This paper presents a new technique for deterministic length reduction. This technique improves the running time of the algorithm presented in [?] for performing fast convolution in sparse data. While the regular fast convolution of vectors V1, V2 whose sizes are N1, N2 respectively, takes O(N1 logN2) using FFT, using the new technique for length reduction, the algorithm proposed in [?] perform...

a. N1 = {(i1, i2, . . . , in) : ik ∈ Ik for all k ∈ {1, 2, . . . , n}} and b. N2 = {(x1, x2, . . . , xn) : ∑n k=1 xk = 0}. Proof. To prove (a), it suffices to show, by Proposition 1, that N1 is nonempty and x + ry ∈ N1 for all r ∈ R and all x, y ∈ N1. For the first condition, (0, 0, . . . , 0) ∈ N1 since Ik is a subgroup of R containing the additive identity 0 for all k ∈ {1, 2, . . . , n}. Tha...

These are the notes I typed during the talks. I haven’t spent much time after the lectures cleaning them up so there are probably lots of typos and even some more serious errors. The hardest thing was getting the quiver diagrams done in real time. I often had to give up! Anyway, there may be some value to this so I’m placing it in the public domain. feedback is welcome. 1. King—Moduli of sheave...

Worldwide, only two types of antiviral inhibitors (M2 ion channel protein inhibitor and Neuraminidase inhibitors) are approved to treat the influenza viral infection. But mutation amino acid sequence in membrane proteins creates resistance existing drugs or inhibitors. So corresponding have be reformulated match these antigenic variations. Fluorination on carbon–based molecule significantly enr...

Let G1 and G2 be simple graphs and let n1 = |V (G1)|, m1 = |E(G1)|, n2 = |V (G2)| and m2 = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G1 G2 of G1 and G2. We show that: τ(G1 G2) ≥ 2(n1−1)(n2−1) n1n2 (τ(G1)n1) n2+1 2 (τ(G2)n2) n1+1 2 and τ(G1 G2) ≤ τ(G1)τ(G2) [ 2m1 n1 − 1 + 2m2 n2 − 1 ](n1−1)(n2−1) . We also characteri...

Vaccinia virus (VACV) protein N1 is an intracellular virulence factor and belongs to a family of VACV B-cell lymphoma (Bcl)-2-like proteins whose members inhibit apoptosis or activation of pro-inflammatory transcription factors, such as interferon (IFN) regulatory factor-3 (IRF-3) and nuclear factor-κB (NF-κB). Unusually, N1 inhibits both apoptosis and NF-κB activation. To understand how N1 exe...