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Decomposable mappings from the space of symmetric$k$-fold tensors over$E$, $\bigotimes_{s,k}E$ , to the space of$k$-fold tensors over$F$, $\bigotimes_{s,k}F$ , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.

Redundant via insertion is highly recommended for improving chip yield and reliability. In this paper, we study the problem of double via insertion with wire bending (DVI/WB) in a post-routing stage, where a single via can have at most one redundant via inserted next to it. Aside from this, we are allowed to bend existing signal wires for enhancing the insertion rate of double vias. The goal of DVI/WB is to primarily insert as many double vias as possible and to minimize the amount of layout perturbation as the secondary objective. We formulate the DVI/WB problem as that of finding a minimum-weight maximum independent set (mWMIS) on an enhanced conflict graph. We propose algorithms to perform wire bending and to construct the enhanced conflict graph from a given design. Moreover, we also propose a zero-one integer linear program (0-1 ILP) based approach to solve mWMIS. Experimental results show that

No abstract uploaded!

No abstract uploaded!