We introduce a new method of proving lower bounds on the depth of algebraic d-degree decision (resp. computation) trees and apply it to prove a lower bound (log N) (resp. (log N= log log N)) for testing membership to an n-dimensional convex polyhedron having N faces of all dimensions, provided that N > (nd) (n) (resp. N > n (n)). This bound apparently does not follow from the methods developed ...

In the standard New Keynesian model, monetary policy is often described by an interest rate rule (e.g. a Taylor rule) that moves the interest rate in response to deviations of inflation and some measure of economic activity from target. Nominal interest rates are bound from below by 0 – since money is storable, one would never accept a negative nominal return. How does the behavior of the NK mo...

Let A and B be two finite subsets of a field F. In this paper we provide a nontrivial lower bound for |{a + b: a ∈ A, b ∈ B, and P (a, b) 6= 0}| where P (x, y) ∈ F[x, y].

It is proved that there is a monotone function in AC(0) 4 which requires exponential size monotone perceptrons of depth 3. This solves the monotone version of a problem which, in the general case, would imply an oracle separation of PPPH.

We will be interested in Boolean functions over the Boolean cube {0, 1} and their polynomial approximations. Razborov’s method for approximating an AC0[p] circuit produces a polynomial over Zp (in fact, this generalizes to prime powers); Smolensky’s lower bounds work for any field F. For the rest of the writeup, we will assume all polynomials are over some fixed field F, unless explicitly noted...

A matrix lower bound is defined that generalizes ideas apparently due to S. Banach and J. von Neumann. The matrix lower bound has a natural interpretation in functional analysis, and it satisfies many of the properties that von Neumann stated for it in a restricted case. Applications for the matrix lower bound are demonstrated in several areas. In linear algebra, the matrix lower bound of a ful...

Motivated by Candes and Donoho′s work (Candés, E J, Donoho, D L, Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Stat. 30, 784-842 (2002)), this paper is devoted to giving a lower bound of minimax mean square errors for Riesz fractional integration transforms and Bessel transforms.