8 Application: differential systems **and** **spectral** **curve**
An important observation of [9] **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 [11]. Here, thanks to **the** result **for** **the** **mixed** **correlation** **function**, we are able to give new representations of those systems, **and** compute explicitely their **spectral** **curve**.

Recently, it has progressively become clear that large N expansion of random **matrix** models has a strong link with algebraic geometry [27]. **The** free energy **and** **correlation** functions have been computed in terms of properties of an algebraic **curve**. **The** large N limit of **the** 1-point **correlation** **function** (called **the** resolvent) is solution of an algebraic equation, which thus defines an algebraic **curve**. There have been many works which computed free energy **and** **correlation** 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 [7], 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.

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SPhT-T06/016 ITEP/TH-05/06
Abstract
We compute **the** complete topological expansion of **the** formal hermitian two- **matrix** **model**. **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 **the** **spectral** **curve**. We extend these rules obtaining a closed formula **for** **correlation** 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.

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models. A promising candidate **for** such an invariant is **the** symplectic class of **the** **matrix** **model** **spectral** **curve**.
**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 **and** **the** 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 [23] in section 3. Section 4 contains our main result: we introduce a chain of matrices **matrix** **model** **and** demonstrate that it reproduces **the** topological string partition **function** on **the** fiducial geometry. By **the** argument above, we thus obtain a **matrix** **model** description **for** **the** 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.

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2iπ t j `
σ ¯
(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 **2** **and** [ 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 **spectral** **curve** 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**

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Let us also remark that **the** **function** 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 [17]:
X(Y (x)) = x (4-1)
**The** consequences of that algebraic equation, can then be studied. This is done **for** instance in [3]. One can also expect to generalize **the** works of [9], or [7], or [8], 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].

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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 **2** **and** **the** filling fractions ǫ i . **The** converse is easier: given a
genus g algebraic **curve**, we can determine V 1 , V **2** **and** **the** ǫ i ’S.

In this article we have explicitly considered only **the** 1 **and** **2**-**matrix** models, al- though section 4 guarantees that it also applies to **the** ”chain of matrices” [7, 8] **matrix** **model**, **and** possibly more. Also, we have written **the** explicit proof **for** 1 **and** **2** 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 [1]), or also **for** **matrix** models with hard edges [1, 9].

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JHEP03(2012)098
In particular loop equations have been known **for** a long time, **and** their recursive solution was recently proposed in [ 17 – 19 , 46 ].
**For** **the** hermitian case, it turned out that a 1-**matrix** **model** was not general enough, **for** instance **the** **spectral** **curve** **for** 1-**matrix** **model** 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**-**matrix** **model** allows to count discrete surfaces carrying colors [ 25 , 54 ].

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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 [25]. 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.

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 [11] . 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. [15] have shown a remarkable enhancement in mixture vapor/gas selectivity of PMP by introducing nanosized silica in **the** **matrix** 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** [16] . 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.

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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.

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Methods
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 **the** **spectral** 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 **for** **the** 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

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