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    Algebra Structures on Hom(C,L)

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    We consider the space of linear maps from a coassociative coalgebra C into a Lie algebra L. Unless C has a cocommutative coproduct, the usual symmetry properties of the induced bracket on Hom(C,L) fail to hold. We define the concept of twisted domain (TD) algebras in order to recover the symmetries and also construct a modified Chevalley-Eilenberg complex in order to define the cohomology of such algebras

    Improved Constructions of Frameproof Codes

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    Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let Mc,l(q)M_{c,l}(q) be the largest cardinality of a qq-ary cc-frameproof code of length ll and Rc,l=lim⁑qβ†’βˆžMc,l(q)/q⌈l/cβŒ‰R_{c,l}=\lim_{q\rightarrow \infty}M_{c,l}(q)/q^{\lceil l/c\rceil}. It has been determined by Blackburn that Rc,l=1R_{c,l}=1 when l≑1Β (β€Šmodβ€ŠΒ c)l\equiv 1\ (\bmod\ c), Rc,l=2R_{c,l}=2 when c=2c=2 and ll is even, and R3,5=5/3R_{3,5}=5/3. In this paper, we give a recursive construction for cc-frameproof codes of length ll with respect to the alphabet size qq. As applications of this construction, we establish the existence results for qq-ary cc-frameproof codes of length c+2c+2 and size c+2c(qβˆ’1)2+1\frac{c+2}{c}(q-1)^2+1 for all odd qq when c=2c=2 and for all q≑4(mod6)q\equiv 4\pmod{6} when c=3c=3. Furthermore, we show that Rc,c+2=(c+2)/cR_{c,c+2}=(c+2)/c meeting the upper bound given by Blackburn, for all integers cc such that c+1c+1 is a prime power.Comment: 6 pages, to appear in Information Theory, IEEE Transactions o

    Bond percolation of polymers

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    We study bond percolation of NN non-interacting Gaussian polymers of β„“\ell segments on a 2D square lattice of size LL with reflecting boundaries. Through simulations, we find the fraction of configurations displaying {\em no} connected cluster which span from one edge to the opposite edge. From this fraction, we define a critical segment density ρcL(β„“)\rho_{c}^L(\ell) and the associated critical fraction of occupied bonds pcL(β„“)p_{c}^L(\ell), so that they can be identified as the percolation threshold in the Lβ†’βˆžL \to \infty limit. Whereas pcL(β„“)p_{c}^L(\ell) is found to decrease monotonically with β„“\ell for a wide range of polymer lengths, ρcL(β„“)\rho_{c}^L(\ell) is non-monotonic. We give physical arguments for this intriguing behavior in terms of the competing effects of multiple bond occupancies and polymerization.Comment: 4 pages with 6 figure
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