 339 3318 S 2-9939(4)7479-9 Article electronically published on May 12, 24 HYPERBOLIC DERIVATIVES AND GENERALIZED SCHWARZ-PICK
More informationThe Polynomial Numerical Index of L p (µ)
KYUNGPOOK Math. J. 53(2013), 117-124 http://dx.doi.org/10.5666/kmj.2013.53.1.117 The Polynomial Numerical Index of L p (µ) Sung Guen Kim Department of Mathematics, Kyungpook National University, Daegu
More informationStanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures
2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon
More informationCorrelation dimension for self-similar Cantor sets with overlaps
F U N D A M E N T A MATHEMATICAE 155 (1998) Correlation dimension for self-similar Cantor sets with overlaps by Károly S i m o n (Miskolc) and Boris S o l o m y a k (Seattle, Wash.) Abstract. We consider
More informationStolz angle limit of a certain class of self-mappings of the unit disk
Available online at www.sciencedirect.com Journal of Approximation Theory 164 (2012) 815 822 www.elsevier.com/locate/jat Full length article Stolz angle limit of a certain class of self-mappings of the
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationv( x) u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently if v exists, v 0.
Sep. 26 The Perron Method In this lecture we show that one can show existence of solutions using maximum principle alone.. The Perron method. Recall in the last lecture we have shown the existence of solutions
More informationEntire functions defined by Dirichlet series
J. Math. Anal. Appl. 339 28 853 862 www.elsevier.com/locate/jmaa Entire functions defined by Dirichlet series Lina Shang, Zongsheng Gao LMIB and Department of Mathematics, Beihang University, Beijing 83,
More informationNOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS
NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS K. Jarosz Southern Illinois University at Edwardsville, IL 606, and Bowling Green State University, OH 43403 kjarosz@siue.edu September, 995 Abstract. Suppose
More informationTwo dimensional exterior mixed problem for semilinear damped wave equations
J. Math. Anal. Appl. 31 (25) 366 377 www.elsevier.com/locate/jmaa Two dimensional exterior mixed problem for semilinear damped wave equations Ryo Ikehata 1 Department of Mathematics, Graduate School of
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationPacking-Dimension Profiles and Fractional Brownian Motion
Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 Packing-Dimension Profiles and Fractional Brownian Motion By DAVAR KHOSHNEVISAN Department of Mathematics, 155 S. 1400 E., JWB 233,
More informationCOMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS
Series Logo Volume 00, Number 00, Xxxx 19xx COMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS RICH STANKEWITZ Abstract. Let G be a semigroup of rational functions of degree at least two, under composition
More informationKey words and phrases. Hausdorff measure, self-similar set, Sierpinski triangle The research of Móra was supported by OTKA Foundation #TS49835
Key words and phrases. Hausdorff measure, self-similar set, Sierpinski triangle The research of Móra was supported by OTKA Foundation #TS49835 Department of Stochastics, Institute of Mathematics, Budapest
More informationPersistence and global stability in discrete models of Lotka Volterra type
J. Math. Anal. Appl. 330 2007 24 33 www.elsevier.com/locate/jmaa Persistence global stability in discrete models of Lotka Volterra type Yoshiaki Muroya 1 Department of Mathematical Sciences, Waseda University,
More informationII - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define
1 Measures 1.1 Jordan content in R N II - REAL ANALYSIS Let I be an interval in R. Then its 1-content is defined as c 1 (I) := b a if I is bounded with endpoints a, b. If I is unbounded, we define c 1
More informationARTICLE IN PRESS. J. Math. Anal. Appl. ( ) Note. On pairwise sensitivity. Benoît Cadre, Pierre Jacob
S0022-27X0500087-9/SCO AID:9973 Vol. [DTD5] P.1 1-8 YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 1 J. Math. Anal. Appl. www.elsevier.com/locate/jmaa Note On pairwise sensitivity Benoît Cadre,
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationThe Hausdorff measure of a Sierpinski-like fractal
Hokkaido Mathematical Journal Vol. 6 (2007) p. 9 19 The Hausdorff measure of a Sierpinski-like fractal Ming-Hua Wang (Received May 12, 2005; Revised October 18, 2005) Abstract. Let S be a Sierpinski-like
More informationDifferential subordination related to conic sections
J. Math. Anal. Appl. 317 006 650 658 www.elsevier.com/locate/jmaa Differential subordination related to conic sections Stanisława Kanas Department of Mathematics, Rzeszów University of Technology, W. Pola,
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationUNIFORMLY DISTRIBUTED MEASURES IN EUCLIDEAN SPACES
MATH. SCAND. 90 (2002), 152 160 UNIFORMLY DISTRIBUTED MEASURES IN EUCLIDEAN SPACES BERND KIRCHHEIM and DAVID PREISS For every complete metric space X there is, up to a constant multiple, at most one Borel
More informationThe small ball property in Banach spaces (quantitative results)
The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence
More informationarxiv: v2 [math.fa] 27 Sep 2016
Decomposition of Integral Self-Affine Multi-Tiles Xiaoye Fu and Jean-Pierre Gabardo arxiv:43.335v2 [math.fa] 27 Sep 26 Abstract. In this paper we propose a method to decompose an integral self-affine Z
More informationMax-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig Uniformly distributed measures in Euclidean spaces by Bernd Kirchheim and David Preiss Preprint-Nr.: 37 1998 Uniformly Distributed
More informationOn the validity of the Euler Lagrange equation
J. Math. Anal. Appl. 304 (2005) 356 369 www.elsevier.com/locate/jmaa On the validity of the Euler Lagrange equation A. Ferriero, E.M. Marchini Dipartimento di Matematica e Applicazioni, Università degli
More informationPROBABLILITY MEASURES ON SHRINKING NEIGHBORHOODS
Real Analysis Exchange Summer Symposium 2010, pp. 27 32 Eric Samansky, Nova Southeastern University, Fort Lauderdale, Florida, USA. email: es794@nova.edu PROBABLILITY MEASURES ON SHRINKING NEIGHBORHOODS
More informationWittmann Type Strong Laws of Large Numbers for Blockwise m-negatively Associated Random Variables
Journal of Mathematical Research with Applications Mar., 206, Vol. 36, No. 2, pp. 239 246 DOI:0.3770/j.issn:2095-265.206.02.03 Http://jmre.dlut.edu.cn Wittmann Type Strong Laws of Large Numbers for Blockwise
More informationAPPROXIMATE WEAK AMENABILITY OF ABSTRACT SEGAL ALGEBRAS
MATH. SCAND. 106 (2010), 243 249 APPROXIMATE WEAK AMENABILITY OF ABSTRACT SEGAL ALGEBRAS H. SAMEA Abstract In this paper the approximate weak amenability of abstract Segal algebras is investigated. Applications
More informationAn Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010
An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 John P. D Angelo, Univ. of Illinois, Urbana IL 61801.
More informationA NEW LINDELOF SPACE WITH POINTS G δ
A NEW LINDELOF SPACE WITH POINTS G δ ALAN DOW Abstract. We prove that implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ1 which has points G δ. In addition, this space has
More informationSHARP INEQUALITIES FOR MAXIMAL FUNCTIONS ASSOCIATED WITH GENERAL MEASURES
SHARP INEQUALITIES FOR MAXIMAL FUNCTIONS ASSOCIATED WITH GENERAL MEASURES L. Grafakos Department of Mathematics, University of Missouri, Columbia, MO 65203, U.S.A. (e-mail: loukas@math.missouri.edu) and
More informationarxiv: v1 [math.ds] 31 Jul 2018
arxiv:1807.11801v1 [math.ds] 31 Jul 2018 On the interior of projections of planar self-similar sets YUKI TAKAHASHI Abstract. We consider projections of planar self-similar sets, and show that one can create
More informationRate of convergence for certain families of summation integral type operators
J Math Anal Appl 296 24 68 618 wwwelseviercom/locate/jmaa Rate of convergence for certain families of summation integral type operators Vijay Gupta a,,mkgupta b a School of Applied Sciences, Netaji Subhas
More informationNatural boundary and Zero distribution of random polynomials in smooth domains arxiv: v1 [math.pr] 2 Oct 2017
Natural boundary and Zero distribution of random polynomials in smooth domains arxiv:1710.00937v1 [math.pr] 2 Oct 2017 Igor Pritsker and Koushik Ramachandran Abstract We consider the zero distribution
More informationIntroduction to Hausdorff Measure and Dimension
Introduction to Hausdorff Measure and Dimension Dynamics Learning Seminar, Liverpool) Poj Lertchoosakul 28 September 2012 1 Definition of Hausdorff Measure and Dimension Let X, d) be a metric space, let
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationOn Bank-Laine functions
Computational Methods and Function Theory Volume 00 0000), No. 0, 000 000 XXYYYZZ On Bank-Laine functions Alastair Fletcher Keywords. Bank-Laine functions, zeros. 2000 MSC. 30D35, 34M05. Abstract. In this
More informationPERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS. A. M. Blokh. Department of Mathematics, Wesleyan University Middletown, CT , USA
PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS A. M. Blokh Department of Mathematics, Wesleyan University Middletown, CT 06459-0128, USA August 1991, revised May 1992 Abstract. Let X be a compact tree,
More informationNOTIONS OF DIMENSION
NOTIONS OF DIENSION BENJAIN A. STEINHURST A quick overview of some basic notions of dimension for a summer REU program run at UConn in 200 with a view towards using dimension as a tool in attempting to
More informationTopological properties of Z p and Q p and Euclidean models
Topological properties of Z p and Q p and Euclidean models Samuel Trautwein, Esther Röder, Giorgio Barozzi November 3, 20 Topology of Q p vs Topology of R Both R and Q p are normed fields and complete
More informationON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES
Commun. Korean Math. Soc. 22 (2007), No. 2, pp. 297 303 ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES Woo Chorl Hong Reprinted from the Communications of the Korean Mathematical Society
More informationA derivative-free nonmonotone line search and its application to the spectral residual method
IMA Journal of Numerical Analysis (2009) 29, 814 825 doi:10.1093/imanum/drn019 Advance Access publication on November 14, 2008 A derivative-free nonmonotone line search and its application to the spectral
More informationJORDAN CONTENT. J(P, A) = {m(i k ); I k an interval of P contained in int(a)} J(P, A) = {m(i k ); I k an interval of P intersecting cl(a)}.
JORDAN CONTENT Definition. Let A R n be a bounded set. Given a rectangle (cartesian product of compact intervals) R R n containing A, denote by P the set of finite partitions of R by sub-rectangles ( intervals
More informationfunctions as above. There is a unique non-empty compact set, i=1
1 Iterated function systems Many of the well known examples of fractals, including the middle third Cantor sets, the Siepiński gasket and certain Julia sets, can be defined as attractors of iterated function
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationUniquely Universal Sets
Uniquely Universal Sets 1 Uniquely Universal Sets Abstract 1 Arnold W. Miller We say that X Y satisfies the Uniquely Universal property (UU) iff there exists an open set U X Y such that for every open
More informationMultiple points of the Brownian sheet in critical dimensions
Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang Ecole Polytechnique Fédérale de Lausanne Based on joint work with: Carl Mueller Multiple points of the Brownian sheet in critical
More informationDENSELY k-separable COMPACTA ARE DENSELY SEPARABLE
DENSELY k-separable COMPACTA ARE DENSELY SEPARABLE ALAN DOW AND ISTVÁN JUHÁSZ Abstract. A space has σ-compact tightness if the closures of σ-compact subsets determines the topology. We consider a dense
More informationEstimates for probabilities of independent events and infinite series
Estimates for probabilities of independent events and infinite series Jürgen Grahl and Shahar evo September 9, 06 arxiv:609.0894v [math.pr] 8 Sep 06 Abstract This paper deals with finite or infinite sequences
More informationAn Asymptotic Property of Schachermayer s Space under Renorming
Journal of Mathematical Analysis and Applications 50, 670 680 000) doi:10.1006/jmaa.000.7104, available online at http://www.idealibrary.com on An Asymptotic Property of Schachermayer s Space under Renorming
More informationORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY
ORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY AND ω-limit SETS CHRIS GOOD AND JONATHAN MEDDAUGH Abstract. Let f : X X be a continuous map on a compact metric space, let ω f be the collection of ω-limit
More informationPACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION
PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION DAVAR KHOSHNEVISAN AND YIMIN XIAO Abstract. In order to compute the packing dimension of orthogonal projections Falconer and Howroyd 997) introduced
More informationSEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE
SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE FRANÇOIS BOLLEY Abstract. In this note we prove in an elementary way that the Wasserstein distances, which play a basic role in optimal transportation
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 433 (200) 867 875 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa On the exponential exponents
More informationarxiv: v2 [math.ca] 4 Jun 2017
EXCURSIONS ON CANTOR-LIKE SETS ROBERT DIMARTINO AND WILFREDO O. URBINA arxiv:4.70v [math.ca] 4 Jun 07 ABSTRACT. The ternary Cantor set C, constructed by George Cantor in 883, is probably the best known
More informationThe Drazin inverses of products and differences of orthogonal projections
J Math Anal Appl 335 7 64 71 wwwelseviercom/locate/jmaa The Drazin inverses of products and differences of orthogonal projections Chun Yuan Deng School of Mathematics Science, South China Normal University,
More informationMeasures and Measure Spaces
Chapter 2 Measures and Measure Spaces In summarizing the flaws of the Riemann integral we can focus on two main points: 1) Many nice functions are not Riemann integrable. 2) The Riemann integral does not
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More informationChapter 1. Measure Spaces. 1.1 Algebras and σ algebras of sets Notation and preliminaries
Chapter 1 Measure Spaces 1.1 Algebras and σ algebras of sets 1.1.1 Notation and preliminaries We shall denote by X a nonempty set, by P(X) the set of all parts (i.e., subsets) of X, and by the empty set.
More informationObserver design for a general class of triangular systems
1st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 014. Observer design for a general class of triangular systems Dimitris Boskos 1 John Tsinias Abstract The paper deals
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationEstimates for derivatives of holomorphic functions in a hyperbolic domain
J. Math. Anal. Appl. 9 (007) 581 591 www.elsevier.com/locate/jmaa Estimates for derivatives of holomorphic functions in a hyperbolic domain Jian-Lin Li College of Mathematics and Information Science, Shaanxi
More informationarxiv: v2 [math.ca] 10 Apr 2010
CLASSIFYING CANTOR SETS BY THEIR FRACTAL DIMENSIONS arxiv:0905.1980v2 [math.ca] 10 Apr 2010 CARLOS A. CABRELLI, KATHRYN E. HARE, AND URSULA M. MOLTER Abstract. In this article we study Cantor sets defined
More informationNonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 20, Number 1, June 2016 Available online at http://acutm.math.ut.ee Nonhomogeneous linear differential polynomials generated by solutions
More informationResearch Article Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients
Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 606149, 15 pages doi:10.1155/2010/606149 Research Article Frequent Oscillatory Behavior of Delay Partial Difference
More informationAW -Convergence and Well-Posedness of Non Convex Functions
Journal of Convex Analysis Volume 10 (2003), No. 2, 351 364 AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy villa@dima.unige.it
More informationPacking Measures of Homogeneous Cantor Sets
Λ45ffΛ4 μ ρ Ω χ Vol.45, No.4 206 7fl ADVANCES IN MATHEMATICS CHINA) July, 206 doi: 0.845/sxjz.204207b Packing Measures of Homogeneous Cant Sets QU Chengqin,, ZHU Zhiwei 2 3,, ZHOU Zuoling. School of Mathematics,
More informationCovering an ellipsoid with equal balls
Journal of Combinatorial Theory, Series A 113 (2006) 1667 1676 www.elsevier.com/locate/jcta Covering an ellipsoid with equal balls Ilya Dumer College of Engineering, University of California, Riverside,
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationIGNACIO GARCIA, URSULA MOLTER, AND ROBERTO SCOTTO. (Communicated by Michael T. Lacey)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 10, October 2007, Pages 3151 3161 S 0002-9939(0709019-3 Article electronically published on June 21, 2007 DIMENSION FUNCTIONS OF CANTOR
More informationShih-sen Chang, Yeol Je Cho, and Haiyun Zhou
J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245 1260 DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract.
More informationLocal semiconvexity of Kantorovich potentials on non-compact manifolds
Local semiconvexity of Kantorovich potentials on non-compact manifolds Alessio Figalli, Nicola Gigli Abstract We prove that any Kantorovich potential for the cost function c = d / on a Riemannian manifold
More informationOn Antichains in Product Posets
On Antichains in Product Posets Sergei L. Bezrukov Department of Math and Computer Science University of Wisconsin - Superior Superior, WI 54880, USA sb@mcs.uwsuper.edu Ian T. Roberts School of Engineering
More informationA note on fixed point sets in CAT(0) spaces
J. Math. Anal. Appl. 320 (2006) 983 987 www.elsevier.com/locate/jmaa Note A note on fied point sets in CAT(0) spaces P. Chaoha,1, A. Phon-on Department of Mathematics, Faculty of Science, Chulalongkorn
More informationHouston Journal of Mathematics. c 2008 University of Houston Volume 34, No. 4, 2008
Houston Journal of Mathematics c 2008 University of Houston Volume 34, No. 4, 2008 SHARING SET AND NORMAL FAMILIES OF ENTIRE FUNCTIONS AND THEIR DERIVATIVES FENG LÜ AND JUNFENG XU Communicated by Min Ru
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationProblem set 1, Real Analysis I, Spring, 2015.
Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n
More informationarxiv:math/ v1 [math.fa] 26 Oct 1993
arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological
More informationOn Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q)
On Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q) Andreas Löhne May 2, 2005 (last update: November 22, 2005) Abstract We investigate two types of semicontinuity for set-valued
More informationarxiv: v1 [math.na] 9 Feb 2013
STRENGTHENED CAUCHY-SCHWARZ AND HÖLDER INEQUALITIES arxiv:1302.2254v1 [math.na] 9 Feb 2013 J. M. ALDAZ Abstract. We present some identities related to the Cauchy-Schwarz inequality in complex inner product
More informationON A PROBLEM RELATED TO SPHERE AND CIRCLE PACKING
ON A PROBLEM RELATED TO SPHERE AND CIRCLE PACKING THEMIS MITSIS ABSTRACT We prove that a set which contains spheres centered at all points of a set of Hausdorff dimension greater than must have positive
More informationFrom now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.
Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x
More informationExercises from other sources REAL NUMBERS 2,...,
Exercises from other sources REAL NUMBERS 1. Find the supremum and infimum of the following sets: a) {1, b) c) 12, 13, 14, }, { 1 3, 4 9, 13 27, 40 } 81,, { 2, 2 + 2, 2 + 2 + } 2,..., d) {n N : n 2 < 10},
More information