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CauchyBinet for pseudo determinants
, 2013
"... Abstract. The pseudodeterminant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basisindependent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We extend here the CauchyBinet formula to pseudodeterminants ..."
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Abstract. The pseudodeterminant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basisindependent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We extend here the CauchyBinet formula to pseudodeterminants. More specifically, after proving some properties for pseudodeterminants, we show that for any two n × m matrices F, G, the formula Det(F T G) = P det(FP)det(GP) holds, where det(FP) runs over all k×k minors of A with k = min(rank(F T G), rank(GF T)). A consequence is the following Pythagoras theorem: for any selfadjoint matrix A of rank k one has Det 2 (A) = ∑ P det2 (AP), where det(AP) runs over all k × k minors of A. 1.
CLASSICAL MATHEMATICAL STRUCTURES WITHIN TOPOLOGICAL GRAPH THEORY
"... Abstract. Finite simple graphs are a playground for classical areas of mathematics. We illustrate this by looking at some results. 1. ..."
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Abstract. Finite simple graphs are a playground for classical areas of mathematics. We illustrate this by looking at some results. 1.
ISOSPECTRAL DEFORMATIONS OF THE DIRAC OPERATOR
"... Abstract. We give more details about an integrable system [26] in which the Dirac operator D = d + d ∗ on a graph G or manifold M is deformed using a Hamiltonian system D ′ = [B, h(D)] with B = d − d ∗ + βib. The deformed operator D(t) = d(t) + b(t) + d(t) ∗ defines a new exterior derivative d(t) ..."
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Abstract. We give more details about an integrable system [26] in which the Dirac operator D = d + d ∗ on a graph G or manifold M is deformed using a Hamiltonian system D ′ = [B, h(D)] with B = d − d ∗ + βib. The deformed operator D(t) = d(t) + b(t) + d(t) ∗ defines a new exterior derivative d(t) and
IF ARCHIMEDES WOULD HAVE KNOWN FUNCTIONS...
"... Abstract. Could calculus on graphs have emerged by the time of Archimedes, if function, graph theory and matrix concepts were available 2300 years ago? 1. Single variable calculus Calculus on integers deals with functions f(x) like f(x) = x2. The difference Df(x) = f ′(x) = f(x+1)−f(x) = 2x+1 as ..."
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Abstract. Could calculus on graphs have emerged by the time of Archimedes, if function, graph theory and matrix concepts were available 2300 years ago? 1. Single variable calculus Calculus on integers deals with functions f(x) like f(x) = x2. The difference Df(x) = f ′(x) = f(x+1)−f(x) = 2x+1 as well as the sum Sf(x) =∑x−1k=0 f(k) with the understanding Sf(0) = 0, Sf(−1) = f(−1) are functions again. We call Df the derivative and Sf the integral. The identities DSf(x) = f(x) and SDf(x) = f(x)− f(0) are the fundamental theorem of calculus. Linking sums and differences allows to compute sums (which is difficult in general) by studying differences (which is easy in general). Studying derivatives of basic functions like xn, exp(a·x) will allow to sum such functions. As operators, Xf(x) = xs∗f(x) and Df(x) = [s, f] where sf(x) = f(x+ 1) and s∗f(x) = f(x − 1) are translations. We have 1x = x,Xx = x(x − 1), X2x = x(x − 1)(x − 2). The derivative operator Df(x) = (f(x + 1) − f(x))s satisfies the Leibniz product rule D(fg) = (Df)g + (Dg)f = Dfg + f+Dg. Since momentum P = iD satisfies the anticommutation relation [X,P] = i we have quantum cal
COUNTING ROOTED FORESTS IN A NETWORK
"... Abstract. If F, G are two n×m matrices, then det(1+xF T G) = P xP  det(FP)det(GP) where the sum is over all minors [18]. An application is a new proof of the ChebotarevShamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in a finite simple graph G with Laplacian ..."
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Abstract. If F, G are two n×m matrices, then det(1+xF T G) = P xP  det(FP)det(GP) where the sum is over all minors [18]. An application is a new proof of the ChebotarevShamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in a finite simple graph G with Laplacian L. We can generalize this and show that det(1 + kL) is the number of rooted edgekcolored spanning forests. If a forest with an even number of edges is called even, then det(1−L) is the difference between even and odd rooted spanning forests in G. 1. The forest theorem A social network describing friendship relations is mathematically described by a finite simple graph. Assume that everybody can chose among their friends a candidate for “president ” or decide not to vote. How many possibilities are there to do so, if cyclic nominations are discarded? The answer is given explicitly as the product of 1 + λj, where λj are the eigenvalues of the combinatorial Laplacian L of G. More generally, if votes can come in k categories, then the number voting situation is the product of 1 + kλj. We can interpret the result as counting rooted spanning forests in finite simple graphs, which is a theorem of ChebotarevShamis. In a generalized setup, the edges can have k colors and get a formula for these rooted spanning forests. While counting subtrees in a graph is difficult [15, 12] in Valiants complexity class #P, ChebotarevShamis show that this is different if the trees are rooted. The forest counting result belongs to spectral graph theory [2, 5, 7, 21, 17] or enumerative combinatorics [10, 11]. Other results relating the spectrum of L with combinatorial properties is Kirchhoff’s matrix tree theorem which expresses the number of spanning trees in a connected graph of n nodes as the pseudo determinant Det(L)/n or the Google determinant det(E + L) with Eij = 1/n2. counting the number