On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)^r = λ(n)^s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.     

Comonotone Jackson's Inequality

Let 2s points y_i=−πy_2s<…<y₁<π be given. Using these points, we define the points y_i for all integer indices i by the equality y_i=y_i+2s+2π. We shall write fΔ⁽¹⁾(Y) if f is a 2π-periodic continuous function and f does not decrease on [y_i, y_i−1], if i is odd; and f does not increase on [y_i, y_i−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. 1. If fΔ⁽¹⁾(Y), then for each nonnegative integer n there is a trigonometric polynomial τ_n(x) of order n such that τ_nΔ⁽¹⁾(Y), and |f(x)−π_n(x)|c(s) ω(f; 1/(n+1)), x , where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

Lacunary Quadrature Formulae and Interpolation Singularity

Birkholl quadrature formulae (q.f.), which have algebraic degree of precision (ADP) greater than the number of values used, are studied. In particular, we construct a class of quadrature rules of ADP = 2n + 2r + 1 which are based on the information {ƒ^(j)(−1), ƒ^(j)(−1), j = 0, ..., r − 1 ; ƒ(x_i), ƒ^(2m)(x_i), i = 1, ..., n}, where m is a positive integer and r = m, or r = m − 1. It is shown that the corresponding Birkhoff interpolation problems of the same type are not regular at the quadrature nodes. This means that the constructed quadrature formulae are not of interpolatory type. Finally, for each In, we prove the existence of a quadrature formula based on the information {ƒ(x_i), ƒ^(2m)(x_i), i = 1, ..., 2m}, which has algebraic degree of precision 4m + 1.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

Matroids with No (q+2)-Point-Line Minors

It is known that a geometry with rankrand no minor isomorphic to the (q+2)-point line has at most (q^r−1)/(q−1) points, with strictly fewer points ifr>3 andqis not a prime power. Forqnot a prime power andr>3, we show thatq^r−1−1 is an upper bound. Forqa prime power andr>3, we show that any rank-rgeometry with at leastq^r−1points and no (q+2)-point-line minor is representable overGF(q). We strengthen these bounds toq^r−1−(q^r−2−1)/(q−1)−1 andq^r−1−(q^r−2−1)/(q−1) respectively whenqis odd. We give an application to unique representability and a new proof of Tutte's theorem: A matroid is binary if and only if the 4-point line is not a minor.     

On the Erd s-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings

A link between Ramsey numbers for stars and matchings and the Erd s-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems. Theorem 5. Let m be an even integer. If c : e (K_2m−1)→{0, 1,…, m−1} is a mapping of the edges of the complete graph on 2m−1 vertices into {0, 1,…, m−1}, then there exists a star K_1,m in K_2m−1 with edges e₁, e₂,…, e_m such that c(e₁)+c(e₂)++c(e_m)≡0 (mod m). Theorem 8. Let m be an integer. If c : e(K^r_(r+1)m−1)→{0, 1,…, m−1} is a mapping of all the r-subsets of an (r+1)m−1 element set S into {0, 1,…, m−1}, then there are m pairwise disjoint r-subsets Z₁, Z₂,…, Z_m of S such that c(Z₁)+c(Z₂)++c(Z_m)≡0 (mod m).     

Extremal Regular Graphs with Prescribed Odd Girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ƒ(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ƒ(k, g) ≥ kg/2 and that ƒ(k, g) = kg/2 if k is even. The exact values of ƒ(k, g) are also known if k = 3 or g = 5. Let x_e denote the smallest even integer no less than x, δ(g) = (−1)^{g − 1}/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ≥ 5 and g ≥ 7 are both odd, then [formula] with the exception that ƒ(5, 7) = 20.     

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (T_L^(s)), Watson statistic (T_W^(s)) and Rao statistic (T_R^(s)) for testing H_0s: μ (a given subspace) against H_1s: μ , based on a sample of size n from a p-variate Langevin distribution M_p(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ⁻¹. For the case when κ is unknown, it is shown that T_R^(s) T_L^(s) T_W^(s) in their powers up to the order κ⁻¹.     

Lower bounds for the cardinality of minimal blocking sets in projective spaces

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r₂(q) be the number such that q+r₂(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most q^m+q^m−1+…+q+1+r₂(q)·(∑_i=2m−n′^m−1qⁱ) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals q^m+q^m−1+…+q+1+r₂(q)(∑_i=2m−n′^m−1qⁱ). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<q^m+q^m−1+…+q+1+r₂(q)(q^m−1+q^m−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. For q=p^3h, p7 and q not a square we show this assertion for |B|q^m+q^m−1+…+q+1+q^2/3·(q^m−1+…+1).     

Remarks on some zero-sum problems

Let Z denote the ring of integers and for a prime p and positive integers r and d, let f_r(P, d) denote the smallest positive integer such that given any sequence of f_r(p, d) elements in (Z/pZ(^d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(^d. That f₁(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f₁(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of f_r(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of f_r(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Rónyai in connection with obtaining good upper bounds for f₁(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.     

Centres of Convex Sets inLMetrics

It is shown that for each convex bodyARⁿthere exists a naturally defined family _AC(Sⁿ⁻¹) such that for everyg _A, and every convex functionf: R→Rthe mappingy∫_Sⁿ⁻¹ f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toL^pmetrics is discussed.     

The Complexity of Indefinite Elliptic Problems with Noisy Data

We study the complexity of second-order indefinite elliptic problems −div(au) +bu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain Ω, the error being measured in theH¹(Ω)-norm. The problem elementsfbelong to the unit ball ofW^{r, p}, (Ω), wherep [2, ∞] andr>d/p. Information consists of (possibly adaptive) noisy evaluations off,a, orb(or their derivatives). The absolute error in each noisy evaluation is at most δ. We find that thenth minimal radius for this problem is proportional ton^−r/d+ δ and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the -complexity (minimal cost of calculating an -approximation) for this problem, said bounds depending on the costc(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation isc(δ) = δ^−s(fors> 0), then the complexity is proportional to (1/)^{d/r + s}.     

A bipartite analogue of Dilworth’s theorem for multiple partial orders

Let r be a fixed positive integer. It is shown that, given any partial orders <₁, …, <_r on the same n-element set P, there exist disjoint subsets A,BP, each with at least n^1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an such that every element of A is larger than every element of B in the partial order <_i, or (2) no element of A is comparable with any element of B in any of the partial orders <₁, …, <_r. As a corollary, we obtain that any family C of n convex compact sets in the plane has two disjoint subfamilies A,BC, each with at least n^1−o(1) members, such that either every member of A intersects all members of B, or no member of A intersects any member of B.     

Some Mean Values Related to Average Multiplicative Orders of Elements in Finite Fields

For any positive integer n let α(n) denote the average order of elements in the cyclic group Z_n. In this note, we investigate the functions α(n)/n and α(n)/φ(n) when n ranges through numbers of the form p−1 with p prime, and when n ranges through numbers of the form 2^m−1 with m a positive integer. In particular, we show that such functions have limiting distributions, and we compute their average values, and their minimal and maximal orders.To Jean-Louis Nicolas at his 60th birthday2000 Mathematics Subject Classification: Primary—11N45; Secondary—05A16, 11N37This work was supported in part by Grant SEP-CONACYT 37259-E.     

Path numbers of tournaments

A family of simple (that is, cycle-free) paths is a path decomposition of a tournament T if and only if partitions the acrs of T. The path number of T, denoted pn(T), is the minimum value of | | over all path decompositions of T. In this paper it is shown that if n is even, then there is a tournament on n vertices with path number k if and only if n/2 k n²/4, k an integer. It is also shown that if n is odd and T is a tournament on n vertices, then (n + 1)/2 pn(T) (n² − 1)/4. Moreover, if k is an integer satisfying (i) (n + 1)/2 k n − 1 or (ii) n < k (n² − 1)/4 and k is even, then a tournament on n vertices having path number k is constructed. It is conjectured that there are no tournaments of odd order n with odd path number k for n k < (n² − 1)/4.     

Typical primitive polynomials over integer residue rings

Let N be a product of distinct prime numbers and Z/(N) be the integer residue ring modulo N. In this paper, a primitive polynomial f(x) over Z/(N) such that f(x) divides x^s−c for some positive integer s and some primitive element c in Z/(N) is called a typical primitive polynomial. Recently typical primitive polynomials over Z/(N) were shown to be very useful, but the existence of typical primitive polynomials has not been fully studied. In this paper, for any integer m1, a necessary and sufficient condition for the existence of typical primitive polynomials of degree m over Z/(N) is proved.     

Finding intersection of rectangles by range search

Three related rectangle intersection problems in k-dimensional space are considered: (1) find the intersections of a rectangle with a given set of rectangles, (2) find the intersecting pairs of rectangles as they are inserted into or deleted from an existing set of rectangles, and (3) find the intersecting pairs of a given set of rectangles. By transforming these problems into range search problems, one need not divide the intersection problem into two subproblems, namely, the edge-intersecting problem and the containment problem, as done by many previous studies. Furthermore, this approach can also solve these subproblems separately, if required. For the first problem the running time is O((log n)^2k−1 + s), where s is the number of intersecting pairs of rectangles. For the second problem the time needed to generate and maintain n rectangles is O(n(log n)^2k) and the time for each query is O((log n)^2k−1 + s). For the third problem the total time is O(n log n + n(log n)^2(k−1) + s) for k ≥ 1.     

Hyperoctahedral Operations on Hochschild Homology

We construct Families {ρ^{l, k}_n} of orthogonal idempotents of the hyperoctahedral group algebras [B_n], which commute with the Hochschild boundary operators b_n=∑ⁿ_i=0 (−1)ⁱd_i. We show that those idempotents are projections onto some hyperoactahedral symmetric powers of the free Lie algebra Lie^{(l, k)}_n( ). The commutations above then decompose the Hochshild homology H_n(C) obtained by any functor C:Δ^op → K-module that factor through Fin′_B, the hyperoctahedral category. Moreover, we show that this decomposition is the finest possible for any such functor. In particular, the Hochschild homology of a commutative algebra equipped with an involutive automorphism splits into components indexed by (l, k) and the corresponding Harrison homology splits into two components indexed by (0, 1) and (1, 0). Generalizing the Harrison complex, we show that H^(l,k)_n(C)H_n(Sh^(l,k)_./Sh^{(l−1, k+1)}_., where Sh^(r,s)_.are some shuffle complexes associated to C. We also give the characters of the representations related to Lie^{(l, k)}_n( ) as a direct sum of induced characters.