# Potential theory with multivariate kernels

@article{Bilyk2022PotentialTW, title={Potential theory with multivariate kernels}, author={Dmitriy Bilyk and Damir Ferizovi'c and Alexey A. Glazyrin and Ryan Matzke and Josiah Park and O. V. Vlasiuk}, journal={Mathematische Zeitschrift}, year={2022} }

In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, n-tuples of particles. Such objects, which arise naturally in various fields, present subtle differences and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results…

## One Citation

### Positive definiteness and the Stolarsky invariance principle

- MathematicsJournal of Mathematical Analysis and Applications
- 2022

## References

SHOWING 1-10 OF 39 REFERENCES

### Positive definiteness and the Stolarsky invariance principle

- MathematicsJournal of Mathematical Analysis and Applications
- 2022

### Optimal Parseval frames: total coherence and total volume

- MathematicsLinear and Multilinear Algebra
- 2022

We introduce three quantities called total coherence, total volume, and nuclear energy, and we show that equiangular Parseval frames maximize all three of these quantities over the set of all…

### Optimal measures for $p$-frame energies on spheres

- MathematicsRevista Matemática Iberoamericana
- 2022

We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel…

### Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling

- Mathematics
- 2015

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.-…

### The Cauchy integral, analytic capacity, and uniform rectifiability

- Mathematics
- 1996

Several explanations concerning notation, terminology, and background are in order. First notation: by 7Hi we have denoted the one-dimensional Hausdorff measure (i.e. length), and A(z,r) stands for…

### Equations aux Dérivées Partielles et Applications

- Mathematics
- 2012

Le cours a porté sur une classe générale d’équations aux dérivées partielles stochastiques, du second ordre, fortement non linéaires et éventuellement dégénérées, de la forme (1) ∂u = F (x, t, ω, u,…

### On the sum of distances determined by a pointset

- Mathematics
- 1956

where d denotes the diameter of the circumcircle of the points. The constant cannot be replaced by a greater one. In order to prove our theorem we denote the points by P~ . . . . ,P~,Fixing the…