When (φn) is a sequence of positive functions in L1(R) such that the convolutions φn ? f converge in L1-norm to f for all f ∈ L1(R), then these convolutions may or may not converge almost everywhere too. There is always a subsequence (φnk ) such that at least φnk ? f converges almost everywhere to f for all f ∈ Lp(R), 1 < p < ∞. A special case of this is when the functions φn are the dilations ...

For α > −1, let A2α be the corresponding weighted Bergman space of the unit ball in C. For a bounded measurable function f , let Tf be the Toeplitz operator with symbol f on A 2 α. This paper describes all the functions f for which Tf commutes with a given Tg, where g(z) = z1 1 · · · z Ln n for strictly positive integers L1, . . . , Ln, or g(z) = |z1| s1 · · · |zn| nh(|z|) for non-negative real...

We propose a new adaptive L1 penalized quantile regression estimator for highdimensional sparse regression models with heterogeneous error sequences. We show that under weaker conditions compared with alternative procedures, the adaptive L1 quantile regression selects the true underlying model with probability converging to one, and the unique estimates of nonzero coefficients it provides have ...

One may think of many ways of combining modal logics representing various aspects of an application domain. Two ‘canonical’ constructions, supported by a well-developed mathematical theory, are fusions [17, 6, 7] and products [8, 7]. The fusion L1⊗ · · ·⊗Ln of n ≥ 2 normal propositional unimodal logics Li with the boxes 2i is the smallest multimodal logic in the language with n boxes 21, . . . ...

Let L = L1 × · · · × Ln be the product of n lattices, each of which has a bounded width. Given a subset A ⊆ L, we show that the problem of extending a given partial list of maximal independent elements of A in L can be solved in quasi-polynomial time.

Let F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 for n ≥ 2 denote the sequence F of Fibonacci numbers. For any modulus m ≥ 2 and residue b (modm), denote by vF (m, b) the number of occurrences of b as a residue in one (shortest) period of F modulo m. Moreover, let vL(m, b) be similarly defined for the Lucas sequence L satisfying L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2. In this paper, completing ...

Restart Model Elimination (RME) has been introduced as a variant of model elimination in [1] as a calculus which avoids contrapositives and which introduces case analysis. In [4] a variant for computing answers to disjunctive logic programs was introduced. RME is implemented as part of the PROTEIN system [2]. One result of this paper is a table of completeness results with respect to the combin...

Let L0 = 2, L1 = 1, and Ln = Ln−1+Ln−2 for n ≥ 2, denote the sequence L of Lucas numbers. For any modulus m ≥ 2, and residue b (mod m), denote by vL(m, b) the number of occurrences of b as a residue in one (shortest) period of L modulo m. In this paper, we completely describe the functions vL(p , .) for k ≥ 1 in the cases p = 2 and p = 5. Using a notion formally introduced by Carlip and Jacobso...

Proposition 1.2 (Splitting Principle). Let E → M be a complex vector bundle of rank n over a manifold X. Then there is a manifold F (E) and a smooth fibration π : F (E) → M such that • The homomorphism π∗ : H∗(M, Z) → H∗(F (E), Z) is injective. • The bundle π∗E splits into a direct sum of complex line bundles, i.e. π∗E = l1 ⊕ · · ·⊕ ln Proof. Consider the projectivisation p : PC(E) → M and the ...

The aim of this study was to assess the significance of programmed cell death 1 ligand 1 (PD-L1) in esophageal squamous cell carcinoma (ESCC) and its association with IL-6 and radiation response. Weretrospectively enrolled 162 patients with ESCC, and examined the correlation between PD-L1 levels and clinical outcomes in esophageal cancer patients. Furthermore, the human esophageal SCC cell line...