Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird

Kirchhoff, G.

doi:10.1002/andp.18471481202

Cited by 1,167 publications

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“…The theory of graph Laplacians goes back to Kirchhoff [22] in his study of electrical networks, and his celebrated matrix-tree theorem (see, e.g., [21]). The spectra of graph Laplacians gained attention in the late 1960s through the independent work of Anderson and Morley [2], Fiedler [13], and Kelmans (see references in [20]).…”

Section: Introductionmentioning

confidence: 99%

Shifted simplicial complexes are Laplacian integral

Duval

Reiner

2002

Trans. Amer. Math. Soc.

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Abstract. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs.We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.

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Section: Introductionmentioning

confidence: 99%

Shifted simplicial complexes are Laplacian integral

Duval

Reiner

2002

Trans. Amer. Math. Soc.

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“…This construction In 1847, Kirchhoff [26] proved that given a unit electric current between the endpoints of an edge e in a finite graph, the current flowing through e equals the (counting)…”

Section: Proof Of Proposition 24mentioning

confidence: 99%

“…[13], p. 248). Therefore, Kirchhoff's Nonunitarizable Representations and Random Forests 4345 characterization [26] in terms of the uniform spanning tree shows i n (e) = π 2 (g n ) (χ e )(e).…”

Section: Proof Of Proposition 24mentioning

confidence: 99%

Nonunitarizable Representations and Random Forests

Epstein

Monod

2009

International Mathematics Research Notices

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Correspondence to be sent to: nicolas.monod@epfl.ch We establish a connection between Dixmier's unitarizability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is nonunitarizable if its first L 2 -Betti number is nonzero or if it is finitely generated with nontrivial cost. Our criterion also applies to torsion groups constructed by Osin, thus providing the first examples of nonunitarizable groups without free subgroups.

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“…In fact the parametric representation follows from the tree matrix theorem of Kirchoff [98], which is a key result of combinatorial theory which in its simplest form may be stated as: Theorem 8.1 (Tree Matrix Theorem). Let A be an n by n matrix such that…”

Section: Ordinary Symanzik Polynomialsmentioning

confidence: 99%

Non-commutative Renormalization

Rivasseau

2007

Quantum Spaces

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Abstract. A new version of scale analysis and renormalization theory has been found on the non-commutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external field. The good news is that quantum field theory is better behaved on non-commutative than on ordinary space: indeed it has no Landau ghost. Noncommutativity might therefore be an alternative to supersymmetry. We review this rapidly growing subject.

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Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird

Cited by 1,167 publications

References 0 publications

Shifted simplicial complexes are Laplacian integral

Shifted simplicial complexes are Laplacian integral

Nonunitarizable Representations and Random Forests

Non-commutative Renormalization

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