We prove an (lg 2 n= lg lg n) lower bound for the depth of n-input sorting networks based on the shuue permutation. The best previously known lower bound was the trivial (lg n) bound, while the best upper bound is given by Batcher's (lg 2 n)-depth bitonic sorting network. The proof technique employed in the lower bound argument may be of independent interest.

This paper provides upper and lower bounds on the information rates achievable with a minimum mean square error (MMSE) Tomlinson-Harashima precoder (THP) assuming ideal interleaving. We also give an exact formula for zero-forcing (ZF) THP achievable information rates. At high SNR, the performance of the MMSE-THP and the ZF-THP identically suuer only the 2.55 bit or 1.53 dB \shaping" loss from c...

Recent work by Bernasconi, Damm and Shparlinski showed that the set of square-free numbers is not in AC, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this note, we show that the Boolean majority function is AC-Turing reducible to the set of prime numbers (represented in binary). From known lower bounds on Maj (due to Razbo...

It is established a linear (thereby, sharp) lower bound on degrees of Positivstellensatz calculus refutations over a real eld introduced in GV99], for the Tseitin tautologies and for the parity (the mod 2 principle). We use the machinery of the Laurent proofs developped for binomial systems in BuGI 98], BuGI 99].

Definition 2.1. Let f : X × Y → V . A subset R of X × Y is a rectangle1 if it is of the form A × B for some A ⊆ X and B ⊆ Y . The rectangle R is said to be monochromatic (wrt. f) if f is constant on R. A monochromatic rectangle R is a 0-rectangle if f(R) = {0}; it is a 1-rectangle if f(R) = {1}. Observation 2.2. A subset S of X × Y is a rectangle iff for all x, x′ ∈ X and y, y′ ∈ Y (x, x′) ∈ S ...

:= F̂ − KL (q(X|Z) ‖ p(X)) . At this point, our variational bound is similar to the one of equation (7), but the first term, here denoted as F̂ , refers to the expanded probability space and, thus, involves the inducing inputs and the additional variational distribution q(U). Since the second term (the KL term) is tractable (because it only involves Gaussian distributions), we are now going to fo...

Let 〈x1, x2, ..., xm〉 be the access sequence. For each access xj compute the following Wilber number. We look at where xj fits among xi, xi+1, ..., xj−1 for all i = j−1, j−2... that is, counting backwards from j − 1 until the previous access to the key xj. Now, we say that ai < xj < bi, where ai and bi are the tightest bounds on xj discovered so far. Each time i is decremented, either ai increa...

Decision Diagrams (DDs) are a data structure for the representation and manipulation of discrete logic functions often applied in VLSI CAD. Common DDs to represent Boolean functions are Binary Decision Diagrams (BDDs). Multiple-valued logic functions can be represented by Multiple-valued Decision Diagrams (MDDs). The efficiency of a DD representation strongly depends on the variable ordering; t...

We show a lower estimate of the Milnor number of an isolated hypersurface singularity, via its Newton number. We also obtain analogous estimate of the Milnor number of an isolated singularity of a similar complete intersection variety. Introduction We study the Newton number of a polyhedron in order to calculate the Milnor number of an isolated singularity defined by an analytic mapping. Sectio...