Let Jα k be a real power of the integration operator Jk defined on Sobolev space W k p [0, 1]. We investigate the spectral properties of the operator Ak = ⊕n j=1 λjJ α k defined on ⊕n j=1W k p [0, 1]. Namely, we describe the commutant {Ak} ′, the double commutant {Ak} ′′ and the algebra AlgAk. Moreover, we describe the lattices LatAk and HypLatAk of invariant and hyperinvariant subspaces of Ak,...

Proof of Proposition 1. Suppose an attribute value ai is not recorded in any of the data sources S1 through Sn for an entity instance. Then, from Assumptions 1 and 2 in the paper, we have P A= ai AS1 = ak AS2 = al ASn = atW i = k i = l i = t = P AS1 = ak A= ai P AS2 = al A= ai × · · ·×P ASn = at A= ai P AS1 = ak AS2 = al ASn = at P A= ai = 1−R A S1 / m− 1 1−RS2 / m− 1 × · · ·× 1−RSn / m− 1 P AS...

Sharpening (a particular case of) a result of Szemerédi and Vu [SV06] and extending earlier results of Sárközy [S89] and ourselves [L97b], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is...

purpose: to investigate confocal scan features of corneal subbasal nerve plexus and epithelial dendritic cells in herpes simplex keratitis (hsk) as compared to acanthamoeba keratitis (ak). methods: confocal scan records of patients with clinical and confocal microscopic features compatible with hsk and ak as well as confocal scan records of normal participants were reviewed. the density and mor...

For positive integers a, k, let Ak(a) denote the sequence ak, ak + 1, ak + a, . . . , ak + ak−1. Let Γ ( Ak(a) ) denote the set of integers that are expressible as a linear combination of elements of Ak(a) with non-negative integer coefficients. We determine g ( Ak(a) ) and n ( Ak(a) ) which denote the largest (respectively, the number of) positive integer(s) not in Γ ( Ak(a) ) . We also determ...

Acanthamoeba keratitis (AK) is a sight-threating infection of the cornea that mostly affects contact lens wearers. Until now, AK treatment remains very difficult due to the existence of a highly resistant cyst stage in the life cycle of Acanthamoeba which is extremely resistant to most of the available anti-amoebic compounds. Moreover, current treatment of AK is usually based in the combination...

Actinic keratosis (AK) lesions are surrounded by field cancerization (areas of subclinical, non-visible sun damage). Existing AK grading tools rely on AK counts, which are not reproducible. An Actinic Keratosis Field Assessment Scale (AK-FAS) for grading the severity of AK/field was developed. Standardized photographs of patients representing the full range of AK severity were collected. Six in...

Acanthamoeba keratitis (AK) is a sight-threating infection of the cornea that mostly affects contact lens wearers. Until now, AK treatment remains very difficult due to the existence of a highly resistant cyst stage in the life cycle of Acanthamoeba which is extremely resistant to most of the available anti-amoebic compounds. Moreover, current treatment of AK is usually based in the combination...