Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Efficient algorithms for learning kernels from multiple similarity matrices with general convex loss functions

In this paper we consider the problem of learning an n × n kernel matrix from m(1) similarity matrices under general convex loss. Past research have extensively studied the m = 1 case and have derived several algorithms which require sophisticated techniques like ACCP, SOCP, etc. The existing algorithms do not apply if one uses arbitrary losses and often can not handle m > 1 case. We present several provably convergent iterative algorithms, where each iteration requires either an SVM or a Multiple Kernel Learning (MKL) solver for m > 1 case. One of the major contributions of the paper is to extend the well knownMirror Descent(MD) framework to handle Cartesian product of psd matrices. This novel extension leads to an algorithm, called EMKL, which solves the problem in O(m2 log n 2) iterations; in each iteration one solves an MKL involving m kernels and m eigen-decomposition of n × n matrices. By suitably defining a restriction on the objective function, a faster version of EMKL is proposed, called REKL,which avoids the eigen-decomposition. An alternative to both EMKL and REKL is also suggested which requires only an SVMsolver. Experimental results on real world protein data set involving several similarity matrices illustrate the efficacy of the proposed algorithms.

Quadratic Lyapunov functions for linear systems Journal

The paper deals with the existence of a quadratic Lyapunov function V = x′P(t)x for an exponentially stable linear system with varying coefficients described by the vector differential equation S0305004100044777_inline1 The derivative dV/dt is allowed to be strictly semi-(F) and the locus dV/dt = 0 does not contain any arc of the system trajectory. It is then shown that the coefficient matrix A(t) of the exponentially stable sy

on quadratic bent functions in polynomial forms

In this correspondence, we construct some new quadratic bent functions in polynomial forms by using the theory of quadratic forms over finite fields. The results improve some previous work. Moreover, we solve a problem left by Yu and Gong in 2006.

On Learning Counting Functions With Queries

We investigate the problem of learning disjunctions of counting functions, which are general cases of parity and modulo functions, with equivalence and membership queries. We prove that, for any prime number p, the class of disjunctions of integer-weighted counting functions with modulus p over the domain Znq (or Zn) for any given integer q ≥ 2 is polynomial time learnable using at most n + 1 equivalence queries, where the hypotheses issued by the learner are disjunctions of at most n counting functions with weights from Zp. The result is obtained through learning linear systems over an arbitrary field. In general a counting function may have a composite modulus. We prove that, for any given integer q ≥ 2, over the domain Zn2, the class of read-once disjunctions of Boolean-weighted counting functions with modulus q is polynomial time learnable with only one equivalence query, and the class of disjunctions of log log n Boolean-weighted counting functions with modulus q is polynomial time learnable. Finally, we present an algorithm for learning graph-based counting functions.

[Book review] Collaborative learning in mathematics

Review of: Collaborative Learning in Mathematics: A challenge to our beliefs and practices by Malcolm Swan, National Institute of Adult Continuing Education, paperback £24.95, ISBN 981-1-86201-311-7; hardback £44.95, ISBN 978-1-86201-316-2.

Some issues on assessment methods and learning in mathematics and statistics

This paper looks at the application of some of the assessment methods in practice with the view to enhance students’ learning in mathematics and statistics. It explores the effective application of assessment methods and highlights the issues or problems, and ways of avoiding them, related to some of the common methods of assessing mathematical and statistical learning. Some observations made by the author on good assessment practice and useful approaches employed at his institution in designing and applying assessment methods are discussed. Successful strategies in implementing assessment methods at different levels are described.