Related papers: Seven open problems in applied combinatorics
The concept of open weak CAD is introduced. Every open CAD is an open weak CAD. On the contrary, an open weak CAD is not necessarily an open CAD. An algorithm for computing projection polynomials of open weak CADs is proposed. The key idea…
Quantum computing is the process of performing calculations using quantum mechanics. This field studies the quantum behavior of certain subatomic particles for subsequent use in performing calculations, as well as for large-scale…
This article explores the current state and future prospects of quantum computing in industrial environments. Firstly, it describes three main paradigms in this field of knowledge: gate-based quantum computers, quantum annealers, and tensor…
We discuss general models of resource-sharing computations, with emphasis on the combinatorial structures and concepts that underlie the various deadlock models that have been proposed, the design of algorithms and deadlock-handling…
I offer a case that quantum query complexity still has loads of enticing and fundamental open problems -- from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element distinctness, to polynomial…
Combinatorial problems are formulated to find optimal designs within a fixed set of constraints. They are commonly found across diverse engineering and scientific domains. Understanding how to best use quantum computers for combinatorial…
Three open problems in analysis are formulated: 1. Injectivity of Radon transform 2. A uniqueness problem for an elliptic system of PDE 3. A problem in operator theory
A selection of open problems in the theory of composites is presented. Particular attention is drawn to the question of whether two-dimensional, two-phase, composites with general geometries have the same set of possible effective tensors…
In this short review, I will summarize my research experience in three fields in applied mathematics: mathematical biology, applied probability, and applied discrete mathematics. Specifically, I will show how each project was initiated, and…
We consider algorithmic problems in the setting in which the input data has been partitioned arbitrarily on many servers. The goal is to compute a function of all the data, and the bottleneck is the communication used by the algorithm. We…
We survey results on the hardness of approximating combinatorial optimization problems.
In this note we list a number of open problems in the fields of number theory, combinatorics, and representation theory: algebraic functions with Fermat property; power product expansion of the generating function for the partition…
Quantum computing and modern tensor-based computing have a strong connection, which is especially demonstrated by simulating quantum computations with tensor networks. The other direction is less studied: quantum computing is not often…
We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference.
This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose…
Motivated by the fact that not all nonconvex optimization problems are difficult to solve, we survey in this paper three widely-used ways to reveal the hidden convex structure for different classes of nonconvex optimization problems.…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
Given a unitary operator in a finite dimensional complex Hilbert space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…