For a system of two parties, the process matrix framework predicts existence causally non-separable structures. Our objective is to provide general condition for identifying such indefinite causal order in terms mutual information exchange. In detail, we show that there are processes which total entropy parties greater than one any separable with same non-signaling part. Furthermore, this not s...

v ∈ ∂f(w) ⇐⇒ ∃wn → w, f(wn)→ f(w),vn ∈ ∂̂f(wn),vn → v. Clearly, ∂̂f(w) ⊆ ∂f(w) for all w. If f is (resp. continuously) differentiable at w, then ∂̂f(w) (resp. ∂f(w)) coincides with the usual derivative. From the definition it follows that if w is a local minimizer, then 0 ∈ ∂̂f(w) and 0 ∈ ∂f(w), which generalizes the familiar Fermat’s rule. In the main text, we are interested in finding some w so t...

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in R and R. We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classic box-spline direction matrices. These n...