Tootfinder

On function-on-function linear quantile regression
Muge Mutis, Ufuk Beyaztas, Filiz Karaman, Han Lin Shang
https://arxiv.org/abs/2510.10792 https://arxiv.o…

On function-on-function linear quantile regression
We present two innovative functional partial quantile regression algorithms designed to accurately and efficiently estimate the regression coefficient function within the function-on-function linear quantile regression model. Our algorithms utilize functional partial quantile regression decomposition to effectively project the infinite-dimensional response and predictor variables onto a finite-dimensional space. Within this framework, the partial quantile regression components are approximated …

Convergence analysis of inexact MBA method for constrained upper-$\mathcal{C}^2$ optimization problems
Ruyu Liu, Shaohua Pan
https://arxiv.org/abs/2511.09940 https://arxiv.org/pdf/2511.09940 https://arxiv.org/html/2511.09940
arXiv:2511.09940v1 Announce Type: new
Abstract: This paper concerns a class of constrained optimization problems in which, the objective and constraint functions are both upper-$\mathcal{C}^2$. For such nonconvex and nonsmooth optimization problems, we develop an inexact moving balls approximation (MBA) method by a workable inexactness criterion for the solving of subproblems. By leveraging a global error bound for the strongly convex program associated with parametric optimization problems, we establish the full convergence of the iterate sequence under the partial bounded multiplier property (BMP) and the Kurdyka-{\L}ojasiewicz (KL) property of the constructed potential function, and achieve the local convergence rate of the iterate and objective value sequences if the potential function satisfies the KL property of exponent $q\in[1/2,1)$. A verifiable condition is also provided to check whether the potential function satisfies the KL property of exponent $q\in[1/2,1)$ at the given critical point. To the best of our knowledge, this is the first implementable inexact MBA method with a full convergence certificate for the constrained nonconvex and nonsmooth optimization problem.
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$L^2$ normal velocity implies strong solution for graphical Brakke flows
Kotaro Motegi
https://arxiv.org/abs/2510.11377 https://arxiv.org/pdf/2510.11377

$L^2$ normal velocity implies strong solution for graphical Brakke flows
We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$ has weak derivatives $\partial_t f, \nabla^2 f \in L^2$, and $v$ coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in $L^{p,q}$ an…

Characterizing Maximal Monotone Operators with Unique Representation
Sotiris Armeniakos, Aris Daniilidis
https://arxiv.org/abs/2510.09368 https://arxiv.org…

Characterizing Maximal Monotone Operators with Unique Representation
We study maximal monotone operators $A : X \rightrightarrows X^*$ whose Fitzpatrick family reduces to a singleton; such operators will be called uniquely representable. We show that every such operator is cyclically monotone (hence, $A=\partial f$ for some convex function $f$) if and only if it is 3-monotone. In Radon-Nikodým spaces, under mild conditions (which become superfluous in finite dimensions), we prove that a subdifferential operator $A=\partial f$ is uniquely representable if and on…

Learning Mean-Field Games through Mean-Field Actor-Critic Flow
Mo Zhou, Haosheng Zhou, Ruimeng Hu
https://arxiv.org/abs/2510.12180 https://arxiv.org/pdf/25…

Learning Mean-Field Games through Mean-Field Actor-Critic Flow
We propose the Mean-Field Actor-Critic (MFAC) flow, a continuous-time learning dynamics for solving mean-field games (MFGs), combining techniques from reinforcement learning and optimal transport. The MFAC framework jointly evolves the control (actor), value function (critic), and distribution components through coupled gradient-based updates governed by partial differential equations (PDEs). A central innovation is the Optimal Transport Geodesic Picard (OTGP) flow, which drives the distributio…

Monte Carlo-Type Neural Operator for Differential Equations
Salah Eddine Choutri, Prajwal Chauhan, Othmane Mazhar, Saif Eddin Jabari
https://arxiv.org/abs/2510.05620 https://

Monte Carlo-Type Neural Operator for Differential Equations
The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions. The kernel is represented as a learnable tensor over sampled in…

An Approach to Anti-Wick Ordering of Bosonic Fields
John E. Gough, Hideyasu Yamasita
https://arxiv.org/abs/2510.06425 https://arxiv.org/pdf/2510.06425

An Approach to Anti-Wick Ordering of Bosonic Fields
We present a new technique for putting general boson fields into ant-Wick ordered form. The anti-Wick map associates an operator with a given function of complex variables, and we show that it may be realized as composition of a mapping to a commutative sub-algebra of a doubled-up boson algebra followed by a partial conditional expectation onto one of the factors.

Partial regularity for parabolic systems of double phase type
Jihoon Ok, Giovanni Scilla, Bianca Stroffolini
https://arxiv.org/abs/2510.03849 https://arxiv…

Partial regularity for parabolic systems of double phase type
We study partial regularity for nondegenerate parabolic systems of double phase type, where the growth function is given by $H(z,s)=s^p+a(z)s^q$, $z=(x,t)\inΩ_T$, with $\tfrac{2n}{n+2}

Spectral results for free random variables
Brian C. Hall, Ching-Wei Ho
https://arxiv.org/abs/2510.03382 https://arxiv.org/pdf/2510.03382

Spectral results for free random variables
Let $(\mathcal{A},\mathrm{tr})$ be a von Neumann algebra with a faithful, normal trace $\mathrm{tr}:\mathcal{A}\rightarrow\mathbb{C}.$ For each $a\in\mathcal{A},$ define \[ S(λ,\varepsilon)=\mathrm{tr}[\log((a-λ)^{\ast}(a-λ)+\varepsilon)],\quadλ\in\mathbb{C},~\varepsilon>0, \] so that the limit as $\varepsilon\rightarrow0^{+}$ of $S$ is the log potential of the Brown measure of $a.$ Suppose that for a fixed $λ\in\mathbb{C},$ the function% \[ \varepsilon\mapsto\frac{\partial S}{\partial\var…

A Function-Sharing Criterion for Normal Functions
Gopal Datt, Ritesh Pal, Ashish Kumar Trivedi
https://arxiv.org/abs/2509.16740 https://arxiv.org/pdf/2509.…

A Function-Sharing Criterion for Normal Functions
In this paper, we present a function-sharing criterion for the normality of meromorphic functions. Let $f$ be a meromorphic function in the unit disc $\mathbb{D}\subset \mathbb{C}$, $ψ_1$, $ψ_2$, and $ψ_3$ be three meromorphic functions in the unit disc $\mathbb{D}$, continuous on $ \partial{\mathbb{D}}:=\{z\in\mathbb{C}\,:\,|z|=1\}$, such that $ψ_i(z)\neqψ_j(z)$ $(1\leq i

Path Integral Derivations Of K-Theoretic Donaldson Invariants
Heeyeon Kim, Jan Manschot, Gregory W. Moore, Runkai Tao, Xinyu Zhang
https://arxiv.org/abs/2509.23042 https://

Path Integral Derivations Of K-Theoretic Donaldson Invariants
We consider 5d $\mathcal{N}=1$ SU(2) super Yang-Mills theory on $X\times S^1$, with $X$ a closed smooth four-manifold. A partial topological twisting along $X$ renders the theory formally independent of the metric on $X$. The theory depends on the spin structure and the circumference $R$ of $S^1$. The coefficients of the $R$-expansion of the partition function are Witten indices, which are identified with $L^2$-indices of Dirac operators on moduli spaces of instantons. The partition function en…

Spatial fluctuation for stochastic heat equation with H\"older coefficients
Carl Mueller, Fei Pu
https://arxiv.org/abs/2510.00807 https://arxiv.org/pd…

Spatial fluctuation for stochastic heat equation with Hölder coefficients
In this paper, we establish associativity, spatial ergodicity and a central limit theorem for certain nonnegative solutions to the stochastic heat equation $\partial_t u=\frac12\partial_x^2 u+ u^γξ$ with $γ\in (0, 1)$. When $γ=\frac12$, we derive a limit for the moment generating function of the spatial integral and provide a lower bound on the spatial growth of the solution.

The properties of general Fourier partial sums of functions $f \in C_L$
Giorgi Tutberidze, Vakhtang Tsagareishvili, Giorgi Cagareishvili
https://arxiv.org/abs/2509.19311 https:/…

The properties of general Fourier partial sums of functions $f \in C_L$
In this paper, we investigate the convergence properties of Fourier partial sums associated with general orthonormal systems, focusing on functions that belong to specific differentiable function classes. While classical Fourier analysis has extensively studied trigonometric systems, our approach considers a broader class of orthonormal systems, including those adapted to weighted function spaces or arising from orthogonal polynomials. The primary objective is to analyze how the smoothness and …

Recurrent Control Barrier Functions: A Path Towards Nonparametric Safety Verification
Jixian Liu, Enrique Mallada
https://arxiv.org/abs/2510.02127 https://…

Recurrent Control Barrier Functions: A Path Towards Nonparametric Safety Verification
Ensuring the safety of complex dynamical systems often relies on Hamilton-Jacobi (HJ) Reachability Analysis or Control Barrier Functions (CBFs). Both methods require computing a function that characterizes a safe set that can be made (control) invariant. However, the computational burden of solving high-dimensional partial differential equations (for HJ Reachability) or large-scale semidefinite programs (for CBFs) makes finding such functions challenging. In this paper, we introduce the notion …

I've probably mentioned that I'm working on switching #Gentoo from our half-broken eselect-ldso logic to #FlexiBLAS. This also involves a transition period where both setups would be supported.
A good thing is that the switch is ABI-compatible with the previous state (or at least it's supposed to be — we're working with upstream on fixing function coverage). Since libblas.so, liblapack.so and the rest are replaced by symlinks, programs that link to them will simply start using FlexiBLAS. So far, so good.
Unfortunately, switching the other way doesn't work as well. Stuff newly built against our libblas.so & co. symlinks naturally reads FlexiBLAS's SONAME from them, and links to libflexiblas directly. So should you decide to switch back, some packages will stay linked to FlexiBLAS and will need to rebuilt.
In order to avoid this, I would have to replace the symlinks with wrapper libraries, having libblas.so.3 and so on SONAMEs, and linking to libflexiblas. Unfortunately, a dummy wrapper isn't going to work — the linker will complain about using indirect symbols from libflexiblas.so. So I would probably have to "reexport" their symbols somehow, and ideally split into appropriate libraries, so that `-Wl,--as-needed` wouldn't drop some of them. But how to do that?
Well, let's look at the existing logic for eselect-ldso — clearly both BLIS and OpenBLAS create some wrappers. So I've spent some time investigating upstream Makefiles, and literally couldn't find the respective targets. I mean, these are quite complex Makefiles, but I'm grepping hard and can't find even a partial match.
As it turns out, these Makefile targets are added by Gentoo-specific patches. And these patches are just horrible. In case of OpenBLAS, they create the wrapper libraries by linking all the relevant .o files from OpenBLAS build, plus the shared OpenBLAS library. So the OpenBLAS symbols relevant to each interface end up duplicated in libblas.so, liblapack.so, etc., and apparently the symbols needed by them are taken from libopenblas.so. The individual interface libraries aren't even linked to one another, so they expose their own duplicate symbols, but use the implementation from OpenBLAS instead.
BLIS is even worse — the patch is simply creating libblas.so and libcblas.so, using all BLIS objects directly, plus symbol visibility to hide symbols irrelevant to the library. So yes, libblis.so, libblas.so and libcblas.so are roughly three separate copies of the same library, differing only in symbol visibility. And of course libcblas.so doesn't use libblas.so.
Truly #GSoC quality.

An older photo of me when I crushed my thumb expressing my opinion of having the cast. It also fits my opinion of the TACO and the maggots who lavish unthinking devotion to him.
I remember being told the doctor that I was extremely lucky not needing surgery and will get most if not all function back. It was crushed by a 35 lb ladder when I was collapsing it, and my thumb slipped when triggering the release. The wife will not allow me to use it unless there is an adult supervising me.…

Nonlocal modeling of spatial fractional diffusion with truncated interaction domains and truncated kernel function singularity
Shiping Zhou, Yanzhi Zhang, Max Gunzburger
https://arxiv.org/abs/2509.16315

Nonlocal modeling of spatial fractional diffusion with truncated interaction domains and truncated kernel function singularity
Parabolic partial differential equations (PDEs) are in ubiquitous, very effective use to model diffusion processes. However, there are many applications (e.g., such as in hydrology, animal foraging, biology, and light diffusion just do name a few) for which results obtained through the use of parabolic PDEs do not agree with observations. In many situations the use of fractional diffusion models has been found to be more faithful to that which is observed. Specifically, we replace the Laplacian…

Partial sharing and cross sharing of entire function with its derivative
Sujoy Majumder, Nabadwip Sarkar, Debabrata Pramanik
https://arxiv.org/abs/2509.20273 https://

Partial sharing and cross sharing of entire function with its derivative
In the paper, we investigate the uniqueness problem of a power of an entire function that share one value partially with it's linear differential polynomial and obtain a result, which improves several previous results in a large scale. Also in the paper we include some applications of our main result. Moreover, we solve the question raised by Wang and Liu (Value cross-sharing of meromorphic functions, Comput. Methods Funct. Theory (2023). https://doi.org/10.1007/s40315-023-00481-9) for a specia…

Universal Scaling Functions of the Gr{\"u}neisen Ratio near Quantum Critical Points
Xuan Zhou, Enze Lv, Wei Li, Yang Qi
https://arxiv.org/abs/2509.17362 https://

Universal Scaling Functions of the Gr{ü}neisen Ratio near Quantum Critical Points
The Grüneisen ratio, defined as $Γ_g \equiv (1/T) (\partial T/\partial g)_S$, serves as a highly sensitive probe for detecting quantum critical points (QCPs) driven by an external feild $g$ and for characterizing the magnetocaloric effect (MCE). Near a QCP, the Grüneisen ratio displays a universal divergence which is governed by a universality-class-dependent scaling function stemming from the scale invariance. In this work, we systematically investigate the universal scaling functions of Gr…

Robin Problems of Elliptic Equations on Rough Domains: H\"older Regularity, Green's Functions, and Harmonic Measures
Jiayi Wang, Dachun Yang, Sibei Yang
https://arxiv.org/abs/2509.23073

Robin Problems of Elliptic Equations on Rough Domains: Hölder Regularity, Green's Functions, and Harmonic Measures
Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $Ω\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $σ$ is the surface measure on the boundary of $Ω$, denoted by $\partial Ω$. Let $β$ be a non-negative measurable function on $\partial Ω$ satisfying $β\in L^{q_0}(\partial Ω,σ)~\text{with}~ q_0 \in(\frac{s}{s+2-n},\infty]~\text{and}\ β\ge a_0~\text{on}~E_0\subset \partial Ω, $ where $a_0$ is a given positive constant and …

Optimal Experimental Design of a Moving Sensor for Linear Bayesian Inverse Problems
Nicole Aretz, Thomas Lynn, Karen Willcox, Sven Leyffer
https://arxiv.org/abs/2509.15961 https…

Optimal Experimental Design of a Moving Sensor for Linear Bayesian Inverse Problems
We optimize the path of a mobile sensor to minimize the posterior uncertainty of a Bayesian inverse problem. Along its path, the sensor continuously takes measurements of the state, which is a physical quantity modeled as the solution of a partial differential equation (PDE) with uncertain parameters. Considering linear PDEs specifically, we derive the closed-form expression of the posterior covariance matrix of the model parameters as a function of the path, and formulate the optimal experimen…

Lyapunov exponents and growth indices for fractional stochastic heat equations with space-time L\'evy white noise
Yuichi Shiozawa, Jian Wang
https://arxiv.org/abs/2509.23534

Lyapunov exponents and growth indices for fractional stochastic heat equations with space-time Lévy white noise
We consider fractional stochastic heat equations with space-time Lévy white noise of the form $$\frac{\partial X}{\partial t}(t,x)={\cal L}_αX(t,x)+σ(X(t,x))\dotΛ(t,x).$$ Here, the principal part ${\cal L}_α=-(-Δ)^{α/2}$ is the $d$-dimensional fractional Laplacian with $α\in (0,2)$, the noise term $\dotΛ(t,x)$ denotes the space-time Lévy white noise, and the function $σ: \R\mapsto \R$ is Lipschitz continuous. Under suitable assumptions, we obtain bounds for the Lyapunov exponents and…

Diffuse Domain Methods with Dirichlet Boundary Conditions
Luke Benfield, Andreas Dedner
https://arxiv.org/abs/2509.25115 https://arxiv.org/pdf/2509.25115…

Diffuse Domain Methods with Dirichlet Boundary Conditions
The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates the problem on a larger, simple domain where the complex geometry is represented by a smooth phase-field function. This paper introduces and analyses several new DDM methods for solving problems with Dirichlet boundary conditions. We derive two new methods…

FedSSG: Expectation-Gated and History-Aware Drift Alignment for Federated Learning
Zhanting Zhou, Jinshan Lai, Fengchun Zhang, Zeqin Wu, Fengli Zhang
https://arxiv.org/abs/2509.13895

FedSSG: Expectation-Gated and History-Aware Drift Alignment for Federated Learning
Non-IID data and partial participation induce client drift and inconsistent local optima in federated learning, causing unstable convergence and accuracy loss. We present FedSSG, a stochastic sampling-guided, history-aware drift alignment method. FedSSG maintains a per-client drift memory that accumulates local model differences as a lightweight sketch of historical gradients; crucially, it gates both the memory update and the local alignment term by a smooth function of the observed/expected p…

Non-parametric estimation of non-linear diffusion coefficient in parabolic SPDEs
Martin Andersson, Benny Avelin, Valentin Garino, Pauliina Ilmonen, Lauri Viitasaari
https://arxiv.org/abs/2509.12921

Non-parametric estimation of non-linear diffusion coefficient in parabolic SPDEs
In this article, we introduce a novel non-parametric predictor, based on conditional expectation, for the unknown diffusion coefficient function $σ$ in the stochastic partial differential equation $Lu = σ(u)\dot{W}$, where $L$ is a parabolic second order differential operator and $\dot{W}$ is a suitable Gaussian noise. We prove consistency and derive an upper bound for the error in the $L^p$ norm, in terms of discretization and smoothening parameters $h$ and $\varepsilon$. We illustrate the a…

The pluricomplex Poisson kernel for convex finite type domains
Leandro Arosio, Filippo Bracci, Matteo Fiacchi
https://arxiv.org/abs/2509.26230 https://arxi…

The pluricomplex Poisson kernel for convex finite type domains
Given a bounded convex domain $D\subset \mathbb C^n$ of finite D'Angelo type and a boundary point $ξ\in \partial D$, we prove that the homogeneous complex Monge-Ampère equation $(dd^cu)^n=0$ possesses a continuous strictly negative solution $Ω_ξ$ that vanishes on $\partial D\setminus \{ξ\}$ and has a simple pole at $ξ$. We establish that $Ω_ξ(z)$ equals (up to sign) the normal derivative at $ξ$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered…

Absolutely Summing Toeplitz operators on Fock spaces
Zhangjian Hu, Ermin Wang
https://arxiv.org/abs/2509.19967 https://arxiv.org/pdf/2509.19967

Absolutely Summing Toeplitz operators on Fock spaces
For $1\le p<\infty$, let $F^p_φ$ be the Fock spaces on ${\mathbb C}^n$ with the weight function $φ$ that \(φ\in {\mathcal{C}}^{2}\left( {\mathbb{C}}^{n}\right)\) is real-valued and satisfies $ m{ω}_{0} \leq d{d}^{c}φ\leq M{ω}_{0} $ for two positive constants \(m\) and \(M\), \({ω}_{0} = d{d}^{c}{\left| z\right| }^{2}\) is the Euclidean Kähler form on \({\mathbb{C}}^{n}\), \({d}^{c} = \frac{\sqrt{-1}}{4}\left( {\bar{\partial } - \partial }\right)\). In this paper, we co…

Integrable Sigma Models and Universal Root $T\bar{T}$ Deformation via Courant-Hilbert Approach
H. Babaei-Aghbolagh, Bin Chen, Song He
https://arxiv.org/abs/2509.17075 https://…

Integrable Sigma Models and Universal Root $T\bar{T}$ Deformation via Courant-Hilbert Approach
We develop a unified Courant--Hilbert framework for constructing two-dimensional integrable sigma models deformed by two couplings: a marginal one $γ$ and an irrelevant one $λ$. The integrability condition is encoded in a nonlinear partial differential equation (PDE) for two invariants $(P_1,P_2)$, whose general solution could be expressed through an arbitrary generating function $\ell(τ)$. This formulation encompasses and extends known models, such as ModMax and Born-Infeld, while introduci…

Knots and variance ordering of sequential Monte Carlo algorithms
Joshua J Bon, Anthony Lee
https://arxiv.org/abs/2510.01901 https://arxiv.org/pdf/2510.0190…

Knots and variance ordering of sequential Monte Carlo algorithms
Sequential Monte Carlo algorithms, or particle filters, are widely used for approximating intractable integrals, particularly those arising in Bayesian inference and state-space models. We introduce a new variance reduction technique, the knot operator, which improves the efficiency of particle filters by incorporating potential function information into part, or all, of a transition kernel. The knot operator induces a partial ordering of Feynman-Kac models that implies an order on the asymptot…

Stochastic diffusive energy balance climate model with a multiplicative noise modeling the Solar variability
Gregorio D\'iaz, Jes\'us Ildefonso D\'iaz
https://arxiv.org/abs/2509.23153

Stochastic diffusive energy balance climate model with a multiplicative noise modeling the Solar variability
We prove the existence, uniqueness, and comparison of solutions for a nonlinear stochastic parabolic partial differential equation that includes the Solar variability in terms of a multiplicative Wiener cylindrical noise in the term of the absorbed radiative energy in a simplified diffusive one-dimensional Energy Balance Model. We introduce a hybrid co-albedo nonlinear term, which has the advantages of both the Sellers model, as it is a continuous function, and the Budyko model, as it has an in…

Physics-informed neural network solves minimal surfaces in curved spacetime
Koji Hashimoto, Koichi Kyo, Masaki Murata, Gakuto Ogiwara, Norihiro Tanahashi
https://arxiv.org/abs/2509.10866

Physics-informed neural network solves minimal surfaces in curved spacetime
We develop a flexible framework based on physics-informed neural networks (PINNs) for solving boundary value problems involving minimal surfaces in curved spacetimes, with a particular emphasis on singularities and moving boundaries. By encoding the underlying physical laws into the loss function and designing network architectures that incorporate the singular behavior and dynamic boundaries, our approach enables robust and accurate solutions to both ordinary and partial differential equations…

A Converse Control Lyapunov Theorem for Joint Safety and Stability
Thanin Quartz, Maxwell Fitzsimmons, Jun Liu
https://arxiv.org/abs/2509.12182 https://arx…

A Converse Control Lyapunov Theorem for Joint Safety and Stability
We show that the existence of a strictly compatible pair of control Lyapunov and control barrier functions is equivalent to the existence of a single smooth Lyapunov function that certifies both asymptotic stability and safety. This characterization complements existing literature on converse Lyapunov functions by establishing a partial differential equation (PDE) characterization with prescribed boundary conditions on the safe set, ensuring that the safe set is exactly certified by this Lyapun…

Remarks on the reinforcement of the spectrum of an elliptic problem with Robin boundary condition
Emanuele Cristoforoni, Federico Villone
https://arxiv.org/abs/2509.22305 https:…

Remarks on the reinforcement of the spectrum of an elliptic problem with Robin boundary condition
We investigate the spectral properties of a differential elliptic operator on $H^1(\barΩ\cup Σ)$, where $Ω$ is a smooth domain surrounded by a layer $Σ$. The thickness of the layer is given by $\varepsilon h$, where $h$ is a positive function defined on the boundary $\partial Ω$ and $\varepsilon$ is the ellipticity constant of the operator in $Σ$. We prove that, in the limit for $\varepsilon$ going to $0$, the spectrum converges to the spectrum of a differential elliptic operator in $H^1(…