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We introduce several methods to define the self-inductance of a single loop as the regularization of divergent integrals obtained from Neumann and Weber formulae for the mutual inductance applied to a single loop. We give generalization of these two formulae in the sense of generalized functions and study the residues.

We generalize the Riesz potential of a compact domain in $\mathbb{R}^{m}$ by introducing a renormalization of the $r^{\alpha-m}$-potential for $\alpha\le0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume., Comment: Adv. Appl. Math. 48 (2012), 365--392 Figure 11 has been corrected after publication. Theorem 3.12 and the exposition of Lemma 2.15 are modified in version 8

We study an energy functional of knots, e p j ( jp > 2), that is finite valued for embedded circles and takes +∞ for circles with double points. We show that for any b ϵ R there are finitely many solid tori T 1 ,…, T m such that any knot with e p j ⩽ b can be contained in some T i in a good manner. Then we can show the existence of a minimizer of e p j in each knot type.

We define energy functionals on the space of embeddings from S1 into R 3 and show the finiteness of knot types under bounded values of those functionals.