Tractability results for weighted Banach spaces of smooth functions

Size: px
Start display at page:

Download "Tractability results for weighted Banach spaces of smooth functions"

Transcription

1 Tractability results for weighte Banach spaces of smooth functions Markus Weimar Mathematisches Institut, Universität Jena Ernst-Abbe-Platz 2, Jena, Germany March 11, 2011 Abstract We stuy the L -approximation problem for weighte Banach spaces of smooth -variate functions, where can be arbitrarily large. We consier the worst case error for algorithms that use finitely many pieces of information from ifferent classes. Aaptive algorithms are also allowe. For a scale of Banach spaces we prove necessary an sufficient conitions for tractability in the case of prouct weights. Furthermore, we show the equivalence of weak tractability with the fact that the problem oes not suffer from the curse of imensionality. 1 Introuction The so-calle curse of imensionality can often be observe for multivariate approximation problems. That is, the minimal number of information operations neee to compute an ε-approximation of a -variate problem epens exponentially on the imension. The phrase curse of imensionality was alreay coine by Bellman in Since the late 1980 s there has been a consierable interest in fining optimal algorithms, also concerning the optimal epenence on an a theory calle information-base complexity (IBC) has been create, see, e.g., [10]. Since there are ifferent ways to measure the lack of exponential behavior, several kins of tractability were introuce. A brief history of the stuies of multivariate problems, as well as general tractability results an many concrete examples can be foun in, e.g., [5, 6, 8]. 1

2 In this paper we especially consier the L -approximation problem efine on some Banach spaces F of real-value -variate functions. In Section 2 we formulate the problem exactly an recall usual error efinitions, as well as notions of tractability. Afterwars, in Section 3, we illustrate the harness of the problem with an example stuie by Novak an Woźniakowski [7] an show how weighte spaces can help to improve this negative result. Thereby, we especially concentrate on so-calle prouct weights. While there exists a wellevelope concept to hanle problems efine on Hilbert spaces, we nee an essentially new approach to conclue results in the general Banach space setting. These new ieas are presente in Section 4. Using this technique we prove a lower error boun for a very small class of functions, i.e. we consier the space P γ of -variate polynomials of egree at most one in each coorinate, equippe with some weighte norm. In Section 5 we recall a known result of Kuo, Wasilkowski an Woźniakowski [3] about upper error bouns on a certain weighte reproucing kernel Hilbert space H γ. Next, in Section 6, we prove the three main theorems of this paper. That is, we show necessary an sufficient conitions for several kins of tractability for a whole scale of weighte Banach function spaces F γ, where P γ F γ Hγ, in terms of the weights γ. In particular, we provie a characterization of weak tractability an the curse of imensionality. It is shown that for these kins of tractability results we can restrict ourselves to linear non-aaptive algorithms. We illustrate our results by applying them to selecte examples an iscuss a typical case of prouct weights. Finally, in Section 7, we a some remarks about possible extensions of the result to other omains. In aition, we briefly consier the L p -approximation problem for 1 p < an correct a small mistake state in [7]. 2 The approximation problem We investigate tractability properties of the approximation problem efine on some Banach spaces F of boune functions f : [0, 1] R. We want to minimize the worst case error e wor (A n, ; F ) = sup f B(F ) f An, (f) L ([0, 1] ) with respect to all algorithms A n, A n that use n pieces of information in imensions from a certain class Λ. Here B(F ) = {f F f F 1} enotes the unit ball of F. Hence, we stuy the n-th minimal error e(n, ; F ) = inf e wor (A n, ; F ) A n, A n of L -approximation on F. An algorithm A n, A n is moele as a mapping φ: R n L ([0, 1] ) an a function N : F R n such that A n, = φ N. In etail, the information 2

3 map N is given by N(f) = (L 1 (f), L 2 (f),..., L n (f)), f F, (1) where L j Λ. Here we istinguish certain classes of information operations Λ. In one case we assume that we can compute arbitrary continuous linear functionals. Then Λ = Λ all coincies with F, the ual space of F. Often only function evaluations are permitte, i.e. L j (f) = f ( t (j)) for a certain fixe t (j) [0, 1]. In this case Λ = Λ st is calle stanar information. If function evaluation is continuous for all t [0, 1] we have Λ st Λ all. If L j epens continuously on f but is not necessarily linear the class is enote by Λ cont. Note that in this case also N is continuous an we obviously have Λ all Λ cont. Furthermore, we istinguish between aaptive an non-aaptive algorithms. The latter case is escribe above in formula (1), where L j oes not epen on the previously compute values L 1 (f),..., L j 1 (f). In contrast, we also iscuss algorithms of the form A n, = φ N with N(f) = (L 1 (f), L 2 (f; y 1 ),..., L n (f; y 1,..., y n 1 )), f F, (2) where y 1 = L 1 (f) an y j = L j (f; y 1,..., y j 1 ) for j = 2, 3,..., n. If N is aaptive we restrict ourselves to the case where L j epens linearly on f, i.e. L j ( ; y 1,..., y j 1 ) Λ all. In all cases of information maps, the mapping φ can be chosen arbitrarily an is not necessarily linear or continuous. The smallest class of algorithms uner consieration is the class of linear, non-aaptive algorithms of the form (A n, f)(x) = n L j (f) g j (x), x [0, 1], with some g j L an L j Λ all or even L j Λ st. We enote the class of all such algorithms by A lin n. On the other han, the most general classes consist of algorithms A n, = φ N, where φ is arbitrary an N either uses non-aaptive continuous or aaptive linear information. We enote the respective classes by A cont n an A aapt n. The minimal number of information operations neee to achieve an error smaller than a given ε > 0, n(ε, ; F ) = min {n N 0 e(n, ; F ) ε}, is calle information complexity. If for a given problem, like the L -approximation (with respect to a given class of algorithms) consiere here, n(ε, ; F ) increases exponentially in the imension we say the 3

4 problem suffers from the curse of imensionality. That is, there exist constants c > 0 an C > 1 such that for at least one ε > 0 we have n(ε, ; F ) c C, for infinitely many N. More generally, if the information complexity epens exponentially on or ε 1 we call the problem intractable. Otherwise we have weak tractability, which can be expresse by ln (n(ε, ; F )) lim ε 1 + ε 1 + We want to stress the point that weak tractability implies the absence of the curse of imensionality, but in general the converse is not true. Since there are many ways to measure the lack of exponential epenence we later istinguish between ifferent types of tractability. The most important type is polynomial tractability. We say that the problem is polynomially tractable if there exist constants c, p, q > 0 such that = 0. n(ε, ; F ) c ε p q for all N, ε > 0. If this inequality hols with q = 0, the problem is calle strongly polynomially tractable. For more specific efinitions an relations between these classes of tractability see, e.g., [6]. 3 The concept of weighte spaces In [7] it is shown that the approximation problem efine on C ([0, 1] ) is intractable. In fact, Novak an Woźniakowski consiere the linear space of all real-value infinitely ifferentiable functions f efine on the unit cube [0, 1] in imensions for which the norm f F = sup D α f α N 0 of f F is finite. Here enotes the usual sup-norm over [0, 1] an D α = α x α xα where α = α j enotes the length of the multi-inex α N 0. The initial error of this problem is given by e(0, ; F ) = 1, the norm of the embeing F L, since A 0, 0 is a vali choice of an algorithm which oes not use any information of f. This means that the problem is well-scale. In etail, Theorem 1 in [7] yiels that for L -approximation efine on F we have e(n, ; F ) = 1 for all n = 0, 1,..., 2 /2 1., 4

5 Therefore, for all N an ε (0, 1), n(ε, ; F ) 2 /2. Hence, the problem suffers from the curse of imensionality; in particular it is intractable. One possibility to avoi this exponential epenence on, i.e. to break the curse, is to shrink the function space F. A closer look at the norm yiels that for f B(F ) we have D α f 1 for all α N 0. (3) Hence, every erivative is equally important. In orer to shrink the space, for each α N 0 we replace the right-han sie of inequality (3) by a weight 0 γ α 1. For α with α = 1 this means that we control the importance of every single variable. So, the norm in the weighte space is now given by f F γ = sup 1 D α f α N 0 γ, α where we eman D α f to be equal to zero if γ α = 0. The iea to introuce weights irectly into the norm of the function space appeare for the first time in a paper of Sloan an Woźniakowski in 1998, see [9]. They stuie the integration problem efine over some Sobolev Hilbert space, equippe with so-calle prouct weights, to explain the overwhelming success of QMC integration rules. Thenceforth, weighte problems attracte a lot of attention. For example it turne out that tractability of approximation of linear operators between Hilbert spaces can be fully characterize in terms of the weights an singular values of the linear operators if we use information operations from the class Λ all. Let us have a closer look at prouct weights. Assume that for every N there exists an orere an boune sequence 1 γ,1 γ,2... γ, 0. Then for N, the prouct weight sequence γ = (γ α ) α N 0 is given by γ α = (γ,j ) α j, α N 0. (4) Note that the epenence of x j on f is now controlle by the so-calle generator weight γ,j. Since γ,j = 0 for some j {1,..., } implies that f oes not epen on x j,..., x we 5

6 assume that γ, > 0 in the rest of the paper. Moreover, the orering of γ,j is without loss of generality. Later on we will see that tractability of our problem will only epen on summability properties of the generator weights. Among other things, it turns out that for the L -approximation problem efine on the Banach space with the norm given above an generator weights γ,j γ j = Θ ( j β) we have intractability for β = 0, weak tractability but no polynomial tractability for 0 < β < 1, strong polynomial tractability if 1 < β. Moreover, we prove that for β = 1 the problem is not strongly polynomially tractable. 4 Lower bouns First, we want to escribe the main ieas use in the Hilbert space setting. Hence, for a moment, consier the problem of L 2 -approximation with respect to linear algorithms efine on a reproucing kernel Hilbert space H(K ) of functions f : [0, 1] R. Let W : H(K ) H(K ), W (g) = g(x) K (, x) x. [0,1] We assume that W is compact. Then the worst case error is fully characterize by the spectrum of W that is also a self-ajoint, an non-negative efinite operator. Let {(λ,j, η,j ) j N} enote a complete orthonormal system of eigenpairs of W, inexe accoring to the nonincreasing orer of the eigenvalues, i.e. W (η,j ) = λ,j η,j an η,i, η,j H(K ) = δ ij with λ,j λ,j+1 0. For λ,n > 0, it is well known that the algorithm A n,(f) = n f, η,j L2 η,j, where η,j = η,j λ,j is optimal. Then the n-th minimal error is given by e(n, ; H(K )) = e wor (A n,; H(K )) = λ,n+1. 6

7 For more etails see, e.g., [4] an [6], as well as the references in there. For a comprehensive introuction to reproucing kernel Hilbert spaces see, for instance, Chapter 1 in the book of Wahba [11]. In the general Banach space setting this approach obviously oesn t work. Our technique is base on the ieas of Werschulz an Woźniakowski [12], as well as Novak an Woźniakowski [7]. Among other things it uses a result from Banach space theory an nonlinear functional analysis, namely, the theorem of Borsuk-Ulam. The proof of the following proposition can be foun in Chapter 1.4.2, [1]. Proposition 1 (Borsuk-Ulam). Let V be a linear norme space over R with im V = m an, moreover, let N : V R n be a continuous mapping for n < m. Then there exists an element f V with f V = 1 such that N(f ) = N( f ). The main tool to conclue lower bouns in the Banach space setting now reas as follows. Lemma 1. Assume that F an G are linear norme spaces such that F G. Furthermore, suppose that V F is a linear subspace of imension m an there exists a constant a > 0 such that Then for every n < m an every A n A cont n f F 1 f G for all f V. (5) a e wor (A n ; F ) = A aapt n sup f A n (f) G a. f B(F ) Proof. For A n A cont n the assertion is a simple conclusion of Proposition 1 an can be foun in [7]. On the other han, if A n A aapt n the proof can be obtaine by arguments from linear algebra, which are inicate in the proof of Theorem 3.1 in [12]. In any case we exclusively use norm properties from the space G, no aitional structure of G is use. Therefore, this tool is available for any kin of approximation problem, not only for L -approximation. In the following we use Lemma 1 to conclue a lower boun for the approximation error for the space } P γ {p = span i : [0, 1] R, p i (x) = (x j ) i j i = (i 1,..., i ) {0, 1} of all real-value -variate polynomials of egree at most one in each coorinate irection, efine on the unit cube [0, 1]. We equip this linear space with the weighte norm f P γ = max 1 D α f α {0,1} γ, f P γ, α where γ is the prouct weight sequence escribe as in Section 3. 7

8 Theorem 1. Let e(n, ; P γ ) be the n-th minimal error of L -approximation on P γ respect to the class A cont n A aapt n of all algorithms escribe in Section 2. Then with e(n, ; P γ ) 1 for all n < 2s, an some integer s [0, ] with ( ) s > 1 3 γ,j 2. (6) Proof. The proof of the lower error boun consists of several steps. At first, we construct a partition of the set {1,..., } into s+1 parts which we will nee later an with s satisfying (6). In a secon step, we efine a special linear subspace V P γ with im V = 2s. Step 3 then shows that V satisfies the assumptions of Lemma 1. The proof is complete in Step 4. Step 1. For k {0,..., }, we efine inuctively m 0 = 0 an m k = inf { t N m k 1 < t, with 2 t j=m k 1 +1 γ,j } with the usual convention inf =. Note that the infimum coincies with the minimum in the finite case, since then m k N. Moreover, we set s = max {k {0,..., } m k < }. We enote I k = {m k 1 + 1, m k 1 + 2,..., m k } for k = 1,..., s. Thus, this gives a uniquely efine isjoint partition of the set ( s ) {1,..., } = I k {m s + 1,..., }, k=1 an m k enotes the last element of the block I k. For all k = 1,..., s, we conclue 2 j I k γ,j < 2 + γ,mk < 3. Finally, summation of these inequalities gives γ,j < s k=1 j I k γ,j + 2 < 3s + 2, 8

9 an (6) follows immeiately. If s = 0 we can stop at this point since the initial error is 1 as the norm of the embeing P γ L an the remaining assertion is trivial. Hence, from now on we can assume that s > 0 an m s 1. Step 2. To apply Lemma 1 we have to construct a linear subspace V of F = P γ such that the conition (5) hols for G = L ([0, 1] ) an a = 1. First, we restrict ourselves to the set F = {f F f epens only on x 1,..., x ms }. By a simple isometric isomorphism we can interpret F as the space Pm γ s. We are reay to construct a suitable space V using the partition from Step 1. We efine V as the span of all functions g i : X = [0, 1] ms R, i = (i 1,..., i s ) {0, 1} s, of the form ( ) ik s g i (x) = γ,j x j, x X. j I k k=1 Obviously, V is a linear subspace of Pm γ s an with the interpretation above also a linear subspace of F. Moreover, it is easy to see that we have by construction g Pm γ s = g F an g L (X) = g L ([0, 1] ) for g V. Finally, we note that im V = #{0, 1} s = 2 s. It remains to show that this subspace is the right choice to prove the claim using Lemma 1. Step 3. The proof of the neee conition (5), g P γ ms g L (X) for all g V, is a little bit technical. Due to the special structure of the functions g V, the left han sie reuces to max {γα 1 D α g L (X) α M}, where the maximum is taken over the set { } M = α {0, 1} ms j I k α j 1 for all k = 1,..., s. This is simply because for α / M we have D α g 0 an the inequality is trivial. To simplify the notation let us efine T : {0, 1} ms N s 0, α T (α) = σ = (σ 1,..., σ s ), where σ k = j I k α j for k = 1,..., s. 9

10 Note that T (M) = {0, 1} s. Moreover, for every g = i {0,1} s a i g i ( ) V efine the function h g : Z = s k=1 [0, j Ik γ,j ] R, h g (z) = Hence, h g (z) = g(x) uner the transformation x z such that i {0,1} s a i s k=1 z i k k. z k = j I k γ,j x j for every k = 1,..., s an every x X. The span W of all functions h: Z R with this structure also is a linear space. Furthermore, easy calculus yiels ( ms ) (Dx α g) (x) = (γ,j ) α j (D ) T (α) z h g (z) for all g V, α M an x X. (7) Here the x an z in Dx α an Dz T (α) inicate ifferentiation with respect to x an z, respectively. Since the mapping x z is surjective we obtain D α g L (X) = γ α D T (α) h g L (Z) by the form of γ given by (4). Hence, max α M 1 γ α D α g L (X) = max σ {0,1} Dσ h g L (Z). s Note that (7) with α = 0 yiels g L (X) = h g L (Z). Therefore, the claim reuces to max σ {0,1} Dσ h g L (Z) h g L (Z) for every g V. s We show this estimate for every h W, i.e., D σ h L (Z) h L (Z) for all σ {0, 1} s. (8) We start with the special case of one erivative, i.e. σ = e k for a certain k {1,..., s}. Since h is affine in each coorinate we can represent it as h(z) = a(z k ) z k + b(z k ) with functions a an b which only epen on z k = (z 1,..., z k 1, z k+1,..., z s ). Thus, we have D e k h(z) = a(z k ) an nee to show that a(z k ) { b(z max k ), a(zk ) 10 } γ,j + b(z k ). (9) j I k

11 This is obviously true for every z Z with a(z k ) = 0. For a(z k ) 0 we can ivie by a(z k ) to get { } 1 max t, γ,j t j I k if we set t = b(z k )/a(z k ). The last maximum is minimal if both of its entries coincie. This is for t = 1 2 j I k γ,j. Hence, we nee to eman 2 j I k γ,j to conclue (9) for all amissible z Z. But this is true for every k {1,..., s} by efinition of the sets I k in Step 1. This proves (8) for the special case σ = e k for all k {1,..., s}. The inequality (8) also hols for every σ {0, 1} s by an easy inuctive argument on the carinality of σ. Inee, if σ 2 then σ = σ + e k with σ = σ 1. We now nee to estimate D σ +e kh L (Z). Since D e kh(z) = a(z k ) has the same structure as the function h itself, we have D σ +e kh L (Z) = D σ a(z k ) L (Z) an the proof is complete by the inuctive step. Step 4. For every g V we have g P γ = g P γ ms = max α {0,1} ms T (α) {0,1} s 1 γ α D α g L (X) = h g L (Z) = g L (X) = g L ([0, 1] ), max σ {0,1} Dσ h g L (Z) s where V is a linear subspace of F = P γ with im V = 2s. Therefore, Lemma 1 with a = 1 yiels that the worst case error of any algorithm A n, we consier, with n < im V pieces of information, is boune from below by one. That is, e wor (A n, ; P γ ) 1. We complete the proof by taking the infimum with respect to A n, A cont n A aapt n. 5 Upper bouns The approximation problem has been stuie in many ifferent settings. We restrict ourselves to the case of L -approximation efine on a special weighte anchore Sobolev Hilbert space H γ = H(Kγ ). For = 1 an γ > 0, this is the space of all absolutely continuous functions f : [0, 1] R whose first erivatives belong to L 2 ([0, 1]). The inner prouct in the space H γ 1 is efine as 1 f, g H γ = f(0)g(0) + γ 1 f (x)g (x) x, f, g H γ 1 1, 0 11

12 where the erivatives have to be unerstoo in the weak sense. For γ = 0 the space consists of only constant functions. It turns out that H γ 1 is a reproucing kernel Hilbert space H(K γ 1 ) whose kernel is K γ 1 (x, y) = 1 + γ min {x, y} for x, y [0, 1]. For > 1, the space H γ = H(Kγ ) is efine as the -fol tensor prouct of H(Kγ,j 1 ), where we once again assume prouct weights, see (4), with 1 γ,1 γ,2... γ, 0. Due to the prouct structure of γ α, the corresponing reproucing kernel of H γ Wiener sheet kernel, is a weighte K γ (x, y) = (1 + γ,j min {x j, y j }), x, y [0, 1]. The associate inner prouct is given by f, g H γ = 1 α f (x α, 0) α g (x α, 0) x α, f, g H γ γ α α {0,1} [0,1] x α α x. α Here the term (x α, a) means the -imensional vector with (x α, a) j = x j for all coorinates j with α j = 1 an (x α, a) j = a j otherwise. For α = 0 we replace the integral by f(a)g(a). Therefore, the point a = 0 [0, 1] is sometimes calle an anchor of the space. A closer look at the respective norm justifies to refer to H(K γ ) as a Sobolev space of ominating mixe smoothness. For γ, > 0, the space H(K γ ) algebraically coincies with the space { } f : [0, 1] R D α f L 2 ([0, 1] ) for all α = (α 1,..., α ) with max α j 1,,..., where D α f once again enotes the weak erivative in the Sobolev sense. Equippe with the usual norm, this space is often enote by W (1,...,1) 2,mix ([0, 1] ), or S2W 1 ([0, 1] ), respectively. If γ,j = 0 for some j {1,..., } we obtain a proper subspace of functions that are constant with respect to x j,..., x. Therefore, we always assume γ, > 0. Kuo, Wasilkowski an Woźniakowski [3, 8. Example] showe 12

13 Proposition 2. There exists a linear algorithm A n, for L -approximation on H γ such that it uses n non-aaptively chosen linear functionals an for every τ (1/2, 1) there are constants a τ, b τ > 0 inepenent of γ an such that e wor (A n,; H γ ) b τ n (1 τ)/(2τ) ( ) 1 + aτ γ,j τ 1/(2τ). Furthermore, A n, is close to be optimal in the class Alin n. 6 Conclusions an applications We now combine lower an upper bouns presente before an prove general results for L -approximation on weighte Banach function spaces. More precisely, consier a sequence of Banach spaces F γ of functions f : [0, 1] R which fulfills the following simple assumptions: (A1) P γ F γ with an embeing factor C 1, 1 for all, (A2) F γ Hγ with an embeing factor C 2, for all an ( ) C 2, a exp b (γ,j ) t for some constants a, b 0 an a parameter t (0, 1], inepenent of an γ. By A B with an embeing factor C, we mean that the norme linear space A is continuously embee in the norme linear space B an f B C f A for all f A. That is, we can take C = i L(A, B) as the (operator-) norm of the ientity i: A B. Moreover, γ is once again a prouct weight sequence given by formula (4). The spaces P γ an H γ are efine in Section 4 an Section 5, respectively. To simplify the notation for necessary an sufficient conitions of tractability, we use the commonly known efinitions of the so-calle sum exponents for the weight sequence γ, p(γ) = inf { κ 0 P κ (γ) = lim sup } (γ,j ) κ < 13

14 an q(γ) = inf { κ 0 Q κ (γ) = lim sup (γ,j) κ ln( + 1) } <, with the convention that inf =. Theorem 2 (Necessary conitions). Assume that (A1) hols. Consier L -approximation over F γ with respect to the class of algorithms Acont n A aapt n. Then ( n(ε, ; F γ ) > exp 1 3 ln 2 ( γ,j 2 )) for all N an ε (0, 1). (10) Therefore, if the problem is polynomially tractable then q(γ) 1, strongly polynomially tractable then p(γ) 1. Proof. Due to (A1), every algorithm A n, A cont n A aapt n for L -approximation efine on F γ also applies to the embee space Pγ. Furthermore, C 1, 1 implies that the unit ball B(P γ ) is containe in the unit ball B(F γ ). Therefore, From Theorem 1 we have e wor (A n, ; F γ ) ewor (A P n, γ; P γ ) e(n, ; Pγ ). e(n, ; P γ ) 1 for n < 2s, where s = s(γ, ) [0, ] satisfies (6). Hence, for N an ε (0, 1) we conclue n(ε, ; F γ ) 2s > 1 21/3 γ,j, 41/3 as claime in (10). Suppose now that the problem is polynomially tractable. Then there are non-negative constants C, p an q such that n(ε, ; F γ ) Cε p q for all N, ε > 0. 14

15 Take now an arbitrarily fixe ε in (0, 1). Then (10) implies that there is a positive C such that 2 1/3 γ,j C q for all N. This is equivalent to the bouneness of γ,j/ ln( + 1), an therefore q(γ) 1, as claime. Suppose that the problem is strongly polynomially tractable. Then q = 0 in the boun above, an γ,j is uniformly boune in. Hence, p(γ) 1, as claime. Of course, the conitions q(γ) 1 an p(γ) 1 are also necessary for polynomial an strong polynomial tractability with respect to smaller classes of algorithms. We next assume (A2) an show that slightly stronger conitions on the weights γ than in Theorem 2 are sufficient for polynomial an strong polynomial tractability. Theorem 3 (Sufficient conitions). Assume that (A2) hols with a parameter t (0, 1]. Consier L -approximation over F γ with respect to the class of linear algorithms Alin n. Then q(γ) < t implies that the problem is polynomially tractable, p(γ) < t implies that the problem is strongly polynomially tractable. Proof. Due to (A2), the restriction of the algorithm A n, in Proposition 2 from Hγ to F γ is a vali linear algorithm for L -approximation over F γ. Furthermore, ue to linearity of A n, for all f Hγ, we have f A n, f L ([0, 1] ) e wor (A n,; H γ ) f Hγ ewor (A n,; H γ ) C 2, f F γ. Therefore, we can estimate the n-th minimal error by e(n, ; F γ ) ewor (A n, F γ; F γ ) C 2, e wor (A n,; H γ ) ( ) a exp b (γ,j ) t b τ n (1 τ)/(2τ) ( ) 1 + aτ γ,j τ 1/(2τ), where τ is an arbitrary number from (1/2, 1). Using 1 + x e x for x 0, we have ( ) e(n, ; F γ ) a b τ n (1 τ)/(2τ) exp b (γ,j ) t + a τ (γ,j ) τ. 2τ 15

16 Choosing n such that the right-han sie is at most ε, we obtain an estimate for the information complexity with respect to the class of linear algorithms, ( ) n(ε, ; F γ ) c 1 ε 2τ/(1 τ) exp (γ,j ) t + c 3 (γ,j ) τ, (11) where the positive constants c 1, c 2 an c 3 only epen on τ, a an b. Suppose that q(γ) < t. Then Q κ (γ) is finite for every κ > q(γ). Taking κ = t we obtain c 2 (γ,j) t ln( + 1) ln( + 1) (Q t (γ) + δ) ln( + 1) = ln( + 1) Qt(γ)+δ for every δ > 0 whenever is larger than a certain δ. ) exp (c 2 (γ,j) t in (11) is polynomially epenent on. This means that the factor On the other han, we can choose τ (max {q(γ), 1/2}, 1) such that Q τ (γ) is finite an ) the factor exp (c 3 (γ,j) τ in (11) is also polynomially epenent on. So, for this value of τ we can rewrite (11) as n(ε, ; F γ ) = O ( ε 2τ/(1 τ) ( + 1) c 4 ), with c 4 inepenent of an ε. This means that the problem is polynomially tractable, as claime. Suppose finally that p(γ) < t. Then the sums (γ,j) t an (γ,j) τ for τ (max {p(γ), 1/2}, 1) are both uniformly boune in. Therefore (11) yiels strong polynomial tractability, an completes the proof. The conitions in Theorem 3 are obviously also sufficient if we consier larger classes of algorithms. Moreover, the proof of Theorem 3 also provies explicit upper bouns for the exponents of tractability. We now iscuss the role of assumptions (A1) an (A2). They are quite ifferent. The assumption (A1) is use to fin a lower boun on the information complexity for the space F γ as long the space P γ is continuously embee in F γ with an embeing factor at most one. Such an embeing can be shown for several ifferent classes of functions. The assumption (A2) is use to fin an upper boun on the information complexity for the space F γ as long as it is continuously embee in the space Hγ with an embeing factor epening exponentially on the sum of some power of the prouct weights. This consierably restricts the choice of F γ. We nee this assumption in orer to use the linear algorithm A n, efine on the space Hγ ue to Kuo et al. [3] an the error boun they 16

17 prove. Obviously, we can replace the space H γ in (A2) by some other space which contains at least P γ an for which we know a linear algorithm using n linear functionals whose worst case error is polynomial in n 1 with an explicit epenence on the prouct weights. We now show that the assumptions (A1) an (A2) allow us to characterize weak tractability an the curse of imensionality. Theorem 4 (Weak tractability an the curse of imensionality). Suppose that (A1) an (A2) with a parameter t (0, 1] hol. Then for L -approximation efine on the space F γ the following statements are equivalent: (i) The problem is weakly tractable with respect to the class A lin n. (ii) The problem is weakly tractable with respect to the class A cont n (iii) There is no curse of imensionality for the class A lin n. (iv) There is no curse of imensionality for the class A cont n 1 (v) For all κ > 0 we have lim (γ,j) κ = 0. 1 (vi) There exists κ (0, t) such that lim (γ,j) κ = 0. Proof. We start by showing that (vi) implies (i), i.e., ln (n(ε, ; F γ lim )) ε 1 + ε 1 + = 0, A aapt n. A aapt n. where the information complexity is taken with respect to linear algorithms A lin n. By the arguments use in the proof of Theorem 3 we obtain estimate (11) for all ε > 0, as well as for every N an all τ (1/2, 1), ue to assumption (A2). Clearly, for κ (0, t) as in the hypothesis an t (0, 1] as in the embeing conition, we can fin τ (1/2, 1) such that κ < min {t, τ}. So, since γ,j 1, we can estimate both sums on the right-han sie of (11) from above by (γ,j) min{t,τ} (γ,j) κ. Thus, ln (n(ε, ; F γ )) ε 1 + ln(c 1) ε τ 1 τ ln (ε 1 ) ε max {c 2, c 3 } (γ,j) κ ε 1 + tens to zero when ε 1 + approaches infinity, as claime. 17

18 Clearly, (i) (ii) (iii) (iv) an (v) (vi). Hence, we only nee to show that (iv) (v). From (A1) we have estimate (10). Then no curse of imensionality implies 1 lim γ,j = 0. Now, Jensen s inequality yiels 1 γ,j ( 1 ) 1/κ (γ,j ) κ for 0 < κ 1, since f(y) = y κ is a concave function for y > 0. Thus, 1 lim (γ,j ) κ = 0 for all 0 < κ 1. Finally, for every κ 1 we can estimate γ,j (γ,j ) κ since γ,j 1 for j = 1,...,. Therefore, lim 1 (γ,j) κ = 0 also hols for κ > 1, an the proof is complete. In the last part of this section, we give some examples to illustrate the results. In the following we only have to prove the embeings, i.e. assumptions (A1) an (A2) from the beginning of this section. Example 1 (Limiting cases P γ an Hγ ). To begin with, we check the case F γ = Pγ. Then (A1) obviously hols with C 1, = 1. To prove (A2), note that the algebraical inclusion F γ Hγ is trivial by arguments given in Section 5. For f F γ = Pγ we calculate f H γ 2 1 D α f 2 γ x α f F γ 2 γ α. α α {0,1} [0,1] α α {0,1} Hence, the norm of the embeing F γ Hγ α {0,1} γ α 1/2 is boune by ( 1/2 ( 1 = (1 + γ,j )) exp 2 ) γ,j. So, with a = 1, b = 1/2 an t = 1 also assumption (A2) is fulfille an we can apply the state theorems for the space F γ = Pγ. 18

19 We now turn to the case F γ = Hγ. Unfortunately, the estimate above inicates that (A1) may not hol for F γ = Hγ with C 1, 1. Nevertheless, in this case assumption (A2) is true with C 2, = 1, i.e., a = 1, b = 0 an t = 1. Therefore, we can apply Theorem 3 for this space. Then the problem is strongly polynomially tractable if p(γ) < 1. Moreover, we have polynomial tractability if q(γ) < 1. It is known that these conitions are also necessary, see, e.g., Theorem 12 in [3]. Example 2 (C (1,...,1) ). Consier the space F γ = {f : [0, 1] R f C (1,...,1), where f F γ = max 1 D α f α {0,1} γ < }. α Since P γ is a linear subset of F γ an Pγ is simply the restriction of F γ we have P γ F γ with an embeing factor C 1, = 1 an (A1) hols. For the factor C 2, of the embeing F γ Hγ, the same estimates hol exactly as in the previous example an, moreover, the set inclusion is obvious. Therefore, also assumption (A2) is fulfille an we can apply the theorems of this section to the space F γ. Finally, the last example shows that even very high smoothness oes not improve the conitions for tractability. Example 3 (C ). Assume F γ = {f : [0, 1] R f C, where f F γ = sup 1 D α f α N γ < }. 0 α Obviously, P γ C, an functions from P γ are at most linear in each coorinate. Hence, D α f 0 for all α N 0 \ {0, 1}. Therefore, once again we have f P γ = max 1 D α f α {0,1} γ = f F γ for all f Pγ. α This yiels P γ F γ with an embeing factor C 1, = 1. In aition, also (A2) can be conclue as in the examples above. So, even infinite smoothness leas to the the same conitions for tractability an the curse of imensionality as before. Note that in the last example we o not nee to claim a prouct structure for the weights accoring to multi-inices α N 0 \ {0, 1}. Moreover, this example is a generalization of the space consiere in [7]. For γ α 1 we reprouce the intractability result state there. In conclusion, we iscuss the tractability behavior of L -approximation efine on one of the spaces F γ above using prouct weights which are inepenent of the imension, i.e., γ,j γ j = Θ(j β ) for some β 0. 19

20 This is a typical example in the theory of prouct weights, an p(γ) is finite if an only if β > 0. If so then p(γ) = 1/β. See, e.g., Section in [6]. If β = 0 then the problem is intractable ue to Theorem 4, assertion (v), since 1 γ,j oes not ten to zero. For β (0, 1), easy calculus yiels q(γ) > 1. So, using Theorem 2 we conclue polynomial intractability in this case. On the other han, for all δ an κ with 0 < δ < κ 1, we have j κ = j κ κ (1+δ) κ δ j (1+δ) κ δ 0 with an if κ > 1 then the fraction obviously tens to zero, too. Hence, conition (vi) of Theorem 4 hols an the problem is weakly tractable if β > 0. For β = 1, we use inequality (10) from Theorem 2 an estimate γ,j = j 1 c ln( + 1) for some positive c. Therefore, n(ε, ; F γ ) 1 2 2/3 ( + 1)c/3 ln(2) for all N, ε (0, 1). Hence, strong polynomial tractability oes not hol. Moreover, it is easy to show that for β = 1 the sufficient conition q(γ) < 1 for polynomial tractability is not fulfille. So, we o not know if polynomial tractability hols. If β > 1 we easily see that p(γ) = 1 < 1 = t. Hence, Theorem 3 provies strong β polynomial tractability in this case. 7 Final remarks Note that the main result of this paper, the lower boun given in Theorem 1, can be easily transfere from [0, 1] to more general omains Ω. Inee, the case Ω = [c 1, c 2 ], where c 1 < c 2, can be immeiately obtaine using our techniques. It turns out that in this case we have to moify estimate (6) by a constant which epens only on the length of the interval [c 1, c 2 ]. Thus, the general tractability behavior oes not change. Another extension of the results is possible if we consier the L p -norms (1 p < ) instea of the L -norm. We want to briefly iscuss these norms for the unweighte case. Then 20

21 the moifications for the weighte case are obvious. Following Novak an Woźniakowski [7] let { } F,p = f : [c 1, c 2 ] R f C with f F,p = sup D α f L p < α N 0 for 1 p < an N. Let l = c 2 c 1 > 0. We want to approximate f F,p in the norm of L p, i.e., we consier the n-th minimal error e p (n, ; F,p ) = sup f A n, (f) L p ([c 1, c 2 ] ). inf e wor p (A n, ; F,p ) = inf A n, A n A n, A n f B(F,p ) Without loss of generality we restrict ourselves to the case [c 1, c 2 ] = [0, l]. In orer to conclue a lower boun analogue to Theorem 1, i.e., e p (n, ; F,p ) 1 for n < 2 s, we once again use Lemma 1 with F = F,p an G = L p ([0, l] ). The authors of [7] suggest to use the subspace V (k) F,p efine as V (k) = span g i : [c 1, c 2 ] R, x g i (x) = s jk m=(j 1)k+1 x m i j i {0, 1} s, where s = /k an k N such that kl 2(p + 1) 1/p. Hence, if l < 2(p + 1) 1/p we have to use blocks of variables with size k > 1 in orer to guarantee (5), i.e., to fulfill the conition g F,p g L p for all g V (k). (12) Therefore, Novak an Woźniakowski efine k = 2(p + 1) 1/p /l, but this is too small as the following example shows. Take l = 1, i.e. [c 1, c 2 ] = [0, 1], an p = 1. Then k = 4 shoul be a proper choice, but for g (x) = (x 1 + x 2 + x 3 + x 4 ) 2 we obtain g L 1 = 7/15 by using Maple, while g / x 1 L 1 = 1. This contraicts (12). Proposition 3. Let 1 p < an k N with k 8(p + 1) 2/p /l 2. (13) Then conition (12) hols for V (k) e p (n, ; F,p ) 1 for all n < 2 /k. F,p. Hence, the problem remains intractable since 21

22 Proof. Step 1. Due to the structure of functions g from V (k), it suffices to show D α g L p ([0, l] ks ) g Lp ([0, l] ks ) for all g V (k) an for every α M (k), where the set of multi-inices M (k) is efine by M (k) = α {0, 1}ks α m 1, for all j = 1,..., s m I j an I j = {(j 1)k + 1,..., jk}. Similar to the proof of Theorem 1, we only consier the case α = e t {0, 1} ks with t I j. The rest then follows by inuction. We can represent g V (k), as well as D et g, by functions a, b: [0, l] k(s 1) R such that g(x) = a( x) k y m + b( x) an D et g(x) = a( x), m=1 where x = (x I1,..., x Ij 1, y, x Ij+1,..., x Is ) [0, l] ks an x = (x I1,..., x Ij 1, x Ij+1,..., x Is ) [0, l] k(s 1), as well as y = (y 1,..., y k ) [0, l] k. Here x Ij enotes the k-imensional vector of components x m with coorinates m I j. Therefore, we can rewrite the inequality D et g L p ([0, l] ks ) g L p ([0, l] ks ) as p a( x) p k y x [0,l] k(s 1) [0,l] k [0,l] k(s 1) [0,l] a( x) y m + b( x) y x k such that it is enough to prove a point wise estimate of the inner integrals for fixe x [0, l] k(s 1) with a = a( x) 0. Easy calculus yiels p k a y m + b y = l p+k k p a z m + b z [0,l] k m=1 [ 1/2,1/2] k for some constant b R. The right-han sie is minimize for b = 0. So, we can estimate this integral from below by p k [0,l] a y m + b y l p+k a p k p z k m=1 [ 1/2,1/2] m z k m=1 = l p a p k p y z m z. [0,l] k [ 1/2,1/2] k 22 m=1 m=1 m=1

23 Hence, it remains to show that the choice of k implies that k p z m z l p. m=1 [ 1/2,1/2] k m=1 Step 2. In this last part, we will show by arguments from Banach space geometry that k p ( ) p/2 ( ) p/2 k 1 k 1/2 z m z [ 1/2,1/2] 2 2 k p (1 + p) = x p x. (14) 2 1/2 Obviously, we only nee to prove the inequality for k 2 since the equation on the right, as well as the case k = 1, are trivial. To abbreviate the notation, we efine f : R k R, k z = (z 1,..., z k ) m=1 z m for fixe k 2. For given vectors z, ξ R k, let z, ξ enote the scalar prouct k m=1 z mξ m. In the special case ξ = 1/ k (1,..., 1) S k 1 it is z, ξ = t for a given t R, if an only if, f(z) = t k. Furthermore, note that every ξ in the k-imensional unit sphere S k 1 uniquely efines a hyperplane ξ = {z R k z, ξ = 0} perpenicular to ξ which contains zero. Therefore, for every t [0, ), the set ξ + tξ = {z R k z, ξ = t} escribes a parallel shifte hyperplane with istance t to the origin. Using Fubini s theorem, this leas to the following representation f(z) p z = 2 f(z) p z = 2 k p/2 t p 1 z t. [ 1/2,1/2] k 0 [ 1/2,1/2] k z,ξ 0 [ 1/2,1/2] k z,ξ =t Now we see that the inner integral escribes the (k 1)-imensional volume v(t) = λ k 1 ( [ 1/2, 1/2] k (ξ + tξ) ) of the parallel section of the unit cube with the hyperplane efine above. Because of Ball s famous theorem we know v(0) 2, inepenent of k, see, e.g., [2, Chapter 7]. Moreover, ξ provies a central hyperplane section of the unit cube such that we have 0 v(t) t = 1 2 λ k([ 1/2, 1/2] k ) =

24 an, by Brunn s theorem (see Theorem 2.3 in [2]), v 0 is non-increasing on [0, ). Thus, v is relate to the istribution function of a certain non-negative real-value ranom variable X, up to a normalizing factor, i.e. v(t) = v(0) P(X t). Using Höler s inequality we obtain E(X 1+p ) (EX) 1+p an, respectively, 0 t p v(t) t ( 1 ) 1+p v(0) p (1 + p) v(t) t 0 by integration by parts. Altogether we conclue inequality (14) an, with k boune from below by (13), even [ 1/2,1/2] k f(z) p z l p. Therefore, the proof is complete. Using other methos, we can improve inequality (14) in Step 2 of the last proof. In etail, we can represent the integral on the left as an expectation E( f(y ) p ) with a suitable ranom vector Y. For p = 2N with N N this can be calculate exactly. Finally, it turns out that it is enough to take { 12/l 2, if 2 p < 4 k 8/l 2, if 4 p in orer to conclue the claime intractability result for the L p -approximation problem. Nevertheless, we want to stress the point that also with this improvements the lower bouns on k are not sharp since we know from [7] that in the limit case p = we can take k = 2/l. On the other han, upper bouns for the k-imensional integral, conclue using Hoeffing s inequality, yiel that k p/2 is the right orer. Acknowlegments The author thanks A. Hinrichs, E. Novak an H. Woźniakowski for their useful hints an valuable comments on this paper. References [1] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin,

25 [2] A. Kolobsky, Fourier Analysis in Convex Geometry, Amer. Math. Soc., Provience, RI, [3] F. Y. Kuo, G. W. Wasilkowski, an H. Woźniakowski, Multivariate L approximation in the worst case setting over reproucing kernel Hilbert spaces, J. Approx. Theory, 152, , [4] F. Y. Kuo, G. W. Wasilkowski, an H. Woźniakowski, On the power of stanar information for multivariate approximation in the worst case setting, J. Approx. Theory, 158, , [5] E. Novak, I. H. Sloan, J. F. Traub, an H. Woźniakowski, Essays on the Complexity of Continuous Problems, Europ. Math. Soc., Zürich, [6] E. Novak, an H. Woźniakowski, Tractability of Multivariate Problems. Vol. I: Linear Information, Europ. Math. Soc., Zürich, [7] E. Novak, an H. Woźniakowski, Approximation of infinitely ifferentiable multivariate functions is intractable, J. Complexity, 25, , [8] E. Novak, an H. Woźniakowski, Tractability of Multivariate Problems. Vol. II: Stanar Information for Functionals, Europ. Math. Soc., Zürich, [9] I. H. Sloan, an H. Woźniakowski, When are quasi-monte Carlo algorithms efficient for high-imensional integrals?, J. Complexity, 14, 1 33, [10] J. F. Traub, G. W. Wasilkowski, an H. Woźniakowski, Information-base Complexity, Acaemic Press Inc., Boston, [11] G. Wahba, Spline Moels for Observational Data, Soc. Inust. Appl. Math. (SIAM), Philaelphia, [12] A. G. Werschulz, an H. Woźniakowski, Tractability of multivariate approximation over a weighte unanchore Sobolev space, Constr. Approx., 30, ,

Generalized Tractability for Multivariate Problems

Generalized Tractability for Multivariate Problems Generalize Tractability for Multivariate Problems Part II: Linear Tensor Prouct Problems, Linear Information, an Unrestricte Tractability Michael Gnewuch Department of Computer Science, University of Kiel,

More information

ON THE OPTIMAL CONVERGENCE RATE OF UNIVERSAL AND NON-UNIVERSAL ALGORITHMS FOR MULTIVARIATE INTEGRATION AND APPROXIMATION

ON THE OPTIMAL CONVERGENCE RATE OF UNIVERSAL AND NON-UNIVERSAL ALGORITHMS FOR MULTIVARIATE INTEGRATION AND APPROXIMATION ON THE OPTIMAL CONVERGENCE RATE OF UNIVERSAL AN NON-UNIVERSAL ALGORITHMS FOR MULTIVARIATE INTEGRATION AN APPROXIMATION MICHAEL GRIEBEL AN HENRYK WOŹNIAKOWSKI Abstract. We stuy the optimal rate of convergence

More information

On lower bounds for integration of multivariate permutation-invariant functions

On lower bounds for integration of multivariate permutation-invariant functions arxiv:1310.3959v1 [math.na] 15 Oct 2013 On lower bouns for integration of multivariate permutation-invariant functions Markus Weimar October 16, 2013 Abstract In this note we stuy multivariate integration

More information

Function Spaces. 1 Hilbert Spaces

Function Spaces. 1 Hilbert Spaces Function Spaces A function space is a set of functions F that has some structure. Often a nonparametric regression function or classifier is chosen to lie in some function space, where the assume structure

More information

Lecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012

Lecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012 CS-6 Theory Gems November 8, 0 Lecture Lecturer: Alesaner Mąry Scribes: Alhussein Fawzi, Dorina Thanou Introuction Toay, we will briefly iscuss an important technique in probability theory measure concentration

More information

6 General properties of an autonomous system of two first order ODE

6 General properties of an autonomous system of two first order ODE 6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x

More information

LECTURE NOTES ON DVORETZKY S THEOREM

LECTURE NOTES ON DVORETZKY S THEOREM LECTURE NOTES ON DVORETZKY S THEOREM STEVEN HEILMAN Abstract. We present the first half of the paper [S]. In particular, the results below, unless otherwise state, shoul be attribute to G. Schechtman.

More information

Lower bounds on Locality Sensitive Hashing

Lower bounds on Locality Sensitive Hashing Lower bouns on Locality Sensitive Hashing Rajeev Motwani Assaf Naor Rina Panigrahy Abstract Given a metric space (X, X ), c 1, r > 0, an p, q [0, 1], a istribution over mappings H : X N is calle a (r,

More information

Lower Bounds for the Smoothed Number of Pareto optimal Solutions

Lower Bounds for the Smoothed Number of Pareto optimal Solutions Lower Bouns for the Smoothe Number of Pareto optimal Solutions Tobias Brunsch an Heiko Röglin Department of Computer Science, University of Bonn, Germany brunsch@cs.uni-bonn.e, heiko@roeglin.org Abstract.

More information

Least-Squares Regression on Sparse Spaces

Least-Squares Regression on Sparse Spaces Least-Squares Regression on Sparse Spaces Yuri Grinberg, Mahi Milani Far, Joelle Pineau School of Computer Science McGill University Montreal, Canaa {ygrinb,mmilan1,jpineau}@cs.mcgill.ca 1 Introuction

More information

Discrete Mathematics

Discrete Mathematics Discrete Mathematics 309 (009) 86 869 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/isc Profile vectors in the lattice of subspaces Dániel Gerbner

More information

PDE Notes, Lecture #11

PDE Notes, Lecture #11 PDE Notes, Lecture # from Professor Jalal Shatah s Lectures Febuary 9th, 2009 Sobolev Spaces Recall that for u L loc we can efine the weak erivative Du by Du, φ := udφ φ C0 If v L loc such that Du, φ =

More information

Agmon Kolmogorov Inequalities on l 2 (Z d )

Agmon Kolmogorov Inequalities on l 2 (Z d ) Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,

More information

Linear First-Order Equations

Linear First-Order Equations 5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)

More information

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,

More information

Computing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions

Computing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5

More information

LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION

LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische

More information

Convergence of Random Walks

Convergence of Random Walks Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of

More information

FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction

FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN 1 Introuction In this note, we stuy the fluctuations in the number

More information

SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES

SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES Communications on Stochastic Analysis Vol. 2, No. 2 (28) 289-36 Serials Publications www.serialspublications.com SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES

More information

Sturm-Liouville Theory

Sturm-Liouville Theory LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory

More information

Monte Carlo Methods with Reduced Error

Monte Carlo Methods with Reduced Error Monte Carlo Methos with Reuce Error As has been shown, the probable error in Monte Carlo algorithms when no information about the smoothness of the function is use is Dξ r N = c N. It is important for

More information

A new proof of the sharpness of the phase transition for Bernoulli percolation on Z d

A new proof of the sharpness of the phase transition for Bernoulli percolation on Z d A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition

More information

CHAPTER 1 : DIFFERENTIABLE MANIFOLDS. 1.1 The definition of a differentiable manifold

CHAPTER 1 : DIFFERENTIABLE MANIFOLDS. 1.1 The definition of a differentiable manifold CHAPTER 1 : DIFFERENTIABLE MANIFOLDS 1.1 The efinition of a ifferentiable manifol Let M be a topological space. This means that we have a family Ω of open sets efine on M. These satisfy (1), M Ω (2) the

More information

Lecture 2 Lagrangian formulation of classical mechanics Mechanics

Lecture 2 Lagrangian formulation of classical mechanics Mechanics Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,

More information

Acute sets in Euclidean spaces

Acute sets in Euclidean spaces Acute sets in Eucliean spaces Viktor Harangi April, 011 Abstract A finite set H in R is calle an acute set if any angle etermine by three points of H is acute. We examine the maximal carinality α() of

More information

Problem Sheet 2: Eigenvalues and eigenvectors and their use in solving linear ODEs

Problem Sheet 2: Eigenvalues and eigenvectors and their use in solving linear ODEs Problem Sheet 2: Eigenvalues an eigenvectors an their use in solving linear ODEs If you fin any typos/errors in this problem sheet please email jk28@icacuk The material in this problem sheet is not examinable

More information

Separation of Variables

Separation of Variables Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical

More information

An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback

An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback Journal of Machine Learning Research 8 07) - Submitte /6; Publishe 5/7 An Optimal Algorithm for Banit an Zero-Orer Convex Optimization with wo-point Feeback Oha Shamir Department of Computer Science an

More information

Robust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k

Robust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine

More information

Introduction to the Vlasov-Poisson system

Introduction to the Vlasov-Poisson system Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its

More information

GLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS

GLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS Electronic Journal of Differential Equations, Vol. 015 015), No. 99, pp. 1 14. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu ftp eje.math.txstate.eu GLOBAL SOLUTIONS FOR D COUPLED

More information

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent

More information

Discrete Operators in Canonical Domains

Discrete Operators in Canonical Domains Discrete Operators in Canonical Domains VLADIMIR VASILYEV Belgoro National Research University Chair of Differential Equations Stuencheskaya 14/1, 308007 Belgoro RUSSIA vlaimir.b.vasilyev@gmail.com Abstract:

More information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control 19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior

More information

ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS

ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS MARK SCHACHNER Abstract. When consiere as an algebraic space, the set of arithmetic functions equippe with the operations of pointwise aition an

More information

NOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,

NOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy, NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which

More information

Table of Common Derivatives By David Abraham

Table of Common Derivatives By David Abraham Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec

More information

Euler equations for multiple integrals

Euler equations for multiple integrals Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................

More information

Permanent vs. Determinant

Permanent vs. Determinant Permanent vs. Determinant Frank Ban Introuction A major problem in theoretical computer science is the Permanent vs. Determinant problem. It asks: given an n by n matrix of ineterminates A = (a i,j ) an

More information

REAL ANALYSIS I HOMEWORK 5

REAL ANALYSIS I HOMEWORK 5 REAL ANALYSIS I HOMEWORK 5 CİHAN BAHRAN The questions are from Stein an Shakarchi s text, Chapter 3. 1. Suppose ϕ is an integrable function on R with R ϕ(x)x = 1. Let K δ(x) = δ ϕ(x/δ), δ > 0. (a) Prove

More information

DIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10

DIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10 DIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10 5. Levi-Civita connection From now on we are intereste in connections on the tangent bunle T X of a Riemanninam manifol (X, g). Out main result will be a construction

More information

MAT 545: Complex Geometry Fall 2008

MAT 545: Complex Geometry Fall 2008 MAT 545: Complex Geometry Fall 2008 Notes on Lefschetz Decomposition 1 Statement Let (M, J, ω) be a Kahler manifol. Since ω is a close 2-form, it inuces a well-efine homomorphism L: H k (M) H k+2 (M),

More information

Lecture 5. Symmetric Shearer s Lemma

Lecture 5. Symmetric Shearer s Lemma Stanfor University Spring 208 Math 233: Non-constructive methos in combinatorics Instructor: Jan Vonrák Lecture ate: January 23, 208 Original scribe: Erik Bates Lecture 5 Symmetric Shearer s Lemma Here

More information

Proof of SPNs as Mixture of Trees

Proof of SPNs as Mixture of Trees A Proof of SPNs as Mixture of Trees Theorem 1. If T is an inuce SPN from a complete an ecomposable SPN S, then T is a tree that is complete an ecomposable. Proof. Argue by contraiction that T is not a

More information

Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations

Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations Characterizing Real-Value Multivariate Complex Polynomials an Their Symmetric Tensor Representations Bo JIANG Zhening LI Shuzhong ZHANG December 31, 2014 Abstract In this paper we stuy multivariate polynomial

More information

Witt#5: Around the integrality criterion 9.93 [version 1.1 (21 April 2013), not completed, not proofread]

Witt#5: Around the integrality criterion 9.93 [version 1.1 (21 April 2013), not completed, not proofread] Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5: Aroun the integrality criterion 9.93 [version 1.1 21 April 2013, not complete, not proofrea In [1, section 9.93, Hazewinkel states

More information

Some Examples. Uniform motion. Poisson processes on the real line

Some Examples. Uniform motion. Poisson processes on the real line Some Examples Our immeiate goal is to see some examples of Lévy processes, an/or infinitely-ivisible laws on. Uniform motion Choose an fix a nonranom an efine X := for all (1) Then, {X } is a [nonranom]

More information

MARKO NEDELJKOV, DANIJELA RAJTER-ĆIRIĆ

MARKO NEDELJKOV, DANIJELA RAJTER-ĆIRIĆ GENERALIZED UNIFORMLY CONTINUOUS SEMIGROUPS AND SEMILINEAR HYPERBOLIC SYSTEMS WITH REGULARIZED DERIVATIVES MARKO NEDELJKOV, DANIJELA RAJTER-ĆIRIĆ Abstract. We aopt the theory of uniformly continuous operator

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises

More information

Homotopy colimits in model categories. Marc Stephan

Homotopy colimits in model categories. Marc Stephan Homotopy colimits in moel categories Marc Stephan July 13, 2009 1 Introuction In [1], Dwyer an Spalinski construct the so-calle homotopy pushout functor, motivate by the following observation. In the category

More information

Analysis IV, Assignment 4

Analysis IV, Assignment 4 Analysis IV, Assignment 4 Prof. John Toth Winter 23 Exercise Let f C () an perioic with f(x+2) f(x). Let a n f(t)e int t an (S N f)(x) N n N then f(x ) lim (S Nf)(x ). N a n e inx. If f is continuously

More information

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides

Make graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides Reference 1: Transformations of Graphs an En Behavior of Polynomial Graphs Transformations of graphs aitive constant constant on the outsie g(x) = + c Make graph of g by aing c to the y-values on the graph

More information

A. Incorrect! The letter t does not appear in the expression of the given integral

A. Incorrect! The letter t does not appear in the expression of the given integral AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question

More information

Robustness and Perturbations of Minimal Bases

Robustness and Perturbations of Minimal Bases Robustness an Perturbations of Minimal Bases Paul Van Dooren an Froilán M Dopico December 9, 2016 Abstract Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important

More information

On the number of isolated eigenvalues of a pair of particles in a quantum wire

On the number of isolated eigenvalues of a pair of particles in a quantum wire On the number of isolate eigenvalues of a pair of particles in a quantum wire arxiv:1812.11804v1 [math-ph] 31 Dec 2018 Joachim Kerner 1 Department of Mathematics an Computer Science FernUniversität in

More information

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix

More information

SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA. where L is some constant, usually called the Lipschitz constant. An example is

SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA. where L is some constant, usually called the Lipschitz constant. An example is SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA. Uniqueness for solutions of ifferential equations. We consier the system of ifferential equations given by x = v( x), () t with a given initial conition

More information

The Generalized Incompressible Navier-Stokes Equations in Besov Spaces

The Generalized Incompressible Navier-Stokes Equations in Besov Spaces Dynamics of PDE, Vol1, No4, 381-400, 2004 The Generalize Incompressible Navier-Stokes Equations in Besov Spaces Jiahong Wu Communicate by Charles Li, receive July 21, 2004 Abstract This paper is concerne

More information

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:

More information

arxiv: v2 [math.dg] 16 Dec 2014

arxiv: v2 [math.dg] 16 Dec 2014 A ONOTONICITY FORULA AND TYPE-II SINGULARITIES FOR THE EAN CURVATURE FLOW arxiv:1312.4775v2 [math.dg] 16 Dec 2014 YONGBING ZHANG Abstract. In this paper, we introuce a monotonicity formula for the mean

More information

1. Aufgabenblatt zur Vorlesung Probability Theory

1. Aufgabenblatt zur Vorlesung Probability Theory 24.10.17 1. Aufgabenblatt zur Vorlesung By (Ω, A, P ) we always enote the unerlying probability space, unless state otherwise. 1. Let r > 0, an efine f(x) = 1 [0, [ (x) exp( r x), x R. a) Show that p f

More information

Perfect Matchings in Õ(n1.5 ) Time in Regular Bipartite Graphs

Perfect Matchings in Õ(n1.5 ) Time in Regular Bipartite Graphs Perfect Matchings in Õ(n1.5 ) Time in Regular Bipartite Graphs Ashish Goel Michael Kapralov Sanjeev Khanna Abstract We consier the well-stuie problem of fining a perfect matching in -regular bipartite

More information

Schrödinger s equation.

Schrödinger s equation. Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of

More information

SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS

SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS GEORGE A HAGEDORN AND CAROLINE LASSER Abstract We investigate the iterate Kronecker prouct of a square matrix with itself an prove an invariance

More information

Math 1B, lecture 8: Integration by parts

Math 1B, lecture 8: Integration by parts Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores

More information

The Exact Form and General Integrating Factors

The Exact Form and General Integrating Factors 7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily

More information

Well-posedness of hyperbolic Initial Boundary Value Problems

Well-posedness of hyperbolic Initial Boundary Value Problems Well-poseness of hyperbolic Initial Bounary Value Problems Jean-François Coulombel CNRS & Université Lille 1 Laboratoire e mathématiques Paul Painlevé Cité scientifique 59655 VILLENEUVE D ASCQ CEDEX, France

More information

Similar Operators and a Functional Calculus for the First-Order Linear Differential Operator

Similar Operators and a Functional Calculus for the First-Order Linear Differential Operator Avances in Applie Mathematics, 9 47 999 Article ID aama.998.067, available online at http: www.iealibrary.com on Similar Operators an a Functional Calculus for the First-Orer Linear Differential Operator

More information

Approximation numbers of Sobolev embeddings - Sharp constants and tractability

Approximation numbers of Sobolev embeddings - Sharp constants and tractability Approximation numbers of Sobolev embeddings - Sharp constants and tractability Thomas Kühn Universität Leipzig, Germany Workshop Uniform Distribution Theory and Applications Oberwolfach, 29 September -

More information

Iterated Point-Line Configurations Grow Doubly-Exponentially

Iterated Point-Line Configurations Grow Doubly-Exponentially Iterate Point-Line Configurations Grow Doubly-Exponentially Joshua Cooper an Mark Walters July 9, 008 Abstract Begin with a set of four points in the real plane in general position. A to this collection

More information

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments 2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor

More information

Abstract A nonlinear partial differential equation of the following form is considered:

Abstract A nonlinear partial differential equation of the following form is considered: M P E J Mathematical Physics Electronic Journal ISSN 86-6655 Volume 2, 26 Paper 5 Receive: May 3, 25, Revise: Sep, 26, Accepte: Oct 6, 26 Eitor: C.E. Wayne A Nonlinear Heat Equation with Temperature-Depenent

More information

On colour-blind distinguishing colour pallets in regular graphs

On colour-blind distinguishing colour pallets in regular graphs J Comb Optim (2014 28:348 357 DOI 10.1007/s10878-012-9556-x On colour-blin istinguishing colour pallets in regular graphs Jakub Przybyło Publishe online: 25 October 2012 The Author(s 2012. This article

More information

THE GENUINE OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP

THE GENUINE OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP THE GENUINE OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP ALESSANDRA PANTANO, ANNEGRET PAUL, AND SUSANA A. SALAMANCA-RIBA Abstract. We classify all genuine unitary representations of the metaplectic

More information

All s Well That Ends Well: Supplementary Proofs

All s Well That Ends Well: Supplementary Proofs All s Well That Ens Well: Guarantee Resolution of Simultaneous Rigi Boy Impact 1:1 All s Well That Ens Well: Supplementary Proofs This ocument complements the paper All s Well That Ens Well: Guarantee

More information

12.11 Laplace s Equation in Cylindrical and

12.11 Laplace s Equation in Cylindrical and SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering

More information

7.1 Support Vector Machine

7.1 Support Vector Machine 67577 Intro. to Machine Learning Fall semester, 006/7 Lecture 7: Support Vector Machines an Kernel Functions II Lecturer: Amnon Shashua Scribe: Amnon Shashua 7. Support Vector Machine We return now to

More information

Notes on Lie Groups, Lie algebras, and the Exponentiation Map Mitchell Faulk

Notes on Lie Groups, Lie algebras, and the Exponentiation Map Mitchell Faulk Notes on Lie Groups, Lie algebras, an the Exponentiation Map Mitchell Faulk 1. Preliminaries. In these notes, we concern ourselves with special objects calle matrix Lie groups an their corresponing Lie

More information

Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners Lower Bouns for Local Monotonicity Reconstruction from Transitive-Closure Spanners Arnab Bhattacharyya Elena Grigorescu Mahav Jha Kyomin Jung Sofya Raskhonikova Davi P. Wooruff Abstract Given a irecte

More information

Witten s Proof of Morse Inequalities

Witten s Proof of Morse Inequalities Witten s Proof of Morse Inequalities by Igor Prokhorenkov Let M be a smooth, compact, oriente manifol with imension n. A Morse function is a smooth function f : M R such that all of its critical points

More information

TRAJECTORY TRACKING FOR FULLY ACTUATED MECHANICAL SYSTEMS

TRAJECTORY TRACKING FOR FULLY ACTUATED MECHANICAL SYSTEMS TRAJECTORY TRACKING FOR FULLY ACTUATED MECHANICAL SYSTEMS Francesco Bullo Richar M. Murray Control an Dynamical Systems California Institute of Technology Pasaena, CA 91125 Fax : + 1-818-796-8914 email

More information

Interconnected Systems of Fliess Operators

Interconnected Systems of Fliess Operators Interconnecte Systems of Fliess Operators W. Steven Gray Yaqin Li Department of Electrical an Computer Engineering Ol Dominion University Norfolk, Virginia 23529 USA Abstract Given two analytic nonlinear

More information

Diophantine Approximations: Examining the Farey Process and its Method on Producing Best Approximations

Diophantine Approximations: Examining the Farey Process and its Method on Producing Best Approximations Diophantine Approximations: Examining the Farey Process an its Metho on Proucing Best Approximations Kelly Bowen Introuction When a person hears the phrase irrational number, one oes not think of anything

More information

Lecture 6: Calculus. In Song Kim. September 7, 2011

Lecture 6: Calculus. In Song Kim. September 7, 2011 Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear

More information

On the Cauchy Problem for Von Neumann-Landau Wave Equation

On the Cauchy Problem for Von Neumann-Landau Wave Equation Journal of Applie Mathematics an Physics 4 4-3 Publishe Online December 4 in SciRes http://wwwscirporg/journal/jamp http://xoiorg/436/jamp4343 On the Cauchy Problem for Von Neumann-anau Wave Equation Chuangye

More information

Calculus and optimization

Calculus and optimization Calculus an optimization These notes essentially correspon to mathematical appenix 2 in the text. 1 Functions of a single variable Now that we have e ne functions we turn our attention to calculus. A function

More information

Calculus of Variations

Calculus of Variations 16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t

More information

A nonlinear inverse problem of the Korteweg-de Vries equation

A nonlinear inverse problem of the Korteweg-de Vries equation Bull. Math. Sci. https://oi.org/0.007/s3373-08-025- A nonlinear inverse problem of the Korteweg-e Vries equation Shengqi Lu Miaochao Chen 2 Qilin Liu 3 Receive: 0 March 207 / Revise: 30 April 208 / Accepte:

More information

Qubit channels that achieve capacity with two states

Qubit channels that achieve capacity with two states Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March

More information

arxiv: v2 [math.pr] 27 Nov 2018

arxiv: v2 [math.pr] 27 Nov 2018 Range an spee of rotor wals on trees arxiv:15.57v [math.pr] 7 Nov 1 Wilfrie Huss an Ecaterina Sava-Huss November, 1 Abstract We prove a law of large numbers for the range of rotor wals with ranom initial

More information

Quantum mechanical approaches to the virial

Quantum mechanical approaches to the virial Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from

More information

II. First variation of functionals

II. First variation of functionals II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent

More information

SYMPLECTIC GEOMETRY: LECTURE 3

SYMPLECTIC GEOMETRY: LECTURE 3 SYMPLECTIC GEOMETRY: LECTURE 3 LIAT KESSLER 1. Local forms Vector fiels an the Lie erivative. A vector fiel on a manifol M is a smooth assignment of a vector tangent to M at each point. We think of M as

More information

QF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim

QF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.

More information

The Sokhotski-Plemelj Formula

The Sokhotski-Plemelj Formula hysics 25 Winter 208 The Sokhotski-lemelj Formula. The Sokhotski-lemelj formula The Sokhotski-lemelj formula is a relation between the following generalize functions (also calle istributions), ±iǫ = iπ(),

More information

WELL-POSEDNESS OF A POROUS MEDIUM FLOW WITH FRACTIONAL PRESSURE IN SOBOLEV SPACES

WELL-POSEDNESS OF A POROUS MEDIUM FLOW WITH FRACTIONAL PRESSURE IN SOBOLEV SPACES Electronic Journal of Differential Equations, Vol. 017 (017), No. 38, pp. 1 7. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu WELL-POSEDNESS OF A POROUS MEDIUM FLOW WITH FRACTIONAL

More information

arxiv: v1 [math.mg] 10 Apr 2018

arxiv: v1 [math.mg] 10 Apr 2018 ON THE VOLUME BOUND IN THE DVORETZKY ROGERS LEMMA FERENC FODOR, MÁRTON NASZÓDI, AND TAMÁS ZARNÓCZ arxiv:1804.03444v1 [math.mg] 10 Apr 2018 Abstract. The classical Dvoretzky Rogers lemma provies a eterministic

More information