The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line


 Buddy Hunter
 3 years ago
 Views:
Transcription
1 The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni, di Tecnologie Chimiche, Automatica e Modelli Matematici (DITCAM) Università degli Studi di Palermo, Viale delle Scienze, Palermo, Italy mail addresses: Abstract The aim of this paper is to introduce an HenstockKurzweilStieltjes type integration process for real functions on a fractal subset of tha real line. Key words: sset; sderivatives; sintegral Contents 1 Introduction 1 2 Preliminaries The sintegral The shenstockkurzweilstieltjes integral Linearity properties Cauchy Criterion and Cauchy extension SaksHenstock Lemma Relationship between the sintegral and the shenstockkurzweil Stieltjes integral The VitaliCarathéodory Theorem Introduction During last years, some mathematicians have been obliged to define a new concept of derivation (see [1] and [2]) and a new concept of integration (see, [6] and [9]) to solve some physical and engineering problems. In fact even if the geometry of fractal 1
2 sets is a well explored subject (see, [3], [4], [5], [7]), in this sets there are so many irregularities to render inapplicable the standard methods and the technique of the ordinary calculus to give reasonable solutions of practical problems. Just think of the fact that the usual derivative of the classical LebesgueCantor staircase function is zero almost everywhere (see [1] and [6]) and of the fact that the Riemann s integral of functions defined on a fractal sets is undefined. So, in 1991, de Guzmann, Martin and Reyes (see [1]) in order to study the problem of the existence and the uniqueness of the solutions of differential equations of the type: d x(t) dt = f(t, x(t)) in which t or x(t) takes values in a noncontinuous set (i.e. a fractal set) introduced, for functions defined on a fractal set, a new concept of derivative, called the sderivative. Later on, Jiang and Su (see [6]) and more recently Parvate and Gangal (see [9]) introduced, for function defined in a fractal set, a new concept of RiemannStieltjes type integral, called the sintegral and the F s integral, respectively. Both authors defined such integration process in an analogous fashion as the classical Riemann integral but with the Hausdorff measure and the mass function taking over the role of the distance, respectively. Since the Hausdorff measure and the mass function are proportional (see [9], section 4), we can say that Jiang and Su in [6] and Parvate and Gangal in [9] defined indipendently the same integral. Both authors developed in a way analogous to the standard calculus the rest of the integration theory. Therefore properties like linearity and additivity with respect to integration domain are valid. About the Fundamental theorem of calculus, both authors use an extra hypothesis to prove it. In this paper we will compare the hypothesis of the Fundamental theorem of Calculus used by Jiang and Su with the one used by Parvate and Gangal and we will notice that without such additional hypothesis the Fundamental theorem of Calculus lose its classical formulation. Therefore in in the fractal case, it is not possible to prove a theorem like this: If R is a closed fractal set and if F : R is sdifferentiable on with a continuous derivative therefore [a,b] F s dh s = F (b) F (a). Moreover, in this paper we introduce an HenstokKurzweilStieltjes type integration process defined for real functions on a fractal subset of the real line of Hausdorff dimension s with 0 < s 1, since it is well known that numerical algorithms are based very rarely on Lebesgue integrals but are based more often on Riemann sums and in the classical literature the best formulation of the Fundamental theorem of calculus is done by the HenstockKurzweil integral that it is based on Riemann sums too (see [?]. Finally, we will show that the HenstokKurzweilStieltjes integral on the fractal set contains properly the RiemannStieltjes integral defined by Jiang and Su and by Parvate and Gangal. 2
3 2 Preliminaries In all the paper we denote by R the set of all real numbers and by a nonempty closed subset of the interval [a, b] R, with a = min and b = max. Moreover, we denote, by L( ) the Lebesgue measure on R. Now, given s, with 0 s 1, we recall that the sdimensional Hausdorff measure of is defined as: { } H s () = lim inf L(A i ) s : A i, L(A i ) δ. δ 0 H s ( ) is a Borel regular measure (see [7]). Moreover, we recall that, the unique number s for which H t () = 0 if t > s and H t () = + if t < s is called the Hausdorff dimension of. Whenever is H s measurable with 0 < H s () <, therefore is said to be an sset. Without loss of generality we can assume in all the paper, that H s () = 1. Definition 2.1. (see [1], [5], and [8]) Let f : R be a function and let x 0. The sderivative of f, on the right and on the left, at the point x 0 is: f s + (x 0 ) = lim x x + 0 x f(x) f(x 0 ) H s ( [x 0, x]) if H s ( [x 0, x]) > 0 for all x > x 0 f s f(x 0 ) f(x) (x 0 ) = lim x x 0 H s ( [x, if H s ( [x, x 0 ]) > 0 for all x < x 0 x 0 ]) x where these limits exist. It is said that the sderivative of f at x 0 exists if f s + (x 0 ) = f s (x 0 ), we be denote by f s(x 0 ) such common value. Remark. Of course, if f is sderivable at the point x 0 then f is continuous at x 0, according to the topology induced on for the usual topology of R. 2.1 The sintegral Definition 2.2. Let A [a, b] be an interval. The set Ã = A is called an interval of. Definition 2.3. A partition of is any collection P = {(Ãi, x i )} p of pairwise disjoint intervals Ãi and points x i Ãi such that = i Ãi. Definition 2.4. (see [5]) Let f : R be a function. A number σ is said to be the sintegral of f on, if for each ε > 0 there is a δ > 0 such that n f(x i ) H s (Ãi) σ < ε 3
4 for each partition P = {(Ãi, x i )} p of with Hs (Ãi) < δ, for i = 1, 2,..., n. By σ = (R) f(t)dh s (t) we will denote the sintegral and by sr() we will denote the collection of all functions that are sintegrable on. Theorem 2.1. (see [5]) Let f : R be a continuous function at all points of. If F : R is a function H s absolutely continuous on and F s(x) = f(x) at H s  a.e. point x, then (R) f(t) dh s (t) = F (b) F (a). Definition 2.5. (see [5]) Let f : R be a function. We say that f is H s absolutely continuous on, if ε > 0, δ > 0 such that n f(b k ) f(a k ) < ε k=1 whenever n k=1 Hs[ (ak, b k ) ] < δ, a k, b k, k = 1,..., n, and a 1 < b 1 a 2 < b 2... a n < b n. Lemma 2.2. Let f : R be H s absolutely continuous on. contiguous interval of therefore If [α, β] is a f(α) = f(β). Proof. Since [α, β] is a contiguous interval of it follows that H s ([α, β]) = 0. Moreover, since f is H s absolutely continuous on, ε > 0 we have that f(β) f(α) < ε. The assertion follows by the arbitrariness of ε. Definition 2.6. (see, [8]) Let f : R R be a function. A point x R is said a point of change of f if f is not constant over any open interval (c, d) containing x. The set of all points of change of f is called the set of change of f and it is denoted by Sch(f). Theorem 2.3. (see [8]) If F : R R is sdifferentiable on, F s is a continuous function on and Sch(F ) then (R) F s(x)dh s (x) = F (b) F (a). 4
5 xample 2.1. Let [0, 1] be the classical Cantor set. Let F be a function on [0, 1] defined by 0, 0 x F (x) = 3x 1, 3 < x < , 3 x 1. and let f be the restriction of F on the set. Note that: f is not H s absolutely continuous on. Sch(f). f is sdifferentiable at each point of and f s(x) = 0 for all points x. f s is sintegrable on and (R) f s dh s = 0 f(1) f(0) = 1. (i) f is not H s absolutely continuous on. (ii) Sch(f). (iii) f is sdifferentiable at each point of and f s(x) = 0 for all points x. (iv) f s is sintegrable on and (R) f s dh s = 0 f(1) f(0) = 1. 3 The shenstockkurzweilstieltjes integral Definition 3.1. A gauge on is any positive real function δ defined on. Definition 3.2. A partition of is any collection P = {(Ãi, x i )} p disjoint intervals of Ãi and points x i Ãi such that = i Ãi. of a pairwise Definition 3.3. Let P = {(Ãi, x i )} p be a partition of. If δ is a gauge on, then we say that P is a δfine partition of whenever Ãi ]x i δ(x i ), x i + δ(x i )[ for all i = 1, 2,..., p. The following Cousin s lemma, addresses the existence of δfine partitions. Lemma 3.1. If δ is a gauge on, then there exists a δfine partition of. 5
6 Proof. If [α, β] [a, b] and if [α, β] =, we said that [α, β] has a δfine partition of. Let c be the midpoint of [a, b] and let us observe that if P 1 and P 2 are δfine partitions of [a, c] and [c, b], respectively, then P = P 1 P 2 is a δfine partition of. Using this observation, we proceed by contradiction. Let us suppose that does not have a δfine partition then at least one of the intervals [a, c] or [c, b] does not have a δfine partition, let us say [a, c]. Therefore [a, c] is not empty. Let us relabel the interval [a, c] with [a 1, b 1 ] and let us repeat indefinitely this bisection method. So, we obtain a sequence of nested intervals: [a, b] [a 1, b 1 ]... [a n, b n ]... Since the length of the interval [a n, b n ] is (b a)/2 n therefore, for the Nested Intervals Property, there is a unique number ξ [a, b] such that: [a n, b n ] = {ξ}. n=0 It is trivial to notice that the interval [an, b n ] is not empty. Let ξ n [a n, b n ], therefore ξ n ξ < b n a n = (b a)/2 n. So lim n ξ n = ξ. Now since is a closed set, ξ. Since δ(ξ) > 0, we can find k N such that [a k, b k ] [ξ δ(ξ), ξ +δ(ξ)]. Therefore {[a k, b k ], ξ} is a δfine partition of [ak, b k ], contrarily to our assumption. Let P = {(Ãi, x i )} p be a partition of, let f : R be a function and let us consider the following sum S(f, P ) = f(x i )H s (Ãi). Definition 3.4. We say that the function f is shkintegrable on, if there exists I R such that for all ε > 0, there is a gauge δ on with: S(f, P ) I < ε for each δfine partition P = {(Ãi, x i )} p of. The number I is called the shkintegral of f on and we write I = (HK) f dh s. The collection of all functions that are shk integrable on will be denoted by shk(). 6
7 Remark. The number I from Definition 5 is determinate uniquely by the shkintegrable function f. In fact, let us suppose that I and J satisfies Definition 5, let us assume that J I and let us take ε = I J /2. Then we can find two gauges δ 1 and δ 2 on so that f(x i )H s (Ãi) I < ε for each δ 1 fine partition P = {(Ãi, x i )} p of and f(x i )H s (Ãi) J < ε for each δ 2 fine partition P = {(Ãi, x i )} p of. Now let δ = min{δ 1, δ 2 }. Then, for each δfine partition of we have I J I f(x i )H s (Ãi) + f(x i )H s (Ãi) J < 2ε = I J, that is a contradiction. 3.1 Linearity properties Theorem 3.2. (a) If f and g are shkintegrable on, then f + g is also shkintegrable on and (HK) (f + g) dh s = (HK) f dh s + (HK) g dh s. (b) If f is shkintegrable on and k R, then kf is shkintegrable on and (HK) kf dh s = k (HK) f dh s. Proof. (a): Let I and J be the shkintegrals of f and g, respectively. Given ε > 0, let δ and δ be two gauges on such that: f(x i )H s (Ãi) I < ε 2, if the partition P = {(Ãi, x i )} p is δ fine and g(x i )H s (Ãi) J < ε 2 7
8 if the partition P = {(Ãi, x i )} p is δ fine. Now let δ = min{δ, δ } therefore if P is a δfine partition, then it is both δ fine and δ fine. So (f + g)(x i )H s (Ãi) (I + J) f(x i )H s (Ãi) I + g(x i )H s (Ãi) J < ε. for each δfine partition of. The proof follows by arbitrariness of ε. (b): Let I be the shkintegral of f. Then, ε > 0 there is a gauge δ on with: f(x i )H s (Ãi) I < ε for each δfine partition P = {(Ãi, x i )} p of. Therefore, k R we have: kf(x i )H s (Ãi) ki = k f(x i )H s (Ãi) I < k ε. The proof follows by arbitrariness of ε. Theorem 3.3. Let f be shkintegrable on and f(x) 0 for all x, then (HK) f dh s 0. Proof. Let δ be a gauge on such that f(x i )H s (Ãi) (HK) if the partition P = {(Ãi, x i )} p is δfine. Since f(x) 0 for all x, then Therefore ε f(x i )H s (Ãi) 0. f(x i )H s (Ãi) ε < (HK) f dh s < f dh s < ε f(x i )H s (Ãi) + ε, 8
9 by the arbitrariness of ε it follows that (HK) f dh s 0. Corollary 3.4. Let f and g be shkintegrable on. If f g for all x, then (HK) f dh s (HK) g dh s. Proof. Let h := g f. By theorem 3.2 (b) h is shkintegrable and (HK) h dh s = (HK) g dh s (HK) f dh s, and since f g then h(x) 0 for all x and (HK) h dhs 0. Therefore the conclusion is immediate. 3.2 Cauchy Criterion and Cauchy extension Theorem 3.5. A function f : R is shk integrable on if and only if for each ε > 0 there exists a gauge δ on such that S(f, P 1 ) S(f, P 2 ) < ε for each pair δfine partitions P 1 and P 2 of. Proof. ( ) Let ε > 0 be given. Since f shk(), there exists a gauge δ on such that S(f, P ) (HK) f dh s < ε 2 for each δfine partition P of. If P 1 and P 2 are two δfine partitions of, we have that S(f, P 1 ) S(f, P 2 ) S(f, P 1 ) (HK) < ε. ( ) For each n N, let δ n be a gauge on such that f dh s + S(f, P2 ) (HK) f dh s S(f, Q n ) S(f, R n ) < 1 n for each pair δ n fine partitions Q n and R n of. 9
10 Let n (x) = min{δ 1 (x),..., δ n (x)} be a gauge on. For each n N, let P n be a n fine partition on. Clearly, if m > n then P m and P n are n fine partitions on ; hence S(f, P n ) S(f, P m ) < 1 n for m > n. Conseguently, {S(f, P n )} n=1 is a Cauchy sequence of real numbers; therefore this sequence converges to some real numer A = lim n S(f, P n ). Then S(f, P n ) A < 1 n for each n. Let ε > 0 be given and choose N such that 1 N < ε 2. If P is an arbitrary N fine partitions on, then S(f, P ) A S(f, P ) S(f, P N ) + S(f, P N ) A < 1 N + 1 N < ε. Thus f shk() and A = (HK) f dhs. Theorem 3.6. If f shk(), then f shk(ã) for each closed interval Ã. Proof. Let Ã be a closed interval of. Since f shk(), it follows that by Theorem 3.4, for each ε > 0 there exists a gauge δ on such that S(f, P ) S(f, Q) < ε for each pair δfine partitions P and Q of. Let {Ã1, Ã2,..., ÃN } be a finite collection of pairwise nonoverlapping intervals of, such that Ã / {Ã1, Ã2,..., ÃN } and such that N = Ã Ã k. For each k {1,..., N} we fix a δfine partition P k of Ãk. If PÃ and QÃ are δfine partitions of Ã, then it is clear that P Ã N k=1 P k and QÃ N k=1 P k are δfine partitions of. k=1 10
11 Thus S(f, PÃ) S(f, QÃ) N = S(f, P Ã ) + N S(f, P k ) S(f, QÃ) S(f, P k ) k=1 k=1 ( N ) ( N ) = S f, PÃ P k S f, QÃ P k < ε. k=1 k=1 Therefore, by Theorem 3.4, f shk(ã) SaksHenstock Lemma Definition 3.5. A subpartition of is a finite collection P = {(Ãi, x i )} p of a pairwise disjoint intervals Ãi and points x i Ãi for i = 1, 2,..., p. Definition 3.6. Let P = {(Ãi, x i )} p be a subartition of. If δ is a gauge on, then we say that P is a δfine subpartition of whenever Ãi ]x i δ(x i ), x i + δ(x i )[ for all i = 1, 2,..., p. Lemma 3.7 (The SaksHenstock Lemma). Let f shk() and ε > 0. If δ is a gauge on such that S(f, P ) (HK) for each δfine partition P of, then and f dh s < ε { } f(x i )H s (Ãi) (HK) f dh s ε f(x i)h s (Ãi) (HK) f dh s 2ε for each δfine subpartition P 0 = {(Ãi, x i )} p of. Proof. Let ε > 0. The set \ p Ãi is a finite union of disjoint intervals. Let K 1, K 2,..., K m be the closure of these intervals. 11
12 From theorem 3.5 we have that f shk( K j ) for each j = 1, 2,..., m. Hence for each η > 0 and for each j there exists a gauge δ j on K j such that S(f, P j) (HK) f dh s K j < η m for each δ j fine partition P j of K j. Let P = P 0 m P j=1 j. Then P is a δfine partition of and since S(f, P ) = S(f, P 0 ) + m j=1 S(f, P j) we have that (HK) f dh s = (HK) f dh s + m j=1 (HK) f dh s. K j Therefore { m } = S(f, P ) S(f, P j ) { } f(x i )H s (Ãi) (HK) f dh s j=1 S(f, P ) (HK) < ε + m η m = ε + η. f dh s + = S(f, P 0) (HK) f dh s m s} f dh s (HK) f dh j=1 K j m S(f, P j) (HK) f dh s kj { (HK) j=1 Since η > 0 is arbitrary, the first inequality is proved. To prove the second part, let } P 0 {(Ãi, + = x i ) P 0 : f(x i )H s (Ãi) (HK) f dh s 0 and let P0 = P 0 \ P 0 +. Let us note that both P 0 + and P 0 of, so they satisfy the first part of Lemma. are δfine subpartition 12
13 Thus, f(x i)h s (Ãi) (HK) f dh s { } = f(x i )H s (Ãi) (HK) f dh s (Ãi,xi) P + 0 (Ãi,xi) P 0 ε + ε = 2ε. { } f(x i )H s (Ãi) (HK) f dh s Theorem 3.8 (Cauchy extension). Let f : R be a function shk integrable on [c, b] for each c (a, b). If lim c a +(HK) [c,b] f dh s exists, then f is shk integrable on and (HK) f dh s = lim (HK) f dh s. c a + [c,b] Proof. Let ε > 0 and let {c n } n=0 be a strictly decreasing seguence of real numbers of such that c 0 = b and inf n N c n = a. Let A = lim c a +(HK) [c,b] f dh s. Choose N such that (HK) [x,b] for x (a, ( ) c N ] and f(a) H s [a, cn ] < ε 4. f dh s A < ε 4 From Theorem 3.5, since f is shk integrable on [c, b] for each c (a, b), then f shk( [c k, c k 1 ]). Therefore, for ε > 0 and for each positive integer k, we let δ k be a gauge on [c k, c k 1 ] such that S(f, Q k) (HK) for each δ k fine partition Q k of [ck, c k 1 ]. Define a gauge δ on by setting [c k,c k 1 ] f dh s < ε 4(2 k ) 13
14 c N a if x = a min{δ k (c k ), δ k+1 (c k ), 1 2 (c k 1 c k ), 1 2 (c k c k+1 )} δ(x) = min{δ k (c k ), 1 2 (x c k), 1 2 (c k 1 x)} if x = c k and k=1,2,.. if x (c k, c k 1 ) and k=1,2, (b c 1) if x = b and let P = {(Ãi, x i )} p be a δfine partition of. Let Ãi = [y i 1, y i ] for i = 1, 2,..., p and a = y 0 < y 1 <... < y p = b. Since a / k=1 [c k, c k 1 ], our choice of δ implies that x 1 = a, y 1 < c N and y 1 (c r+1, c r ] for some unique positive integer r. We also observe that for each k {1, 2,..., r}, our choice of δ implies that {(Ã, x) P : Ã [c k, c k 1 ]} is a δ k fine partition of Since [ck, c k 1 ]. S(f, P ) = f(a)h s ( [a, y 1 ]) + (Ã,x) P Ã [y 1,c r] f(x)h s (Ã) + ( r k=1 (Ã,x) P Ã [c k,c k 1 ] f(x)h s (Ã) ) and (HK) f dh s = (HK) [y 1,b] [y 1,c r] f dh s + ( r (HK) k=1 [c k,c k 1 ] f dh s ) thus 14
15 S(f, P ) lim c a +(HK) f dh s [c,b] f(a) H s ( [a, y 1 ]) + f(x)h s (Ã) (HK) f dh s (Ã,x) P [y 1,c r] Ã [y 1,c r] ( r ) + f(x)h s (Ã) (HK) f dh s k=1 (Ã,x) P [c k,c k 1 ] Ã [c k,c k 1 ] + (HK) f dh s lim [y 1,b] c a +(HK) f dh s [c,b] < ε. 4 Relationship between the sintegral and the shenstock KurzweilStieltjes integral 4.1 The VitaliCarathéodory Theorem Theorem 4.1 (The VitaliCarathéodory Theorem). Let be an sset, let f be L integrable on and let ε > 0. Then there exist functions u and v on such that u f v, u is upper semicontinuous and bounded above, v is lower semicontinuous and bounded below, and (L) (v u) dh s < ε. (1) Proof. Assume first that f 0 and that f is not identically 0. Since f is the pointwise limit of an increasing sequence of simple functions s n (see [9]), f is the sum of the simple functions t n = s n s n 1 (taking s 0 = 0), and since t n is a linear combination of characteristic functions, we see that there are measurable sets M i (not necessarily disjoint) and constants c i > 0 such that f(x) = c i χ Mi (x) (x ). 15
16 Since (L) f dh s = c i H s (M i ) (2) the series in (3) converges. Since H s is Borel regular (see [7], pag. 57) and is an sset, therefore H s L is a Radon measure. So, there are compacts sets K i and open sets V i such that K i M i V i and Put where N is chosen so that c i H s (V i K i ) < 2 i 1 ε (i = 1, 2, 3,...). (3) N v = c i χ Vi, u = c i χ Ki, c i H s (M i ) < ε 2. (4) N+1 Then v is lower semicontinuous, u is upper semicontinuous, u f v, and v u = N c i (χ Vi χ Ki ) + c i χ Vi c i (χ Vi χ Ki ) + N+1 c i χ Vi so that (4) and (5) imply (2). In the general case, write f = f + f, attach u 1 and v 1 to f +, attach u 2 and v 2 to f, as above, and put u = u 1 u 2, v = v 1 v 2. Since v 2 is upper semicontinuous and since the sum of two upper semicontinuous functions is upper semicontinuous, u and v have the desired properties. We now show that the shk integral is more general than the Lebesgue integral. Theorem 4.2. If is an sset, then L() shk() and (L) f dh s = (HK) f dh s for each f L(). Proof. Let f L() and ε > 0. For the VitaliCarathéodory Theorem, there are upper and lower semicontinuos functions g and h, respectively, such that g f h + and (L) (h g) dhs < ε. Find a gauge δ on so that g(t) f(t) + ε and h(t) f(t) ε for each x, t with x t < δ(x). Now if P = {(Ãi, x i )} p is a δfine partition of, then (L) g dh s (L) f dh s (L) h dh s. (5) N+1 16
17 Since g(t) f(x i ) + ε, then g(t) ε f(x i ), for each t Ãi we have that (L) (g ε) dh s (L) f(x i ) dh s and therefore (L) g dh s ε H s (Ãi) f(x i ) H s (Ãi). Moreover, since h(t) f(x i ) + ε, then f(x i ) h(t) + ε, for each t Ãi and therefore f(x i ) H s (Ãi) (L) h dh s + ε H s (Ãi). The last two results lead to (L) g dh s ε H s (Ãi) f(x i ) H s (Ãi) (L) h dh s + ε H s (Ãi), (6) i = 1,..., p. It follows that: Since S(f, P ) (L) f dh s f(x i)h s (Ãi) (L) f dh s For the (7), we have that f(x i)h s (Ãi) (L) f dh s (L) h dh s + ε H s (Ãi) (L) f dh s Ã i = ε Hs (Ãi) + (L) (h f) dh s. and therefore we have that S(f, P ) (L) (L) h dh s (L) f dh s (L) g dh s and then the theorem is proved. 0 (L) (h f) dh s (L) (h g) dh s f dh s f(x i)h s (Ãi) (L) f dh s [ ] ε H s (Ãi) + (L) (h g) dh s < ε (H s () + 1) 17
18 Remark. The converse of previus theorem is not true. The following example explains this assertion. xample 4.1. Let be the classical Cantor set and let F : R be the function defined as follow F (x) = ( 1)n+1 2 n n if x [ ] 2 3 n, 1 3 n 1, n = 1, 2, 3,... We will prove that F is shkintegrable on, but it is not slintegrable on. (HK) [ 2 3 k,1] F dh s = = = k n=1 k n=1 n=1 For the Cauchy extension, we have: (HK) [ 2 3 n, 1 ( 1) n+1 2 n n F dh s = 3 n 1 ] ([ 2 H s 3 n, 1 3 n 1 n=1 ]) = k ( 1) n+1 2 n 1 k n 2 n = ( 1) n+1 n (HK) F dh s = lim (HK) F dh s = [0,1] k [ 2 3 k,1] ( 1) n+1 = log 2 n F is not slintegrable on. In fact, if F were slintegrable on, therefore F would be slintegrable on. Following the previous sequnce of steps, it follows easily that References (HK) F dh s = [0,1] n=1 n=1 1 n = +. [1] M. De Guzman, M. A. Martin and M. Reyes, On the derivation of Fractal functions. Proc. 1st IFIT Conference on Fractals in the Fundamental and Applied Sciences, Lisboa, Portugal (NorthHolland, 1991), [2] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (2003). [3] K. Falconer, The geometry of fractal sets. Cambridge University Press, Cambridge, (1985). 18
19 [4] K Falconer, Techniques in fractal geometry. John Wiley and Sons (1997). [5] H. Jiang and W. Su, Some fundamental results of calculus on fractal sets, Commun. Nonlinear Sci. Numer. Simul. 3, No.1, (1998) [6] B. B. Mandelbrot, The fractal geometry of nature. Freeman and company (1977). [7] P. Mattila, Geometry of Sets and Measures in uclidean Spaces. Fractal and rectificability, Cambridge University Press, Cambridge, (1995). [8] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line  I: formulation. Fractals, Vol. 17, No. 1, (2009) [9] W. Rudin, Real and Complex Analysis, McGrawHill Book Company (1987). 19
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationGambling Systems and MultiplicationInvariant Measures
Gambling Systems and MultiplicationInvariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More informationCHAPTER 5. Product Measures
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationA new continuous dependence result for impulsive retarded functional differential equations
CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationIrreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients
DOI: 10.2478/auom20140007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationSPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM
SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationLarge induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More informationPSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2
PSEUDOARCS, PSEUDOCIRCLES, LAKES OF WADA AND GENERIC MAPS ON S 2 Abstract. We prove a BrucknerGarg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. We
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationDedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes)
Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) by Jim Propp (UMass Lowell) March 14, 2010 1 / 29 Completeness Three common
More informationUniversity of Miskolc
University of Miskolc The Faculty of Mechanical Engineering and Information Science The role of the maximum operator in the theory of measurability and some applications PhD Thesis by Nutefe Kwami Agbeko
More informationScheduling a sequence of tasks with general completion costs
Scheduling a sequence of tasks with general completion costs Francis Sourd CNRSLIP6 4, place Jussieu 75252 Paris Cedex 05, France Francis.Sourd@lip6.fr Abstract Scheduling a sequence of tasks in the acceptation
More informationChapter 7. Continuity
Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationLebesgue Measure on R n
8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationDetermination of the normalization level of database schemas through equivalence classes of attributes
Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationPricing of Limit Orders in the Electronic Security Trading System Xetra
Pricing of Limit Orders in the Electronic Security Trading System Xetra Li Xihao Bielefeld Graduate School of Economics and Management Bielefeld University, P.O. Box 100 131 D33501 Bielefeld, Germany
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationSMALL SKEW FIELDS CÉDRIC MILLIET
SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite
More informationTOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS
TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationMinimal R 1, minimal regular and minimal presober topologies
Revista Notas de Matemática Vol.5(1), No. 275, 2009, pp.7384 http://www.saber.ula.ve/notasdematematica/ Comisión de Publicaciones Departamento de Matemáticas Facultad de Ciencias Universidad de Los Andes
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationLocal periods and binary partial words: An algorithm
Local periods and binary partial words: An algorithm F. BlanchetSadri and Ajay Chriscoe Department of Mathematical Sciences University of North Carolina P.O. Box 26170 Greensboro, NC 27402 6170, USA Email:
More informationSolutions of Equations in One Variable. FixedPoint Iteration II
Solutions of Equations in One Variable FixedPoint Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationA domain of spacetime intervals in general relativity
A domain of spacetime intervals in general relativity Keye Martin Department of Mathematics Tulane University New Orleans, LA 70118 United States of America martin@math.tulane.edu Prakash Panangaden School
More informationON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationNonparametric adaptive age replacement with a onecycle criterion
Nonparametric adaptive age replacement with a onecycle criterion P. CoolenSchrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK email: Pauline.Schrijner@durham.ac.uk
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationDiscernibility Thresholds and Approximate Dependency in Analysis of Decision Tables
Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables YuRu Syau¹, EnBing Lin²*, Lixing Jia³ ¹Department of Information Management, National Formosa University, Yunlin, 63201,
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationExtension of measure
1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.
More informationGENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY
GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY CARL G. JOCKUSCH, JR. AND PAUL E. SCHUPP Abstract. Generic decidability has been extensively studied in group theory, and we now study it in
More informationON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction
ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove
More informationRend. Istit. Mat. Univ. Trieste Suppl. Vol. XXX, 111{121 (1999) Fuzziness in Chang's Fuzzy Topological Spaces Valentn Gregori and Anna Vidal () Summary.  It is known that fuzziness within the concept
More informationFuzzy Probability Distributions in Bayesian Analysis
Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationConvex Programming Tools for Disjunctive Programs
Convex Programming Tools for Disjunctive Programs João Soares, Departamento de Matemática, Universidade de Coimbra, Portugal Abstract A Disjunctive Program (DP) is a mathematical program whose feasible
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationFixed Point Theorems in Topology and Geometry
Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationINTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES
INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More informationwhere D is a bounded subset of R n and f is bounded. We shall try to define the integral of f over D, a number which we shall denote
ntegration on R n 1 Chapter 9 ntegration on R n This chapter represents quite a shift in our thinking. t will appear for a while that we have completely strayed from calculus. n fact, we will not be able
More informationRANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis
RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationThe degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a kuniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More information