Nick Lord

Bio: Nick Lord is an academic researcher from Tonbridge School. The author has contributed to research in topics: Euler's formula & Product (mathematics). The author has an hindex of 12, co-authored 176 publications receiving 1156 citations.

Linear algebra problem book, by Paul R. Halmos. Pp. 335. $35. 1995. ISBN 0-88385-322-1 (Mathematical Association of America).

Abstract: 1. Scalars 2. Vectors 3. Bases 4.Transformations 5. Duality 6. Similarity 7. Canonical forms 8. Inner product spaces 9. Normality 10. Hints and solutions.

Quantum graphs: I. Some basic structures

Abstract: A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.

Estimating groundwater recharge

Abstract: Understanding groundwater recharge is essential for successful management of water resources and modeling fluid and contaminant transport within the subsurface. This book provides a critical evaluation of the theory and assumptions that underlie methods for estimating rates of groundwater recharge. Detailed explanations of the methods are provided - allowing readers to apply many of the techniques themselves without needing to consult additional references. Numerous practical examples highlight benefits and limitations of each method. Approximately 900 references allow advanced practitioners to pursue additional information on any method. For the first time, theoretical and practical considerations for selecting and applying methods for estimating groundwater recharge are covered in a single volume with uniform presentation. Hydrogeologists, water-resource specialists, civil and agricultural engineers, Earth and environmental scientists and agronomists will benefit from this informative and practical book. It can serve as the primary text for a graduate-level course on groundwater recharge or as an adjunct text for courses on groundwater hydrology or hydrogeology. For the benefit of students and instructors, problem sets of varying difficulty are available at http://wwwbrr.cr.usgs.gov/projects/GW_Unsat/Recharge_Book/

Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents

Abstract: We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: ( −4pu = λ|u|r−2u+ μ |u| q−2 |x|s u in Ω, u|∂Ω = 0, where λ and μ are two positive parameters and Ω is a smooth bounded domain in Rn containing 0 in its interior. The variational approach requires that 1 < p < n, p ≤ q ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ p ∗ ≡ p∗(0) = np n−p , which we assume throughout. However, the situations differ widely with q and r, and the interesting cases occur either at the critical Sobolev exponent (r = p∗) or in the Hardy-critical setting (s = p = q) or in the more general Hardy-Sobolev setting when q = n−s n−pp. In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case p = 2, especially those corresponding to singularities (i.e., when 0 < s ≤ p).

Measure and Integral

Abstract: Mass und Integral und ihre Algebraisierung Von Prof C Caratheodory Herausgegeben von P Finsler, A Rosenthal und R Steuerwald (Lehrbucher und Monographien aus dem Gebiete der Exakten Wissenschaften Mathematische Reihe, Band 10) Pp 337 (Basel und Stuttgart: Birkhauser Verlag, 1956) 3850 francs; 3850 DM

On the numerical integration of ordinary differential equations by symmetric composition methods

Abstract: Differential equations of the form $\dot x = X = A + B$ are considered, where the vector fields A and B can be integrated exactly, enabling numerical integration of X by composition of the flows of A and B. Various symmetric compositions are investigated for order, complexity, and reversibility. Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order. A new, more accurate way of applying the methods thus obtained to compositions of an arbitrary first-order integrator is described and tested. The determining equations are explored, and new methods up to 100 times more accurate (at constant work) than those previously known are given.