G08CDF also returns the standardized statistic Z 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 þ n2 n1n2 r D, where D may be Dn1;n2 , D þ n1;n2 or D n1;n2 depending on the choice of the alternative hypothesis. The distribution of this statistic converges asymptotically to a distribution given by Smirnov as n1 and n2 increase; see Feller (1948), Kendall and Stuart (1973), Kim and Jenri...

We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Ω(n1/6), and that any O(m)-size shortcut set cannot bring it below Ω(n1/11). These improve Hesse’s [15] lower bou...

Motivated by the Erdős multiplication table problem we study the following question: Given numbers N1, . . . , Nk+1, how many distinct products of the form n1 · · ·nk+1 with 1 ≤ ni ≤ Ni for i ∈ {1, . . . , k + 1} are there? Call Ak+1(N1, . . . , Nk+1) the quantity in question. Ford established the order of magnitude of A2(N1, N2) and the author of Ak+1(N, . . . , N) for all k ≥ 2. In the presen...

We calculate the asymptotic value of the choice number of complete multi-partite graphs, given certain limitations on the relation between the sizes of the different sides. In the bipartite case, we prove that if n0 ≤ n1 and log n0 ≫ log log n1, then ch(Kn0,n1) = (1 + o(1)) log2 n1 log2 x0 , where x0 is the unique root of the equation x − 1 − x k−1 k = 0 in the interval [1,∞) and k = log2 n1 lo...

Image processing is a science that uncovers information about images. Enhancement of an image is necessary to improve appearance or to highlight some aspect of the information contained in the image. Whenever an image is converted from one form to another, e.g., acquired, copied, scanned, digitized, transmitted, displayed, printed, or compressed, many types of noise or noiselike degradations ca...

Tumour, node, metastasis staging is essential for lung cancer management. However, similarly staged cancers may have markedly different prognoses, indicating that stage cannot completely explain tumour behaviour. While ipsilateral hilar node involvement is designated N1, the current authors hypothesised that primary tumours involving nodes by direct extension are biologically distinct from thos...

Multidrug resistance (MDR) is a barrier to successful cancer chemotherapy. Although MDR is associated with overexpression of ATP-binding cassette (ABC) membrane transporters, mechanisms behind their up-regulation are not entirely understood. The cleaved form of the Notch1 protein, intracellular Notch1 (N1(IC)), is involved in transcriptional regulation of genes. To test whether Notch1 is involv...

The neural cell-specific N1 exon of the c-src pre-mRNA is both negatively regulated in nonneural cells and positively regulated in neurons. We previously identified conserved intronic elements flanking N1 that direct the repression of N1 splicing in a nonneural HeLa cell extract. The upstream repressor elements are located within the polypyrimidine tract of the N1 exon 3' splice site. A short R...

Humans have two nearly identical copies of the Survival Motor Neuron (SMN) gene, SMN1 and SMN2. In spinal muscular atrophy (SMA), SMN2 is not able to compensate for the loss of SMN1 due to exclusion of exon 7. Here we describe a novel inhibitory element located immediately downstream of the 5' splice site in intron 7. We call this element intronic splicing silencer N1 (ISS-N1). Deletion of ISS-...