Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.

Let A„{a, b) = {ban+(a-l)/b)2+4an with n > 1 and ¿>|a-l . If W is a finite set of primes such that for each n > 1 there exists some q £W for which the Legendre symbol {A„{a, b)/q) ^ -1 , we call <£ a quadratic residue cover (QRC) for the quadratic fields K„{a, b) = Q{^jA„{a, b)). It is shown how the existence of a QRC for any a, b can be used to determine lower bounds on the class number of K„{...

Finding the shortest path in a directed graph is one of the most important combinatorial optimization problems, having applications in a wide range of fields. In its basic version, however, the problem fails to represent situations in which the value of the objective function is determined not only by the choice of each single arc, but also by the combined presence of pairs of arcs in the solut...

Building on evidence from neurobiology and neuroscience, we model the physiological limitations faced by individuals in the process of decision-making that starts with sensory perception and ends in action selection. The brain sets a neuronal threshold, observes whether the neuronal cell firing activity reaches the threshold or not, and takes the optimal action conditional on that (limited) inf...

By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 0-1 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the primal and corresponding dual problems. Both global and local optimality conditions are given. An algorithm is p...

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly...

The support vector machine (SVM) is a popular classifier in machine learning, but it is not robust to outliers. In this paper, based on the Correntropy induced loss function, we propose the rescaled hinge loss function which is a monotonic, bounded and nonconvex loss that is robust to outliers. We further show that the hinge loss is a special case of the proposed rescaled hinge loss. Then, we d...

Positive unlabeled learning (PU learning) refers to the task of learning a binary classifier from only positive and unlabeled data [1]. This problem arises in various practical applications, like in multimedia/information retrieval [2], where the goal is to find samples in an unlabeled data set that are similar to the samples provided by a user, as well as for applications of outlier detection ...

An iteration of the sequential quadratically constrained quadratic programming method (SQCQP) consists of minimizing a quadratic approximation of the objective function subject to quadratic approximation of the constraints, followed by a linesearch in the obtained direction. Methods of this class are receiving attention due to the development of efficient interior point techniques for solving s...