8 Application: differential systems andspectralcurve
An important observation of  and [10, 32, 29] is that windows of consecutive bi- orthogonal polynomials satisfy some integrable differential systems, and a Riemann– Hilbert problem. An explicit representation of those differential systems was found in . Here, thanks to the result forthemixedcorrelationfunction, we are able to give new representations of those systems, and compute explicitely their spectralcurve.
Recently, it has progressively become clear that large N expansion of random matrix models has a strong link with algebraic geometry . The free energy andcorrelation functions have been computed in terms of properties of an algebraic curve. The large N limit of the 1-point correlationfunction (called the resolvent) is solution of an algebraic equation, which thus defines an algebraic curve. There have been many works which computed free energy andcorrelation functions in terms of that algebraic curve. The leading order resolvent and free energy were computed in the 1-cut case (algebraic curve of genus zero) in the pioneering work of , then some recursive method for computing correlation functions and free energy to all orders in 1/N were invented by [3, 4]. Those methods were first limited to 1-matrix case and 1-cut.
We compute the complete topological expansion of the formal hermitian two- matrixmodel. For this, we refine the previously formulated diagrammatic rules for computing the N 1 expansion of the nonmixed correlation functions and give a new formulation of thespectralcurve. We extend these rules obtaining a closed formula forcorrelation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.
models. A promising candidate for such an invariant is the symplectic class of thematrixmodelspectralcurve.
The outline of this paper is as follows. In section 2, after a very brief review of toric geometry basics, we introduce the fiducial geometry andthe notation that we will use in discussing it throughout the paper. We also review the transformation properties of the topological string partition function under flop transitions, which will relate the fiducial to an arbitrary toric geometry, in this section. We recall the topological vertex formalism and its application to geometries on a strip  in section 3. Section 4 contains our main result: we introduce a chain of matrices matrixmodeland demonstrate that it reproduces the topological string partition function on the fiducial geometry. By the argument above, we thus obtain a matrixmodel description forthe topological string on an arbitrary toric Calabi-Yau manifold, in the large radius limit. We discuss implications of this result in section 5, and point towards avenues for future work in section 6.
2iπ t j `
(4.26) And, there may exist residual terms corresponding to irreducible flat connections [ 36 , 38 ]. Since the measure in ( 2.1 ) is now complex, we cannot apply stricto sensu the arguments of asymptotic analysis raised in Section 2and [ 15 ]. Nevertheless, we can take the saddle point equation ( 2.18 ) with complex valued right-hand side as a starting point, and compute the corresponding spectralcurve with the methods of Section 3 . The only difference in the result is a rescaling of the Newton polygon, and now the coefficients inside the Newton polygon depend on the collection filling fractions l “ N ` ~. This dependence is in general transcendental, since the I are periods of the 1-form ln y d ln x on the
Let us also remark that thefunction eq.2-27 E(x, y) is unchanged under the exchange x ↔ y, V1 ↔ V2, s ↔ ˜ s, which is the generalization of Matytsin’s duality property :
X(Y (x)) = x (4-1)
The consequences of that algebraic equation, can then be studied. This is done for instance in . One can also expect to generalize the works of , or , or , i.e. the computation of all correlation functions and their 1/N 2 expansion, and further, compute the expansion of the free energy [10, 11].
Forthe hermitian case, it turned out that a 1-matrixmodel was not general enough, for instance thespectralcurvefor 1-matrixmodel is always quadratic, it can’t have higher degree, the critical behaviors are related to only a small subset of all possible conformal field theories. A 2-matrix hermitian model turns out to be much more general, it allows to reach spectral curves of any degrees, and any conformal minimal model [21, 26, 27, 56]. Moreover, as applications to counting discretized surfaces, a 2-matrixmodel allows to count discrete surfaces carrying colors [25, 54].
proportion ǫ i of the total number of eigenvalues, this why ǫ i is called a filling fraction.
In practice, we don’t have a closed expression for P (0) (x, y) as a function of the
coefficients of V 1 and V 2andthe filling fractions ǫ i . The converse is easier: given a
genus g algebraic curve, we can determine V 1 , V 2andthe ǫ i ’S.
In this article we have explicitly considered only the 1 and2-matrix models, al- though section 4 guarantees that it also applies to the ”chain of matrices” [7, 8] matrixmodel, and possibly more. Also, we have written the explicit proof for 1 and2 ma- trix model only for polynomial potentials, and again section 4 guarantees that the same works for potentials whose derivative is a rational function (called semi-classical potentials ), or also formatrix models with hard edges [1, 9].
In particular loop equations have been known for a long time, and their recursive solution was recently proposed in [ 17 – 19 , 46 ].
Forthe hermitian case, it turned out that a 1-matrixmodel was not general enough, for instance thespectralcurvefor 1-matrixmodel is always quadratic, it can’t have higher degree, the critical behaviors are related to only a small subset of all possible conformal field theories. A 2-matrix hermitian model turns out to be much more general, it allows to reach spectral curves of any degrees, and any conformal minimal model [ 21 , 26 , 27 , 56 ]. Moreover, as applications to counting discretized surfaces, a 2-matrixmodel allows to count discrete surfaces carrying colors [ 25 , 54 ].
The paper extends to the case of the two-matrixmodelthe work contained in Refs. 9 , 12 , and
10 ; it uses, however, a different approach closer to a recent one. 6
In Refs. 9 , 12 , 10 , and 19 the partition function of the one-matrixmodel 共and certain shifted Töplitz determinants兲 were identified as isomonodromic functions by using spectral residue formulas in terms of thespectralcurve of the differential equation. Such spectralcurve has interesting properties inasmuch as—in the one-matrix case—thespectral invariants can be related to the expectation values of thematrixmodel. Recently thespectralcurve of the two-matrixmodel 8 has been written explicitly in terms of expectation values of the two-matrixmodeland hence one could use their result and follow a similar path forthe proof as the one followed in Ref.
The sum over all partitions is dominated by partitions close to a typical equilibrium partition, i.e. a saddle point. The typical partition has a certain typical length referred to as its equilibrium length ¯ n. All partitions with a length very different from the equilibrium length contribute only in an expo- nentially small way (and thus non-perturbatively) to the full partition function. Introducing a cutoﬀ on the length of partitions which is larger than the equi- librium length hence does not change the perturbative part of the partition function. Now recall that when we defined the h i (γ) of a representation γ in
The “mixed actuation and power flow control” (MAPFC) configuration proposed in this paper is able to contribute to the HVDC voltage quality: As will be detailed, it introduces new possible power transfers which can be used to contribute posi- tively to the two last items (transient behavior and availability). In the same time, it satisfies mechanical needs, as the proposed structure is basically a mechanical drive. Indeed, as can be seen in Fig. 1, the windings of an open-ended AC machine are connected between two different DC buses through two voltage source inverters.
andthe observations belonging to different groups are the most dissimilar from each other. The most commonly used such algorithms are k-means, the hierarchical clustering and EM [ 12 , 15 ]. Spectral clustering techniques (see e.g. [ 20 , 17 ]) make use of the spectrum of a similarity matrix of the data or of a more involved derived version of it (such as the as- sociated Laplacian matrix ) to perform dimensionality reduction before clustering in fewer dimensions, usually thanks to one of the previously mentioned algorithms. More specifi- cally, given n observations x 1 , . . . , x n ∈ R p one wants to cluster, one chooses a similarity
Fig. 6. Validation of actuation function of the MAPFC.
Fig. 7. Notations used for power sharing analysis.
IV. E MBEDDED N ETWORK O PERATIONS
The use of MAPFC is very attractive in an embedded net- work to replace a classical PMSM—inverter association like in an electric environmental cooling system application . In- deed, these machines are important electric loads and have a fair inertia, which may be interesting during transient operations. Here, we choose to discuss the feasibility of two application cases of the MAPFC concept.
The Gram matrix Ψ of a system is very useful in system identification, power density spectrum modeling andmodel reduction. The Gram ma- trix contains elements that are inner products of repeated integrals and/or derivatives of the im- pulse response of the system. In model reduction applications, the elements of this matrix Ψ have to be computed from the mathematical model of the original system. Note that only stable sys- tems will be considered.
Recently, Pinnau and He reported that the addition of non- porous nanosized-fumed silica, which has opposing properties with porous inorganic fillers, can alter polymer chain packing in glassy, high-free-volume polymers  . This resulted in an increase in free volume leading to a significant enhancement in permeability. Merkel et al. [12–14] extended this approach and demonstrated enhancement in permeabilities of a variety of high-free-volume polymers such as poly (4-methyl-2-pentyne) (PMP) and poly (1-trimethylsilyl-1-propyne) (PTMSP), as well as poly (2,2-bis(trifluoromethyl)-4,5-difluoro-1,3-dioxole-co- tetrafluoroethylene) (Teflon AF2400). Moreover, He et al.  have shown a remarkable enhancement in mixture vapor/gas selectivity of PMP by introducing nanosized silica in thematrix of high-free-volume glassy polymers. This attractive result is in complete contradiction to the Maxwell model prediction of a reduction in permeability and little effect on selectivity by the addition of nonporous fillers to a polymer matrix  . In another study, the interface between polymer and silica agglomerates was investigated by observation of the morphology of fumed silica in amorphous, low-free-volume glassy poly(ether imide) [17,18] . It was shown that void formation occurs during film fabrication, which has a significant effect on the gas transport properties of the hybrid membrane.
its own canonical model. Rosenlicht’s proofs involve relating global invariants. We give similar proofs of these statements in our Theorem 4.3.
Unaware of Rosenlicht’s work, several authors have reproved various form of (1). The first proofs were given by Deligne and Mumford in 1969, by Sakai in 1977, and by Catanese in 1982, according to Catanese [4, p. 51]. Their work was motivated by the study of families of curves, and they allowed C to be reducible, but required it to be connected in a strong sense, and to have only mild Gorenstein singularities. In 1973, Mumford and Saint-Donat proved (1) for a smooth C, using the Jacobian of C. In 1983, Fujita [9, p. 39] asserted their proof works virtually without change for any Gorenstein C. Then Fujita [9, Thm. (A1), p. 39] gave another proof for any Gorenstein C, involving ideas from Mumford’s version of Castelnuovo Theory.
The statistical physics problem of counting plane partition configurations of some do- main, as well as its various equivalent formulations, has become a very active and fas- cinating area of mathematical physics in the past years, culminating with Okounkov’s renowned work. Beyond a beautiful combinatorics problem, it has also many indirect applications, like a tiling problem similar to a discrete version of TASEP, i.e. the sim- plest model of out of equilibrium statistical physics, and algebraic geometry, as it plays a key role in the computation of Gromov-Witten invariants of some toric Calabi-Yau 3-folds, through the topological vertex method .
Experimental set-up. It is based on a standard mi-
crophotoluminescence experiment operating at a temper- ature of T=4K. The sample is excited by a continuous wave diode laser emitting at 405 nm. The luminescence is split by a 50/50 beamsplitter and each beam is sent to a monochromator (resolution δE = 0.2 meV or δλ = 0.05 nm) whose output slit is imaged on an avalanche photo- diode (APD). The width of the output slit can be var- ied allowing us to choose thespectral window within the SD inhomogeneously broadened line. The voltage pulses of each APD are sent to a time-correlated single pho- ton module that builds an histogram of the time delays between photons. This allows us to perform either auto- correlation when the two monochromators are tuned to the same wavelength or cross-correlation otherwise. Ex- cept forthe results of fig. 4, the work presented here has been obtained with high quantum efficiency APDs (η = 60% at 550 nm). High detection efficiency is im- portant when performing correlation experiments since