Recent efficient pairings such as Ate pairing use two efficient rational point subgroups such that π(P ) = P and π(Q) = [p]Q, where π, p, P , and Q are the Frobenius map for rational point, the characteristic of definition field, and torsion points for pairing, respectively. This relation accelerates not only pairing but also pairing–related operations such as scalar multiplications. It holds i...

We consider the first passage percolation model on Z2. In this model, we assign independently to each edge e a passage time t(e) with a common distribution F . Let T (u, v) be the passage time from u to v. In this paper, we show that, whenever F (0) < pc, σ 2 (T ((0, 0), (n, 0))) ≥ C log n for all n ≥ 1. Note that if F satisfies an additional special condition, infsupp(F ) = r > 0 and F (r) > ~...

We prove a number of results related to problem Po-Shen Loh, which is equivalent in Ramsey theory. Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ be two triples integers. Define $a$ 2-less than $b$ if $a_i<b_i$ for at least values $i$, define sequence $a^1,\dots,a^m$ 2-increasing $a^r$ $a^s$ whenever $r<s$. Loh asks how long can all the take $\{1,2,\dots,n\}$, gives $\log_*$ improvement over trivi...

This paper studies the vanishing of $Ext$ modules over group rings. Let $R$ be a commutative noetherian ring and $ga$ a group. We provide a criterion under which the vanishing of self extensions of a finitely generated $Rga$-module $M$ forces it to be projective. Using this result, it is shown that $Rga$ satisfies the Auslander-Reiten conjecture, whenever $R$ has finite global dimension and $ga...

Let k and n be positive integers, n > k. Define r(n, k) to be the minimum positive value of | √ a1 + · · ·+ √ ak − √ b1 − · · · − √ bk| where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. Define R(n, k) to be − log r(n, k). It is important to find tight bounds for r(n, k) and R(n, k), in connection to the sum-of-square-roots problem, a famous open problem in com...

Let d ≥ 1, and let u, v : R × R → C be Schwartz space solutions to the Schrödinger equation iut + ∆xu = 0; ivt + ∆xv = 0. The standard L bilinear Strichartz estimate for the Schrödinger equation asserts that ‖uv‖L2t,x(R×Rd) .d M N‖u(0)‖L2x(Rd)‖v(0)‖L2x(Rd) whenever u, v are supported in frequency regions {ξ ∈ R : N ≤ |ξ| ≤ 2N}, {η ∈ R : M ≤ |η| ≤ 2M}, and N ≥ 4M . This estimate has been used ex...

Let $R$ be a commutative ring with $1 \ne 0$ and let $m$ and&#x0D; $n$ integers $1\leq n &lt; m$. A proper ideal $I$ of is&#x0D; called an $(m, n)$-closed if whenever $a^m \in I$ for&#x0D; some $a\in R$ implies $a^n I$. $ f:A\rightarrow B$ a&#x0D; homomorphism $J$ $B.$ This paper&#x0D; investigates the concept $(m,n)$-closed ideals in the&#x0D; amalgamation $A$ $B$ along respect $f$ denoted by&...

In this paper we revisit the classical regular expression matching problem, namely, given a regular expression R and a string Q, decide if Q matches one of the strings specified by R. Let m and n be the length of R and Q, respectively. On a standard unit-cost RAM with word length w ≥ log n, we show that the problem can be solved in O(m) space with the following running times:    O(nm log w...