Related papers: Newman's theorem via Carath\'eodory
Muchnik's theorem about simple conditional descriptions states that for all strings $a$ and $b$ there exists a short program $p$ transforming $a$ to $b$ that has the least possible length and is simple conditional on $b$. In this paper we…
I show that a simple multi-party communication task can be performed more efficiently with quantum communication than with classical communication, even with low detection efficiency $\eta$. The task is a communication complexity problem in…
An technically interesting proof of a known theorem.
We present a dynamical approach to the classical Perron-Frobenius theory by using some elementary knowledge on linear ODEs. It is completely self-contained and significantly different from those in the literature. As a result, we develop a…
Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept…
Proofs of Tychonoff's theorem often seem to require a bit of magic. Machinery such as ultrafilters, nets or maximal families with the finite intersection property are employed to give proofs that can be very neat, but not the kind of thing…
In this paper the Neciporuk method for proving lower bounds on the size of Boolean formulae is reformulated in terms of one-way communication complexity. We investigate the scenarios of probabilistic formulae, nondeterministic formulae, and…
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
We study Cantor's powerset theorem from a graph-theoretic perspective, consider some alternative proofs to Cantor's original, and provide a new proof.
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic.…
Information complexity is the interactive analogue of Shannon's classical information theory. In recent years this field has emerged as a powerful tool for proving strong communication lower bounds, and for addressing some of the major open…
We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows: 1. We show that the classical adversary bound lifts to a lower bound on randomized communication…
Communication complexity problems (CCPs) are tasks in which separated parties attempt to compute a function whose inputs are distributed among the parties. Their communication is limited so that not all inputs can be sent. We show that…
Noether's theorem is an elegant and powerful tool of classical mechanics, but it is of little to no consequence in discrete theories. Here we define and explore a discrete approach to covariant mechanics and show that within this framework…
We give a short proof of the well-known Knuth's old sum and provide some generalizations. Our approach utilizes the binomial theorem and integration formulas derived using the Beta function. Several new polynomial identities and…
We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f:{0,1}^n->{0,1} and let A_f be the matrix whose columns are each an application of f to some…
We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared…
Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank,…
We derive a theory for transmission through disordered finite superlattices in which the interface roughness scattering is treated by disorder averaging. This procedure permits efficient calculation of the transmission thr ough samples with…
We give a formula for the Carath\'eodory distance on the Neil parabola, the variety ${z^2=w^3}$ restricted to the bidisk; thus making it the first variety with a singularity to have its Carath\'eodory distance explicitly computed. In…