Boundary effects on convergence rates for Tikhonov regularization

We consider the Tikhonov regularizer f_λ of a smooth function f ε H^2m[0, 1], defined as the solution (see [1]) to We prove that if f^(j)(0) = f^(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − f_λ¦_j² Rλ^{(2k − 2j + 3)/2m}, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates.     

An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

On the spectral function of the Poisson–Voronoi cells

Denote by (t)=∑_n1e^−λ_nt, t>0, the spectral function related to the Dirichlet Laplacian for the typical cell of a standard Poisson–Voronoi tessellation in . We show that the expectation E(t), t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E(t), when t→0⁺,+∞. In particular, we prove that the law of the first eigenvalue λ₁ of satisfies the asymptotic relation lnP{λ₁t}−2^dω_dj_(d−2)/2^d·t^−d/2 when t→0⁺, where ω_d and j_(d−2)/2 are respectively the Lebesgue measure of the unit ball in and the first zero of the Bessel function J_(d−2)/2.     

On Some Subfactors of Integer Index Arising from Vertex Models

This paper is devoted to the study of subfactors arising out of commuting squares constructed out of the simplest possible vertex models. After setting up the necessary general background, we start with two classes of commuting squares and compute the principal graphs of the resulting subfactors, one of them being related to the group dual of a suitable (closed) subgroup ofU(N), and the other to the Cayley graph of a (non-closed) group, modulo scalars, generated byNelements ofU(N). In the last section, we “classify” the commuting squares in dimension 2, and identify the possible resulting principal graphs asA⁽¹⁾_(2n−1), 1n∞     

Weak Asymptotics for the Generating Polynomials of the Stirling Numbers of the Second Kind

For the horizontal generating functions P_n(z)=∑ⁿ_k=1 S(n, k) z^k of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Q_n(z)=P_n(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of \[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

On Chebyshev–Markov Rational Functions over Several Intervals

Chebyshev–Markov rational functions are the solutions of the following extremal problem

Full-size image

withKbeing a compact subset of andω_n(x) being a fixed real polynomial of degree less thann, positive onK. A parametric representation of Chebyshev–Markov rational functions is found forK=[b₁, b₂]…[b_2p−1, b_2p], −∞<b₁b₂<…<b_2p−1b_2p<+∞ in terms of Schottky–Burnside automorphic functions.     

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).     

Onα-Symmetric Multivariate Characteristic Functions

Ann-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the form((|u₁|^α+…+|u_n|^α)^1/α). We study the classesΦ_n(α) of all admissible functions: [0, ∞)→ . It is known that members ofΦ_n(2) andΦ_n(1) are scale mixtures of certain primitivesΩ_nandω_n, respectively, and we show thatω_nis obtained fromΩ_2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: If(0)=1,is continuous, lim_t→∞ (t)=0, and⁽²ⁿ⁻²⁾(t) is convex, thenΦ_n(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution.     

Ramsey functions involving with large

For fixed integers m,k2, it is shown that the k-color Ramsey number r_k(K_m,n) and the bipartite Ramsey number b_k(m,n) are both asymptotically equal to k^mn as n→∞, and that for any graph H on m vertices, the two-color Ramsey number is at most (1+o(1))n^m+1/(logn)^m-1. Moreover, the order of magnitude of is proved to be n^m+1/(logn)^m if H≠K_m as n→∞.     

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.     

Compact causal data interpolation

Consider a Hilbert space equipped with a time-structure, i.e., a resolution E of the identity on defined on subsets of some linearly ordered set Λ. For which x and y in is it possible to find a causal (time respecting) compact operator T, so that Tx = y? When T is required to be a Hilbert-Schmidt operator and (Λ, E) is sufficiently regular, this question is answered in terms of the “time-densities” of x and y. The condition is that the integral ∝_gLμ_x({s t})⁻¹ dμ_y(t) should be finite, where μ_x and μ_y are the measures on Λ given by μ_x(Ω) = ¦|E(Ω)x¦|² and μ_y(Ω) = ¦|E(Ω)y¦|². Further a solution is given for the related problem of minimizing the sum of ¦|Tx − y¦|² and the squared Hilbert-Schmidt norm ¦|R¦|₂² of T.     

Two New Classes of Difference Families

A Z-cyclic triplewhist tournament for 4n+1 players, or briefly a TWh(4n+1), is equivalent to a n-set {(a_i, b_i, c_i, d_i) | i=1, …, n} of quadruples partitioning Z_4n+1−{0} with the property that ⁿ_i=1 {±(a_i−c_i), ±(b_i−d_i)}=ⁿ_i=1 {±(a_i−b_i), ±(c_i−d_i)}=ⁿ_i=1 {±(a_i−d_i), ±(b_i−c_i)}=Z_4n+1−{0}. The existence problem for Z-cyclic TWh(p)'s with p a prime has been solved for p1 (mod 16). I. Anderson et al. (1995, Discrete Math.138, 31–41) treated the case of p≡5 (mod 8) while Y. S. Liaw (1996, J. Combin. Des.4, 219–233) and G. McNay (1996, Utilitas Math.49, 191–201) treated the case of p≡9 (mod 16). In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes p≡1 (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all≡1 (mod 4) and distinct from 5, 13, and 17.     

Analysis of Carry Propagation in Addition: An Elementary Approach

Our goal in this paper is to analyze carry propagation in addition using only elementary methods (that is, those not involving residues, contour integration, or methods of complex analysis). Our results concern the length of the longest carry chain when two independent uniformly distributed n-bit numbers are added. First, we show using just first- and second-moment arguments that the expected length C_n of the longest carry chain satisfies C_n = log₂n + O(1). Second, we use a sieve (inclusion–exclusion) argument to give an exact formula for C_n. Third, we give an elementary derivation of an asymptotic formula due to Knuth, C_n = log₂n + Φ(log₂ n) + O((logn)⁴/n), where Φ(ν) is a bounded periodic function of ν, with period 1, for which we give both a simple integral expression and a Fourier series. Fourth, we give an analogous asymptotic formula for the variance V_n of the length of the longest carry chain: V_n = Ψ(log₂ n) + O((logn)⁵/n), where Ψ(ν) is another bounded periodic function of ν, with period 1. Our approach can be adapted to addition with the “end-around” carry that occurs in the sign-magnitude and 1s-complement representations. Finally, our approach can be adapted to give elementary derivations of some asymptotic formulas arising in connection with radix-exchange sorting and collision-resolution algorithms, which have previously been derived using contour integration and residues.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations

Stability of the Rossby–Haurwitz (RH) wave of subspace H₁H_n in an ideal incompressible fluid on a rotating sphere is analytically studied (H_n is the subspace of homogeneous spherical polynomials of degree n). It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets M₋ⁿ, M₊ⁿ, H_n and M₀ⁿ−H_n depending on the value of their mean spectral number χ(t)=η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M₋ⁿ and M₊ⁿ due to a hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced using the invariant subspace H_n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H_n, the factor norm controls the perturbation part orthogonal to H_n. It is shown that in the set M₋ⁿ (χ(t)<n(n+1)), any nonzonal RH wave of subspace H₁H_n (n2) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M₀ⁿ−H_n. A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n+1), where n is the RH-wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave is shown. The instability in the invariant set M₊ⁿ of small-scale perturbations (χ(t)>n(n+1)) is still open problem.     

Bijections for ternary trees and non-crossing trees

The number N_n of non-crossing trees of size n satisfies N_n+1=T_n where T_n enumerates ternary trees of size n. We construct a new bijection to establish that fact. Since , it follows that 3(3n−1)(3n−2)T_n−1=2n(2n+1)T_n. We construct two bijections “explaining” this recursion; one of them easily extends to the case of t-ary trees.     

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 ).     

On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).     

Decomposable graphs and definitions with no quantifier alternation

Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D₀(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q₀(n) to be the minimum of D₀(G) over all graphs G of order n.We prove that for all n we have

log^*n−log^*log^*n−2≤q₀(n)≤log^*n+22,