We explore the degrees of freedom (DoF) of the finite state compound multiple-input single-output (MISO) broadcast channel (BC) in this paper. In prior work, Weingarten et. al. have found inner and outer bounds on the DoF with 2 users. The bounds have a different character. While the inner bound collapses to unity as the number of states increases, the outer bound does not diminish with the inc...

We work on an algebraically closed field of characteristic zero. By Lefschetz’s theorem, a smooth codimension two subvariety X ⊂ P, n ≥ 4, which is not a complete intersection, lying on a hypersurface Σ, verifies dim(X ∩ Sing(Σ)) ≥ n− 4. In this paper we deal with a situation in which the singular locus of Σ is as large as can be, but, at the same time, the simplest possible: we assume Σ is an ...

We prove that the determinantal complexity of a hypersurface of degree d > 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3×3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in Pn of degree d that has an expression as a dete...

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface Z in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the se...

We consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this form...

We describe a new method for the fitting of differentiable fuzzy model functions to crisp data. The model functions can be either scalar or multidimensional and need not be linear. The data are n-component vectors. An efficient algorithm is achieved by restricting the fuzzy model functions to sets which depend on a fuzzy parameter vector and assuming that the vector has a conical membership fun...