On the critical points of harmonic functions

On the Critical Points of Harmonic Functions

Critical Points of Harmonic Functions on Domains In/?3

It is shown that the critical point relations of Morse theory, together with the maximum principle, comprise a complete set of critical point relations for harmonic functions of three variables. The proof proceeds by first constructing a simplified example and then developing techniques to modify this example to realize all admissible possibilities. Techniques used differ substantially from tho...

the effect of using critical discourse analytical tools on the improvement of the learners level of critical thinking in reading comprehension

?it is of utmost priority for an experienced teacher to train the mind of the students, and enable them to think critically and correctly. the most important question here is that how to develop such a crucial ability? this study examines a new way to the development of critical thinking utilizing critical discourse analytical tools. to attain this goal, two classes of senior english la...

Note on the Location of the Critical Points of Harmonic Functions.

THEOREM 1. Let the region R of the extended (x, y)-plane be bounded by a finite number of mutually disjoint Jordan curves Co, G., C2, • • • , Cn. Let the function u(x, y) be harmonic in R, continuous in the corresponding closed region, equal to zero on Co and to unity on G, C2, • • • , Cn. Denote by Ro the region bounded by Co containing the curves Ci, C2, • • • , Cn in its interior; define non...

Critical Points of Functions on Singular Spaces

We compare and contrast various notions of the “critical locus” of a complex analytic function on a singular space. After choosing a topological variant as our primary notion of the critical locus, we justify our choice by generalizing Lê and Saito’s result that constant Milnor number implies that Thom’s af condition is satisfied.