Papers

Reconstructing an infinite-dimensional signal from a finite set of measurements is a fundamental problem in approximation theory and signal processing. While the generalized sampling (GS) framework provides a robust methodology for recovering elements in arbitrary separable Hilbert spaces, deterministic approaches suffer from severe basis-dependent dimensionality constraints, often requiring a quadratic sample complexity $m \gtrsim n^2$ to avoid numerical instability. In this paper, we introduce a fully stochastic framework for GS that natively overcomes these deterministic barriers. By drawing measurements according to an optimal leverage-score probability distribution, we prove that stable recovery is guaranteed with high probability at a near-linear sample complexity of $m \gtrsim n\log n$. Crucially, this optimal rate is universal-independent of the specific choice of measurement and reconstruction bases-and holds even when the sensing system is a highly redundant frame. To establish these guarantees, we derive a novel matrix Bernstein inequality for random rectangular operators, allowing us to rigorously control the aliasing error governed by the empirical cross-term. Finally, we demonstrate the practical efficacy of our approach on the classical problem of recovering analytic functions from continuous Fourier measurements via Legendre polynomials, where our randomized method achieve near-exponential convergence rates.

Generalized stochastic approximation replaces the log-likelihood ratio of a sequential change-point problem by an affine function of a fixed feature dictionary, whose coefficients solve a normal system in the dictionary's moments under the pre- and post-change laws. The increment is a rescaled orthogonal projection of the bounded score, and the information it carries is bounded by the triangular discrimination between the laws. The dictionary decides what is detectable: a symmetry shared by both laws annihilates a whole parity at every order, and one further column raises the information by an explicit bordered-metric increment. A single efficiency factor, at most one, governs all three procedures the increment drives – cumulative-sum, Shiryaev-Roberts and Shiryaev – through false-alarm and delay bounds valid at every threshold. Estimated coefficients are consistent and asymptotically normal. A Monte-Carlo study measures the factor against the delay ratio it predicts, and locates the order at which the dictionary stops paying.

Polytopal methods provide a flexible framework for the numerical approximation of partial differential equations on general meshes. Their convergence analysis raises specific challenges due to their inherently non-conforming nature and, in many cases, the fully discrete nature of their solution. Two main techniques are considered: the virtual-function approach, used, e.g., in the context of Virtual Element Methods, and the fully discrete approach, which underlies, e.g., the Discrete de Rham method. We introduce here a novel framework based on the notion of conforming liftings, namely bounded and consistent mappings from the discrete space into the continuous space. This approach bridges the virtual and fully discrete viewpoints, clarifies the role of norm equivalence for virtual functions, and leads to a decomposition of the consistency error usable for polytopal methods. The three approaches are demonstrated on a model problem, which provides the opportunity to discuss relevant technical points. Bridges with the convergence properties of discrete differential complexes are also built.

We propose a definition of the category $\mathit{Mo}^{\mathrm{mult}}(P)$ of multi-valued Morse homotopy on $P$ consisting of multi-valued functions associated to Lagrangian multi-sections. We then show that a full subcategory $\mathit{Mo}^{\mathrm{mult}}_{\mathcal{E}}(P)$ of $\mathit{Mo}^{\mathrm{mult}}(P)$ is $A_\infty$-equivalent to a full subcategory $\mathit{DG}_{\mathcal{E}}^{\mathrm{vect}}(\mathbb{C}P^2)$ of the category $\mathit{DG}^{\mathrm{vect}}(\mathbb{C}P^2)$ consisting of holomorphic vector bundles over the complex projective plane $\mathbb{C}P^2$. As an application, we study the mirror description for global sections of the holomorphic tangent bundle over $\mathbb{C}P^2$.

Physics-informed neural networks (PINNs) offer a mesh-free route to solving coupled multiphysics systems, but their accuracy degrades systematically as inter-equation coupling strengthens, and inverse-gradient-norm loss balancing alone does not reliably prevent this failure. This study explains why coupling degrades PINN training and identifies an optimizer structure that removes the dependence, replacing case-by-case tuning with a principled remedy. Through a neural tangent kernel analysis, we prove that the standard kernel's spectral radius grows as $\Omega(\gamma^2)$ with coupling strength $\gamma$, whereas block-diagonal Gauss-Newton (GN) preconditioning bounds it by the number of networks $S$, independent of $\gamma$; no diagonal preconditioner recovers this bound for any coupling type or loss weighting. We realize block-diagonal GN preconditioning through the Kronecker-preconditioned optimizer SOAP combined with inverse-gradient-norm loss balancing (SOAP+GradNorm) and evaluate it across 222 experiments on four benchmarks of increasing difficulty. Across all systems, SOAP+GradNorm is the only configuration whose degradation remains bounded in every regime tested: it preserves weak-coupling accuracy in linear problems and limits degradation to $2.3\times$ in the nonlinear Nernst-Planck-Poisson system, whereas Adam-based training leaves the $L_2$ error above the 0.1 failure threshold. The same effect applies to a six-residual, four-network, 2D electro-osmotic flow where the electric double layer is resolved down to a Debye length of $\varepsilon = 0.01$ on an $x$-invariant reference solution. These results recast coupling-induced accuracy loss as a problem of the preconditioner's structure rather than loss weighting and identify Kronecker preconditioning as a structural lever for training PINNs on strongly coupled, stiff multiphysics systems.

Conventional communication systems are mainly designed to reduce error rates and increase transmission rates, and therefore usually provide uniform protection to all transmitted messages. However, in intent-oriented applications, different messages may have different semantic meanings and importance levels, requiring different levels of reliability. This paper proposes a layered construction of message-level unequal error protection (UEP) codes for short-blocklength communication. Instead of appending an explicit protection tag to each codeword, the proposed method embeds the protection structure directly into the Hamming-distance structure of the codebook. By assigning larger minimum intra-level distances to higher-importance message groups and imposing suitable inter-level distance constraints, the proposed codebook provides differentiated error-correction capabilities while enabling reliable importance-level classification at the receiver. Theoretical conditions for correct group classification are derived, and simulations over AWGN and VLC-ISI channels show that the proposed scheme improves BER performance and group classification accuracy compared with a tag-based ECC baseline.

In this paper, we completely classify the rational weights $k$ for which the Kaneko-Zagier (KZ) differential equation admits a fundamental system of solutions consisting of modular forms for a principal congruence subgroup $\Gamma(N)$. By transforming the KZ equation into a hypergeometric differential equation, we study the global analytic continuation of its solutions, adopting an approach analogous to Stiller's work on Picard-Fuchs equations. We explicitly construct the monodromy representation matrices corresponding to the elements of the principal congruence subgroups and completely determine the algebraic conditions under which these connection matrices commute. Leveraging these stringent commutativity constraints, we prove that the weights $k$ yielding modular solutions are strictly limited to $k \equiv 1/2, 7/2, 1, 2, 3 \pmod{6}$ and $k = (6n+1)/5$, thereby demonstrating that no modular solutions exist beyond those previously discovered by Kaneko and Koike. Furthermore, the commutative algebras generated by these connection matrices reveal a profound analogy with commuting transfer matrices in quantum integrable systems.

Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $\lambda_{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence $\lambda_k = \lambda_{\max}(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that $\lambda_k$ converges to a limit $\lambda_\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: \[ \lambda_k = \lambda_\infty + \frac{\alpha}{d+k} + O\!\left(\frac{1}{(d+k)^2}\right), \] where $d$ is the degree of the original vertex, and the coefficient $\alpha$ is given by a spectral projection. As a corollary, when $\alpha \neq 0$, $\lambda_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.

Ambulance response is time-critical in out-of-hospital cardiac arrest (OHCA), where dispatchers must balance timely arrivals with limited fleet capacity. Static territories and deterministic travel-time estimates are vulnerable to dynamic congestion, while always-dual dispatch adds redundancy but consumes fleet capacity. We propose IDEAL (Intelligent Dual dispatch of Emergency AmbuLances), a selective dual-dispatch framework that sends a second ambulance only when the optimistic gap between primary and secondary paths exceeds a threshold. IDEAL learns context-specific edge travel times from trip-level dispatch records, including unobserved routes, using a weakly supervised bilevel representation network. We train the nonsmooth model with mini-batch conservative gradients and prove an asymptotic convergence guarantee. IDEAL models uncertainty via Burg-divergence perturbations to a shared metric in the learned representation space, thereby inducing correlated changes in edge travel times and learning context-specific radii from historical underprediction errors. For real-time decisions, IDEAL casts optimistic-gap computation as a difference-of-convex program and derives an efficient oracle with complexity guarantees. In collaboration with the Hong Kong Fire Services Department, we evaluate IDEAL using historical OHCA records and real-time adaptive simulations. The results achieve a stronger response-time/resource trade-off relative to all region-based and Google-based baselines.

We study cosmological polytopes induced by Erdős–Rényi random graphs in a high-dimensional regime. These graph-based lattice polytopes form a natural model of random lattice polytopes in which geometric features are determined by the structure of the underlying random graph. Focusing on the number of polytope edges and on the number of edges in unimodular triangulations, we derive asymptotic formulas for expectations and variances and prove quantitative central limit theorems in the relevant parameter regime. The analysis relies on explicit graph-theoretic descriptions of the corresponding edge sets and on the discrete Malliavin–Stein method for normal approximation.

Double (quasi-)Poisson brackets were introduced on associative algebras by Van den Bergh to induce a (quasi-)Poisson structure on their representation spaces naturally equipped with a $\mathrm{GL}$-action (type $\mathtt{A}$). If there exists a compatible involutive anti-automorphism on the underlying associative algebras, Olshanski and Safonkin proved that this construction can be upgraded to induce a Poisson structure on twisted representation spaces (types $\mathtt{B},\mathtt{C},\mathtt{D}$). We provide an analogous result for double quasi-Poisson brackets, and over an arbitrary semisimple base. We also apply our theory to quivers in order to understand the Poisson structure on twisted (localised multiplicative) quiver varieties. The formalism permits that different vertices are assigned different types. As a first application, we recover the framework of Massuyeau and Turaev for Hopf algebras with a Fox pairing, which induces in particular the Poisson structure of character varieties for the orthogonal or symplectic groups. As a second application, we introduce a modified Kontsevich system.

Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as budgeted heteroskedastic multi-judge estimation under independent, unbiased score observations. Given $K$ prompt-response pairs and $J$ judges with known costs but unknown query-judge variances, our goal is to estimate the bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its leading error bound. Since this allocation depends on the unknown variances, we then address the unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and uses optimistic variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a local minimax lower bound matching the leading allocation-dependent rate, establishing asymptotic instance-optimality up to logarithmic factors. An Assouad-type construction based on local, variance-preserving perturbations yields this bound both in expectation and with constant probability. A complementary Fano-type change-of-measure argument captures the logarithmic dependence on the confidence level with a different (suboptimal) allocation geometry. Finally, we numerically validate our approach on synthetic data and resampled HelpSteer2 evaluations, demonstrating improvements over naïve uniform allocation.

Joinings are fundamental global objects in ergodic theory, yet in compact metric models one naturally observes only finite orbit-distance patterns. We bridge this gap by introducing multi-particle distance arrays, which sample finite orbit segments and record their joint metric evolution. In the anchored fixed-model setting, this framework yields a purely finite-observable characterization of Furstenberg disjointness: two systems are disjoint if and only if all their anchored multi-orbit distance-array projections are independent. The structural engine behind this criterion is a marked and colored version of the Gromov–Vershik reconstruction principle for exchangeable arrays; unanchored arrays reconstruct the intrinsic twin-free quotient, while anchors recover the actual joining in a fixed model. To quantify this independence, we introduce Wasserstein dependence coefficients, establishing an all-order zero criterion for disjointness, and show that weak neighborhoods of the product joining always admit finite distance-array certificates. Examples from compact rotations, Bernoulli and reversible Markov shifts, common factors, Kronecker factors, and weak mixing demonstrate the strict necessity of the multi-particle level and the broad scope of this approach.

The ground state of Bose–Einstein condensates can be described as the minimizer of the Gross–Pitaevskii energy functional subject to a mass conservation constraint. In this paper, we study the corresponding discrete optimization problem in nonconforming finite element spaces and establish a priori error estimates for the discrete ground state energy, the discrete eigenvalue, and the discrete ground state. Specifically, we derive explicit convergence rates for the a priori error in the particular case of the $EQ_1^{\mathrm{rot}}$ finite element. Furthermore, we proof that within the $EQ_1^{\mathrm{rot}}$ finite element framework, the discrete ground state energy provides a lower bound estimation to the exact energy. Finally, numerical experiments are presented to validate the theoretical analysis.

This paper contributes to maximum distance separable (MDS) and near MDS (NMDS) properties of the extended generalized twisted Reed-Solomon (TGRS) codes. Firstly, a family of extended TGRS (ETGRS) are constructed by appending three columns to the generator matrix of original TGRS codes. Secondly, the necessary and sufficient conditions for these codes to be MDS or almost MDS (AMDS) codes are derived. Then, by analyzing the AMDS properties of their dual codes, the necessary and sufffcient conditions for them to be NMDS codes are established. Furthermore, some examples are given to verify the main results. Finally, we determine the non-generalized Reed-Solomon (non-GRS) characteristics of them via the Schur product method.

In this paper, we derive quantitative convergence rates for stochastic processes associated with resistance forms. While the qualitative convergence of heat kernels and semigroups under the Gromov-Hausdorff-vague convergence of underlying measured resistance metric spaces has been investigated previously, their quantitative convergence rates have remained unexplored. We establish explicit convergence rates for the associated semigroups and heat kernels under the assumptions of measure regularity and lower resistance estimates. Furthermore, we introduce a new metric that induces the Gromov-Hausdorff-vague topology, and is convenient for evaluation. As applications of our main results, we present two illustrative examples. First, we derive first estimate on the convergence rate for the random walk approximation of Brownian motion on the Sierpinski gasket. Second, we apply our results to the one-dimensional Bouchaud trap model, successfully extending the previously known parameter regime to all cases where homogenization occurs and improving the convergence rate estimates in the existing regime by at least a quadratic factor.

The purpose of this paper is to introduce a curious function of two variables, expressable via the employment of the Lambert W Funtion, which can be generalized to satisfy Euler's Equation of Inviscid Motion over a specific domain, with pressure independent of space variables.

An influential theorem of Nikiforov states that if an $N$-vertex graph $G$ contains at least $\gamma N^h$ copies of some fixed $h$-vertex graph $H$, then $G$ contains an $H$-blowup of order $c_H(\gamma)\log N$. We provide a new proof of this theorem, which in particular improves the best known bound on the constant $c_H(\gamma)$. In contrast to previous proofs, our proof is iterative, finding the blowup one vertex at a time.

We present a collection of observations concerning the peculiar behavior of the Lebesgue function in the setting of the interval $[-1,1]\subset \mathbb{R}$ and the square $[-1,1]^2\subset \mathbb{R}^2$. We provide numerical results and formulate several open problems related to the geometry of the Lebesgue function.

Given a mean zero functional $F$ of a Poisson measure on a metric space, we apply the Malliavin-Stein method to establish sharpened second-order Poincaré inequalities for $F/\sqrt{\operatorname{Var} (F)}$ in terms of fourth moments of difference operators. The rates of normal approximation are expressed in the Kolmogorov and Wasserstein distances and require fewer error terms than corresponding previous results. When $F$ is expressible as a sum of score functions which are distributionally close to scores having short-range structure, then we deduce that $F/\sqrt{\operatorname{Var}(F)}$ satisfies Berry-Esseen bounds. The normal approximation criteria of the scores, here called bounded Lipschitz localization, are more general than stabilization criteria and allow for unbounded interactions of scores. This approach yields Berry-Esseen bounds for local U-statistics on metric measures spaces, localizing functionals on hyperbolic space, as well as for Poisson functionals in a space-time setting, with infinite time horizon, including statistics of spatial birth-growth models and Laguerre tessellations.

Let $M$ be a matroid on a finite ground set $E$, and suppose that the automorphism group of $M$ acts transitively on $E$. We show the following: if $X_1,\ldots,X_K$ are sampled independently from a distribution $p$ on $E$, then the probability that the samples are distinct and that $\{X_1,\ldots,X_K\}$ is an independent set in $M$ is quasi-concave in $p$ and maximized when $p$ is uniform. For $2\le K\le\rank(M)$, we characterize all maximizers by uniform total mass on the parallel classes and, in particular, show that the maximizer is unique and has global quadratic stability when $M$ is simple. As a corollary, for a random $K\times N$ matrix over a finite field with i.i.d.\ rows, uniform sampling from the nonzero vectors maximizes the probability of full row rank. Finally, we show that uniform sampling maximizes $\Prob\big(\rank_M(X_1,\ldots,X_L)\ge r\big)$, and hence also $\E\rank_M(X_1,\ldots,X_L)$, for every $L, r$.

We develop and analyze high-order ensemble schemes for the unsteady Navier–Stokes–Darcy system with uncertain initial conditions, forcing terms, hydraulic conductivity tensors, and Lions-Beavers-Joseph-Saffman interface conditions. The proposed schemes which are called GSAV-GBDF$k$-Ensemble schemes integrate a partitioned decoupling strategy, the generalized scalar auxiliary variable (GSAV) approach, and generalized BDF$k$ discretizations. This framework achieves high-order temporal accuracy and long-time stability, permits explicit treatment of the nonlinear term, and facilitates an efficient ensemble implementation for multiple parameter realizations by sharing a single, unified coefficient matrix at each time step. Moreover, the numerical solutions are shown to satisfy uniform-in-time bounds without time-step restrictions. Owing to the ensemble formulation, the resulting linear systems share common coefficient matrices, which significantly improves computational efficiency. We further establish optimal-order error estimates for the proposed high-order schemes. Numerical results are included to confirm the theoretical analysis and to illustrate the accuracy, stability, and efficiency of the proposed methods.

We introduce and analyze a class of Surface Quasi-Geostrophic (SQG) equations in the presence of moving rigid obstacles. The model is motivated both by vortex-wave type asymptotics for singular structures in active scalar equations and by geophysical phenomena exhibiting rigid-like coherent regions, such as cyclone eyes or long-lived atmospheric dust clouds. We consider the critical SQG equation in a time-dependent exterior domain generated by a prescribed rigid motion and reconstruct the velocity through a nonlocal elliptic formulation adapted to impermeability constraints. The active scalar is assumed to remain constant inside the rigid region and in a neighborhood of its boundary, yielding a plateau structure compatible with the transport dynamics. For a single moving obstacle, we establish local well-posedness of classical solutions in Sobolev spaces $H^k$, $k\geq 4$ together with uniqueness, local stability, and a blow-up criterion. The analysis relies on a reformulation in adapted coordinates reducing the problem to a fixed domain, combined with integral representations for the fractional elliptic operator, regularization procedures, and a nonlinear fixed-point argument. A central difficulty comes from the critical singularity of the SQG Biot-Savart kernel in the case $s=\frac{1}{2}$, for which the velocity reconstruction near the moving boundary requires commutator estimates. We further prove propagation of the plateau property and derive a priori estimates controlling both the support of the scalar gradient and the Sobolev norm of the solution. This work provides, to our knowledge, the first well-posedness theory for SQG equations with moving rigid obstacles and constitutes a first step toward the rigorous derivation of point-vortex type dynamics from shrinking rigid bodies in SQG flows.

The aim of this paper is to develop a $(q,m)$-polymatroidal approach to higher supports and higher rank-weight enumerators of rank-metric codes. In this framework, we establish analogs of several fundamental results known for matroids and linear codes, including the description of minimal supports in terms of cocircuits of truncations and a Greene-type identity relating higher rank-weight enumerators to rank generating functions. We also show that the associated $(q,m)$-polymatroid and the higher support distributions determine each other. As a further application, we derive MacWilliams-type identities for higher rank-weight enumerators.

For a connected graph \(G\), let $q(G)$ denote the $Q$-index of $G$, i.e., the largest eigenvalue of its signless Laplacian matrix. Abreu and Nikiforov (2013) showed that \[ q(G) \leq 2n\left(1-\frac{1}{\omega(G)}\right), \] where $\omega(G)$ denotes the clique number of $G$. We first give a short algebraic proof of this result. For a vertex $v\in V(G)$, let \(c(v)\) denote the order of the largest clique of \(G\) containing \(v\). Our main result is the following vertex localized bound that refines the result of Abreu and Nikiforov: \[ q(G) \leq 2\sum_{v\in V(G)}\left(1-\frac{1}{c(v)}\right). \] Equality holds precisely for complete bipartite graphs when \(\omega(G)=2\), and for regular complete \(\omega(G)\)-partite graphs when \(\omega(G)\geq 3\). As a consequence, we also obtain an analogous localized inequality for the $A_\alpha$-matrix of $G$. Finally, we generalize the above localized inequality to vertex-weighted signed graphs. This contributes to the localization program for spectral Turán-type results.

Let $\Omega \subset \mathbb{R}^N$, $N \ge 2$, be a bounded domain with Lipschitz boundary, divided by a Lipschitz hypersurface $\Sigma$ into two open, disjoint Lipschitz subdomains $\Omega_1$ and $\Omega_2$. We study a nonlinear transmission eigenvalue problem driven by nonhomogeneous operators with $p_i$- growth in each subdomain $\Omega_i$, $i=1,2$, and subject to continuity and flux transmission conditions across the interface $\Sigma$. The real parameter $\lambda$ appears both in the equations and in the nonlinear boundary conditions. Using variational methods, we prove the existence of an unbounded sequence of eigenvalues. Under additional assumptions, we establish that the set of eigenvalues coincides with the entire interval $(0,\infty)$. As a particular case, we obtain the corresponding eigenvalue results for the associated single-domain problem.

Fluid antenna multiple access (FAMA) enables each user to rapidly switch among several closely spaced ports and select the strongest received signal. Although this mechanism offers micro-scale spatial diversity, its behavior in multiuser downlink systems with spatial correlation and linear precoding is not well understood. This paper develops a unified analytical framework for the multiple-input single-output (MISO) downlink with FAMA users served via maximum ratio transmission (MRT) or zero-forcing (ZF). We show that the per-port signal-to-interference ratio (SIR) follows a Beta-prime distribution with parameters \((M_{\mathrm{eff}},L)\), where \(M_{\mathrm{eff}}=M\) under MRT and \(M_{\mathrm{eff}}=M-U+1\) under ZF, and derive closed-form finite-sum cumulative distribution functions (CDFs) for both cases. We further provide the first analytical characterization of cross-port SIR correlation. \textcolor{black}{Furthermore, we derive rigorous outage probability bounds that tightly bracket the exact performance and become exact in the limiting cases of fully correlated and independent ports.} Asymptotic analyses reveal the fundamental diversity orders and tail behavior for each precoder. Numerical results confirm the accuracy of the SIR distributions, correlation model, and outage bounds, and show that MRT achieves weaker port correlation and larger selection gains than ZF when the base station (BS) has ample spatial degrees of freedom. The framework offers explicit guidelines for port configuration and precoder selection in practical FAMA systems.

In this paper, we study Toeplitz operators with a positive symbol on pluriharmonic Fock spaces over $\mathbb{C}^{n}.$ We characterize the conditions under which the Toeplitz operator $T_\mu$ is bounded, compact, or belongs to the Schatten class $S_1$. Furthermore, we give a necessary condition for a Toeplitz operator to belong to $S_p$ for $p \geq 1,$ and under an additional assumption, we prove the sufficiency of the condition.

Given a function $u$ defined on $\mathbb R^n$, its fractional $p$-Laplacian is given by $$(-\Delta)_p^su(\vec x)=C_1(n,s,p)\int_{\mathbb R^n}\frac{|u(\vec x)-u(\vec y)|^{p-2}(u(\vec x)-u(\vec y))}{\|\vec x-\vec y\|_2^{n+sp}}d\vec y,\quad\vec x\in\mathbb R^n,$$where the integral is understood in the principal value sense, $p\in(1,\infty)$, $s\in(0,1)$, and $C_1(n,s,p)$ is a normalization constant. A formally equivalent nonlinear Balakrishnan formulation is given by $$(-\Delta)_p^su(\vec x)=C_4(n,s,p)\int_0^\infty\Delta(t-\Delta)^{-1}\left[\Phi_p(u(\vec x)-u(\cdot))\right](\vec x)\frac{dt}{t^{1-sp/2}},$$ where $C_4(n,s,p)$ is another normalization constant, and $\Phi_p(t)=|t|^{p-2}t$. In this paper, we present a matrix-based spectral method to approximate numerically the fractional Laplacian (i.e., the linear case, where $p = 2$) and the fractional $p$-Laplacian for functions defined on $\mathbb R^n$. Our approach builds on the Balakrishnan representation, where we discretize the 2nd-order derivatives in $\Delta$ using spectrally accurate differentiation matrices. A key advantage is that these matrices can be diagonalized in a well-conditioned manner, enabling a stable and robust numerical scheme that naturally extends to arbitrary spatial dimensions $n$. In particular, this diagonalization allows the fractional operator to act directly on the eigenvalue spectrum, effectively reducing the Balakrishnan integral to an analytical evaluation at the spectral level and thereby avoiding costly multidimensional quadrature. The resulting method also avoids domain truncation and variational formulations, making it both computationally efficient and conceptually straightforward. As a practical application, we simulate the evolution of $$\frac{\partial u}{\partial t}+(-\Delta)^s_pu=0,$$in one and two spatial dimensions, being able to capture the self-similar solutions that arise as $t\to\infty$.

We present a Fourier–Galerkin asymptotic framework for the analysis and computation of subwavelength resonances in two-dimensional scattering problems in finite domains. Starting from the boundary integral formulation, we apply a Fourier–Galerkin discretization to derive an explicit finite-dimensional effective matrix whose kernel characterizes the resonant frequencies. In the subwavelength regime, we obtain asymptotic expansions of this matrix in terms of $\omega$ and the material contrast, identifying the leading-order operators and their kernel structure. This reduction transforms the resonance problem into a low-dimensional nonlinear eigenvalue problem, avoiding large-scale discretizations and global root-search procedures. The entries of the effective matrix are explicitly computable and admit fast evaluation using FFT-based quadrature. The resulting approach provides an efficient and robust computational framework for resonances in general smooth geometries.