In [ 20 ] a new method was found, for rank 2 **systems**, proving condition 4 for WKB- Lax **systems**, not relying on an insertion operator, but only relying on **loop** **equations**. The generalization of this method to higher dimensional representations is still missing. Let us also mention results obtained **from** the opposite end: assuming only topolog- ical recursion, we get **loop** **equations** and Topological Type property, and the goal is to prove that we get a diﬀerential system. In other words, starting **from** topological recur- sion, one builds correlators ˆ W g,n , then deﬁne formal series ˆ W n =

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The TT property is neither expected to hold in general – even among integrable **systems** – nor obvious to establish for a given system. Our proof that it holds for the q-th reduction of the KP hierarchy depends in an essential way on the integrability of the latter, i.e. on the existence of another system ~ B t Ψpx, tq “ Mpx, tqΨpx, tq with rational coefficients in x, which is compatible with ( 1-1 ), but also on the specific form of Mpx, tq. This is clear **from** the technical results of Section 5.7 and 5.8 . Within the TT property, the structure of the asymptotic expansion of correlators is identified in Theorem 3.1 , but when the semiclassical spectral curve has genus g ą 0, it features an unknown ”holomorphic part” H n pgq pz 1 , . . . , z n q, which are basically the moduli of the space of solutions of **loop** **equations**. A given solution Ψpxq knows which H n pgq pz 1 , . . . , z n q is chosen. It thus remains to investi- gate which conditions have to be added to the **loop** **equations** to determine completely the holomorphic part. They probably should take the form of period conditions. Actually, for many interesting so- lutions Ψpxq, we expect the TT property to breakdown if the semiclassical spectral curve has genus g ą 0.

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solving partial **differential** **equations** (PDEs), as they can appear to model natural phenomena.
Our MIS approach for solving PDEs distinguishes it- self **from** classical methods as finite differences meth- ods, finite element method, finite volume method (see [14] for hyperbolic problems), spectral method, meshfree methods, or domain decomposition meth- ods and multigrid methods. It also stands out more recent methods, as for example, the generalized fi- nite difference method (GFDM) [4], the indirect RBFN method [18] and the numerical meshfree technique [24, 8], or (for parabolic problems) monotone Jacobi and Gauss-Seidel methods [7], NIPG/SIPG discon- tinuous Galerkin methods for time-dependent diffu- sion equation [19], and for instance conservative local discontinuous Galerkin methods for time dependent Schrödinger equation [16]. Indeed, all these methods are based on synchronous iterations, so that they are not well adapted to the kind of simulations described above [2].

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x = ± √ (at which they possess a limit) in a spiraling manner. They depend analytically on the parameter taken **from** a ramified sector of opening > 2π (or √
**from** a sector of opening > π), thus covering a full neighborhood of the origin in the parameter space (including those parameters values for which the unfolded system is resonant), and they converge uniformly when tends radially to 0 to a pair of the classical sectoral solutions: Borel sums of the formal power series solution of the limit system, defined on two sectors covering a full neighborhood of the double singularity at the origin. In fact, each such pair of the sectoral Borel sums for = 0 unfolds to a unique above mentioned parametric solution. We state these results in Section 3.2.2, and illustrate them in Section 3.2.3 on the problem of existence of normalizing transformations for families of linear **differential** **systems** unfolding a non-resonant irregular singularity of Poincaré rank 1.

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Pablo Meyer, Thomas Cokelaer, Deepak Chandran, Kyung H Kim, Po-Ru Loh, George Tucker, Mark Lipson, Bonnie Berger, Clemens Kreutz, Andreas Raue, et al. Network topology and parameter estimation: **from** experimental design methods to gene regulatory network kinetics using a community based approach. BMC **systems** biology, 8(1):13, 2014. Minh Quach, Nicolas Brunel, and Florence d’Alch´ e Buc. Estimating parameters and hidden variables in non-linear state-space models based on odes for biological networks inference. Bioinformatics, 23(23):3209–3216, 2007.

The present work leaves wide open the question of refining the strong law of large numbers given by the propagation of chaos result stated in Theorem 2 – Theorem 22 in its full form. A central limit theorem for the fluctuations of the empirical measure of the particle system is expected to hold under reasonable conditions on the common law of the rough drivers. Large and moderate deviation results would also be most welcome. In a different direction, it would be interesting to investigate the propagation of chaos phenomenon for **systems** of interacting rough dynamics subject to a common noise. Very interesting things happen in the Itˆ o setting in relation with mean field games [8, 32]. Also, one would get a more realistic model of natural phenomena by working with **systems** of particles driven by non-independent noises. Individuals with close initial conditions could have drivers strongly correlated while individuals started far apart could have (almost-)independent drivers. Limit mean field dynamics are likely to be different **from** the results obtained here – see [14] for a result in this direction in the Itˆ o setting. We invite the reader to make his own mind about these problems.

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1 Introduction
It is well known that matrix models satisfy both **loop** **equations** [12], and determinantal formulae [19, 13, 9, 20, 17]. However, both notions (**loop** **equations** and determinantal formulae) exist beyond matrix models. In this paper we show that the correlators ob- tained **from** determinants of the Christoffel-Darboux kernel of an arbitrary **differential** system of order 2, do satisfy **loop** **equations**.

||Z(t, x)|| 2 dxdt,
the observation being done on all of the components of the state.
However, unless D is diagonalizable, one cannot simply deduce this inequality **from** ( 1.3 ) and this kind of inequality may even be hard to obtain. For example, in [29], **systems** of heat **equations** with (time and space-varying) zero-order coupling terms are treated by means of Carleman estimates, but the proof only works under the condition that the Jordan blocks of D are of size less that 4 (which seems to be a purely technical condition that until now has not been removed).

Similarly, we study the **systems** of DAE encountered in the numerical modeling of HTS devices using the FEM with edge elements for a 2-D model. We give the discretization of the **equations** in space and identify the resulting system of DAE. Depending on the boundary conditions, the resulting system of DAE can be of index 0 or 2 in Hessenberg form. For the system of DAE of index 2, we cannot conclude if it is better to discretize it directly or to reduce its index. We note that reducing the index **from** 2 to 1 yields a system of ODEs for the Degrees Of Freedom (DOFs) of interest but that a matrix needs to be inverted. We verify the code develop for this project against two analytical solutions for three different problems. The strategy of direct discretization is implemented in the code through the IDAS library. There are no difficulties reported when computing the approximations with the direct discretization strategy. Therefore, this strategy works for the problems considered. We show that the code gives good approximations for all the problems implemented, except where the solution is not linear. The approximations get better when the mesh is refined.

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Resume : Dans cette note, nous etendons les resultats decrits dans une note precedente au cas d'Hamiltoniens non reguliers pour des **equations** aux derivees partielles stochas- tiques completement nonlineaires. Et nous presentons quelques applications de notre theorie au contr^ole stochastique trajectoriel et a la propagation de fronts dans des en- vironnements aleatoires.

medium is a well-known problem that remains unsolved yet. There are several difficulties in this framework that make the classical machinery of diffusions in random media (i.e. without reflection) fall short of determining the limit in (1). In particular, the reflection term breaks the stationarity properties of the process X " so that the method of the environment as seen **from** the particle (see [15] for an insight of the topic) is inefficient. Moreover, the lack of compactness of a random medium prevents **from** using compactness methods. The main resulting difficulties are the lack of invariant probability measure (IPM for short) associated to the process X " and the study of the boundary ergodic problems. The aim of this paper is precisely to investigate the random case and prove the convergence of the process X " towards a reflected Brownian motion. The convergence is established in probability with respect to the random medium and the starting point x.

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The design of free-**from** shapes by mathematical methods is a discipline, named computer-aided-design (CAD). It had its origins slightly later than the computer-aided-engineering (CAE). In fact, CAD is the use of computer technology for design: it allows the creation, modification, analysis and optimization of drawings and geometric modeling. The Bézier curve was the first method used to construct free-form curves and surfaces, and is named according to its inventor, Dr. Pierre Bézier. Bézier was an engineer in the Renault car company and developed this method in 1966. Actually, another French engineer, Paul de Casteljau at Citroën developed the same technology some years earlier. A further development to Bézier’s method were B-splines which provide more flexibility in the modeling of free-form curves and surfaces. Since 1975 non- uniform rational B-splines (NURBS) have been used in CAD programs, as a generalization of B-splines. The development of NURBS provided a technology that can exactly describe circular shapes (cylinders, spheres, etc.) which are basic elements in geometric modeling, but also allows very flexible modeling of free-form surfaces [43].

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4. Discussion and applications In this section, we consider the **differential** equation
f 000 + A(z)f = 0, (4.1)
where A(z) is a meromorphic function of finite order. It is clear that the difficulty of the study of the **differential** polynomial generated by solutions lies in the calculation of the coefficients α i,j . We explain here that by using our method, the calculation

takes values in ¯ Q . Then RBSDE (1.5) has an adapted solution (Y, Z, K, L) in S 2 × M 2 × (N 2 ) 2 .
We first sketch the proof.
Sketch of the Proof: The proof is divided into five subsections. In Subsection 3.1, we introduce the penalized RBSDEs whose existence of solution follows **from** a slightly generalized result in [11]. In Subsection 3.2, we give the (implicit) representation of these solutions. In Subsection 3.3, we state a fundamental lemma and some (uniform) a priori estimates for these solutions. In Subsection 3.4, we prove the (monotone) convergence of these solutions. And the last subsection is devoted to checking out the boundary conditions.

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order have been recently proved to be a valuable tool in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous me- dia, electromagnetics, etc. (see [5, 11, 12, 14, 21, 22, 26]). There has been a significant progress in the investigation of fractional **differential** and partial **differential** **equations** in recent years; see the monographs of Kilbas et al [17], Miller and Ross [23], Samko et al [30] and the papers of Delbosco and Rodino [4], Diethelm et al [5, 6, 7], El-Sayed [8, 9, 10], Kaufmann and Mboumi [15], Kilbas and Marzan [16], Mainardi [21], Momani and Hadid [24], Momani et al [25], Podlubny et al [29], Yu and Gao [31] and Zhang [32] and the references therein. Very recently some basic theory for the initial boundary value problems of fractional **differential** **equations** involving a Riemann–Liouville **differential** op- erator of order 0 < α ≤ 1 has been discussed by Lakshmikantham and Vatsala [18, 19, 20]. In a series of papers (see [1, 2, 3]) the authors considered some classes of initial value problems for functional **differential** **equations** involving Riemann–Liouville and Caputo fractional derivatives of order 0 < α ≤ 1.

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Su, X. (2009). Boundary value problem for a coupled system of nonlinear fractional **differential** **equations**. Applied Mathematics Letters. 22(1), 64–69.
Yang, W. (2012). Positive solutions for a coupled system of nonlinear fractional di fferential **equations** with integral boundary conditions. Comput. Math. Appl. 63, 288–297.

to understand and model. In particular, prior work does not provide closed-form solutions of cache performance, i.e. simple expressions for the miss rate of a specific access pattern. Existing cache models instead use numerical methods that, unlike closed-form solutions, are computationally expensive and yield limited insight. We present cache calculus, a technique that models cache behavior as a system of ordinary **differential** **equations**, letting standard calculus techniques find simple and accurate solutions of cache performance for common access patterns.

2 B ACKGROUND
Cache models try to predict the cache’s hit rate using a probabilistic description of the access pattern and a description of the cache (e.g., its size). Prior models [1, 3, 4, 5, 6, 8] aim to provide an efficient alternative to simulation, to, for example, accelerate design space exploration [1, 5, 8] or perform dynamic cache partitioning [3]. By contrast, our main goal is to augment simulation through simpler, closed-form models. Cache calculus provides simple **equations** that capture the main tradeoffs and aid in the early stages of design. To that end, we focus on common access patterns and simple approximations.

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(4) dX t n = dW t + b n (t, X t n )dt, t ∈ [0, T ],
where b n = b⋆φ n and (φ n ) is a sequence of mollifiers converging to the Dirac
measure.
Diffusions in the generalized sense were studied by several authors begin- ning with, at least in our knowledge [20]; later on, many authors considered special cases of stochastic **differential** **equations** with generalized coefficients, it is difficult to quote them all: in particular, we refer to the case when b is a measure, [4, 7, 18, 22]. [4] has even considered the case when b is a not nec- essarily locally finite signed measure and the process is a possibly exploding semimartingale. In all these cases solutions were semimartingales. In fact, [8] considered special cases of non-semimartingales solving stochastic differ- ential **equations** with generalized drift; those cases include examples coming **from** Bessel processes.

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