A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably) equivalent sumnetwork for any multiple-unicast network, and thus for any directed acyclic communication network. It is also shown that there exists a linear solvably equivalent multiple-unicast network for every sum-network. It is shown that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F . For any finite or cofinite set of prime numbers, a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set. The insufficiency of linear network coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks. Under fractional vector linear network coding, a sum-network and its reverse network are shown to be equivalent. However, under non-linear coding, it is shown that there exists a solvable sum-network whose reverse network is not solvable.

We study a gauge theory model where there is an intermediate symmetry breaking to a metastable vacuum that breaks a simple gauge group to a U (1) factor. Such models admit the existence of meta-stable magnetic monopoles, which we dub false monopoles. We prove the existence of these monopoles in the thin wall approximation. We determine the instantons for the collective coordinate that corresponds to the radius of the monopole wall and we calculate the semi-classical tunneling rate for the decay of these monopoles. The monopole decay consequently triggers the decay of the false vacuum. As the monopole mass is increased, we find an enhanced rate of decay of the false vacuum relative to the celebrated homogeneous tunneling rate due to Coleman [1].

Abstract-We consider the problem of communicating the sum of m sources to n terminals in a directed acyclic network. Recently, it was shown that for a network of unit capacity links with either m = 2 or n = 2, the sum of the sources can be communicated to the terminals if and only if every source-terminal pair is connected in the network. We show in this paper that for any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set. As a corollary, this gives networks where the sum can not be communicated over any finite field even though every source is connected to every terminal.