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1.
This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the
centered L
2-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans.
Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the
initial design. 相似文献
2.
Fractional factorial designs (FFD’s) are no doubt the most widely used designs in the experimental investigations due to their efficient use of experimental runs to study many factors simultaneously. One consequence of using FFD’s is the aliasing of factorial effects. Follow-up experiments may be needed to break the confounding. A simple strategy is to add a foldover of the initial design, the new fraction is called a foldover design. Combining a foldover design with the original design converts a design of resolution r into a combined design of resolution \(r+1\). In this paper, we take the centered \(L_2\)-discrepancy \(({\mathcal {CD}})\) as the optimality measure to construct the optimal combined design and take asymmetrical factorials with mixed two and three levels, which are most commonly used in practice, as the original designs. New and efficient analytical expressions based on the row distance of the \({\mathcal {CD}}\) for combined designs are obtained. Based on these new formulations, we present new and efficient lower bounds of the \({\mathcal {CD}}\). Using the new formulations and lower bounds as the benchmarks, we may implement a new algorithm for constructing optimal mixed-level combined designs. By this search heuristic, we may obtain mixed-level combined designs with low discrepancy. 相似文献
3.
Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Häggkvist in 1978 and, specifically, L(n, K n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K n,n with n colours having all two-coloured cycles of length ≤ 2q 2, for integers n in a set of density 1 ? 3/(q ? 1). One consequence is that L(n, K n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n ? 4. 相似文献
4.
A subalgebra H of a finite dimensional Lie algebra L is said to be a SCAP-subalgebra if there is a chief series 0 = L0 ? L1 ?... ? Lt = L of L such that for every i = 1, 2,..., t, we have H + Li = H + Li-1 or H ∩ Li = H ∩ Li-1. This is analogous to the concept of SCAP-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its SCAP-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable. 相似文献
5.
Let \(\mathcal{N}\) denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field \(\mathbb{F}\), there exists a smallest ideal I of L such that L/I ∈ \(\mathcal{N}\). This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L\(\mathcal{N}\). In this paper, we define the subalgebra S(L) = ∩H≤LIL(H\(\mathcal{N}\)). Set S0(L) = 0. Define Si+1(L)/Si(L) = S(L/Si(L)) for i > 1. By S∞(L) denote the terminal term of the ascending series. It is proved that L = S∞(L) if and only if L\(\mathcal{N}\) is nilpotent. In addition, we investigate the basic properties of a Lie algebra L with S(L) = L. 相似文献
6.
We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L 2(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? N of Hausdorff dimension D. This rate turns out to be of order n ?H/D (log?n)1/2. In the case T=[0,1] N we present a concrete wavelet representation of X H leading to an approximation of X H with the optimal rate n ?H/N (log?n)1/2. 相似文献
7.
V. F. Babenko N. V. Parfinovich 《Proceedings of the Steklov Institute of Mathematics》2012,277(1):9-20
Let L ∞,s 1 (? m ) be the space of functions f ∈ L ∞(? m ) such that ?f/?x i ∈ L s (? m) for each i = 1, ...,m . New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥D α f∥∞ of functions f ∈ L ∞,s 1 (? m ). Stechkin’s problem on approximation of unbounded operators D α by bounded operators on the class of functions f ∈ L ∞,s 1 (? m ) such that ∥?f∥ s ≤ 1 and the problem of optimal recovery of the operator D α on elements from this class given with error δ are solved. 相似文献
8.
V. B. Korotkov 《Siberian Mathematical Journal》2009,50(1):96-99
We consider the class of the continuous L 2,1 linear operators in L 2 that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of L 2 into a compact subset of L 1. We prove that a functional equation with an operator from L 2,1 is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if T ∈ L 2,1 and VTV ?1 ∈ L 2,1 for all unitary operators V in L 2 then T = α1 + C, where α is a scalar, 1 is the identity operator in L 2, and C is a compact operator in L 2. 相似文献
9.
Let L be a lattice of finite length, ξ = (x 1,…, x k )∈L k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x 1)+?+d(y, x k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x 1∨?∨x k . In other words, L satisfies the so-called c 1 -median property. 相似文献
10.
Let L=?Δ+V be a Schrödinger operator on ? d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L p (? d ), for some p>d /2. Let K t be the semigroup generated by ?L. We say that an L 1(? d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?t>0|K t f| belongs to L 1(? d ). We prove that \(f\in H^{1}_{L}\) if and only if R j f∈L 1(? d ) for j=1,…,d, where R j =(?/? x j )L ?1/2 are the Riesz transforms associated with L. 相似文献
11.
Vladimir Eiderman Alexander Reznikov Alexander Volberg 《Journal d'Analyse Mathématique》2018,134(1):55-105
We show that, given a family of discs centered at a chord-arc curve, the analytic capacity of a union of subsets of these discs (one subset in each disc) is comparable with the sum of their analytic capacities. However, we need the discs in question to be separated, and it is not clear whether the separation condition is essential. We apply this result to find families {μj} of measures in ? with the following property. If the Cauchy integral operators \({C_{{\mu _j}}}\) from L2(μj) to itself are bounded uniformly in j, then Cμ, μ = Σμj, is also bounded from L2(μ) to itself. 相似文献
12.
Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I). 相似文献
13.
N. Yu. Makarenko 《Siberian Mathematical Journal》2007,48(1):95-111
We prove that if a (?/n?)-graded Lie algebra L = ? i=0 n?1 L i has d nontrivial components L i and the null component L 0 has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? i=0 n?1 with few nontrivial components L i if the null component L 0 has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? i=0 n?1 L i with trivial component L 0 and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2]. 相似文献
14.
V. D. Stepanov E. P. Ushakova 《Proceedings of the Steklov Institute of Mathematics》2008,260(1):254-278
A new criterion for the weighted L p ?L q boundedness of the Hardy operator with two variable limits of integration is obtained for 0 < q < q + 1 ≤ p < ∞. This criterion is applied to the characterization of the weighted L p ?L q boundedness of the corresponding geometric mean operator for 0 < q < p < ∞. 相似文献
15.
M. Brundin 《Czechoslovak Mathematical Journal》2007,57(1):345-365
If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L p and weak L p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L Φ having the property L ∞ ? L Φ L p , 1 ? p > ∞. The second contains spaces L Φ that resemble L p spaces. 相似文献
16.
G. Amirkhanyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2008,43(3):127-134
Convergence of the greedy algorithm in Walsh system in L p , p > 1 is studied. It is proved that there exists a function in L p , 1 < p < 2, with greedy algorithm not converging in measure to that function. A continuous function with divergent in L p , p > 2, greedy algorithm is constructed and sufficient conditions for convergence of the greedy algorithm in L p , p > 1 are given. 相似文献
17.
T. F. Filippova 《Proceedings of the Steklov Institute of Mathematics》2016,293(1):67-76
In the Banach space L1(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L2(M, τ) is introduced, and its main properties are established. A convergence criterion in L2(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L1(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖1 ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L1(M, τ) is complemented. The convergence in L2(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained. 相似文献
18.
In the current paper, we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension d ≥ 3. In particular, we use dyadic harmonic analysis to prove that the dyadic product BMO and exp(L2/(d?1)) norms of the discrepancy function of so-called digital nets of order two are bounded above by (logN)(d?1)/2. The latter bound has been recently conjectured in several papers and is consistent with the best known low-discrepancy constructions. Such estimates play an important role as an intermediate step between the well-understood Lp bounds and the notorious open problem of finding the precise L∞ asymptotics of the discrepancy function in higher dimensions, which is still elusive. 相似文献
19.
Let L = L 0 + V be a Schrödinger type operator, where L 0 is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L q boundedness of Riesz transform ? m L ?1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e ?t L . Furthermore, the authors impose extra regularity assumptions on V to obtain the L q boundedness of Riesz transform ? m L ?1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients. 相似文献
20.
Goran Djanković 《Archiv der Mathematik》2011,97(6):535-547
In this paper we investigate the sixth moment of the family of L-functions associated to holomorphic modular forms on GL 2 with respect to a congruence subgroup Γ1(q). The bound for central values averaged over the family, consistent with the Lindelöf hypothesis, is obtained for prime levels q. 相似文献