Papers

Positive-semidefinite matrices are most efficiently factored using the Cholesky decomposition. For indefinite matrices, the Cholesky factorization does not exist, and the alternatives face greater challenges in achieving numeric stability and preservation of banded structure. Here we pursue an analogy between the requirement for positive-semidefinite matrices and the solution of the quadratic equation x^2 = c for c <= 0. It is shown that a non-associative algebra, called the hemiplex numbers, allows the Cholesky factorization to be computed for arbitrary symmetric matrices. Crucially, the hemiplex Cholesky factorization does not require pivoting for its existence or stability, allowing it to preserve banded structure. For singular matrices it produces a parametrization of the null space, and provides opportunity for truncation of nearly-null directions in a manner similar to common usage of the singular value decomposition. The hemiplex Cholesky factorization may be a practically useful addition to the tools for solving symmetric linear equations.

In this note we furnish a new result concerning the well known théorème de structure by Leray related to a weak solution to the Navier-Stokes equations. We are able to furnish a new estimate on the left endpoint of the unbounded interval of regularity.

Let $\Gamma$ be a countable discrete sofic group acting by homeomorphisms on a compact metrizable space $X$ and $\Sigma$ a sofic approximation of $\Gamma$. We prove that for every $1\leq p<\infty$, the sofic $p$-metric mean dimension is equivalent to the sofic metric mean dimension, i.e there exists a common value $ D_\Sigma(X,\Gamma)\in\{-\infty\}\cup[0,+\infty]$ such that, $$D_\Sigma(X,\Gamma)=\mdim_{\Sigma,\mathrm M,p}(X,\Gamma) =\mdim_{\Sigma,\mathrm M,\infty}(X,\Gamma),$$ which answers a question of Hayes in \cite[Question 3]{Hayes}. Moreover, a tame-metric variational formula is established. That is for every $1\leq q\leq\infty$, $$D_\Sigma(X,\Gamma) =\inf_{\rho\in\mathcal T(X)} \underline{\mdim}_{\Sigma,q}(X,\rho), $$ where $\mathcal T(X)$ is the set of all compatible metrics on $X$ having tame growth of covering numbers.

Fluid antenna arrays (FAAs) offer a promising means of exploiting spatial degrees of freedom through adaptive port positioning. However, most existing communication models treat antenna ports as independent channel samples and therefore overlook the electromagnetic coupling that fundamentally governs compact apertures. This paper develops an electromagnetic-aware current-domain framework for planar FAAs. The proposed model integrates position-dependent multiport impedance, mutual coupling, radiated and accepted power, source-voltage feasibility, and channel variations into a unified baseband-compatible description. The framework is implementation-agnostic: the closed-form half-wave-dipole model adopted in this paper is only one instance and can be replaced by full-wave, measured, or surrogate impedance and embedded-pattern models. Building on this framework, we formulate two optimization-oriented design problems. The first addresses single-beam superdirective beamforming through the joint optimization of port currents and positions under sidelobe, current, voltage, and geometry constraints. The second maximizes the multi-user weighted sum rate via current-domain precoding and position optimization under accepted-power, current, voltage, and spacing constraints. In both cases, the electromagnetic model is not applied as an after-design correction, but is incorporated directly into tractable alternating algorithms with convex current or precoding subproblems and reduced-gradient geometry updates. Simulation results demonstrate that, when properly modeled, mutual coupling can be exploited as a valuable design resource, enabling lower sidelobes and persistent sum-rate gains over fixed-grid and random fluid-antenna benchmarks.

We consider a fractional Brownian motion $B$ with Hurst index $0<H<1/2$, and its maximiser $\tau$ on $[0,1]$. We show that the rescaled process $a^H(B_{\tau+\,\cdot\,/a}-B_\tau)$ converges in $C_{\mathrm{loc}}(\mathbb R)$ to a limiting tangent law that is $H$-self-similar, supported on nonpositive paths pinned at zero, and rerooting-rescaling invariant: rerooting the limit process at its maximum on any fixed compact interval separated from zero and rescaling again asymptotically reproduces the same law. We also identify the tangent law as the limit of two-sided finite-grid hard-wall laws as the mesh vanishes and both horizons diverge, which can informally be interpreted as conditioning fractional Brownian motion on a nonpositive path. As an application, we consider persistence probabilities for fractional Brownian motion: a tilted variant of $B$ yields a different tangent law with a finite left horizon and an infinite right horizon and we show that \[ \mathbb P(B_t\leq1\text{ for all }0\leq t\leq T) = \big(C+o(1)\big)\,T^{-(1-H)},\quad \text{as }T\to\infty, \] where the leading order coefficient $C\in(0,\infty)$ has an explicit representation in terms of the expected maximum and the tilted tangent law.

Photoacoustic tomography (PAT) is a promising modality for high-resolution biomedical imaging, motivating the need for reliable uncertainty quantification (UQ) of reconstructed images. Bayesian approaches provide a rigorous framework for UQ but remain computationally challenging for realistic three-dimensional PAT and are sensitive to numerical approximations in the governing wave equation. We develop a finite-element Bayesian UQ framework for PAT that accommodates complex computational domains and detector geometries while enabling large-scale three-dimensional inference. The proposed methodology reformulates the randomize-then-optimize (RTO) sampling strategy as a matrix-free algorithm that generates independent posterior samples using only forward and adjoint wave propagations. Particular attention is given to constructing an adjoint discretization that forms an exact transpose pair with the discrete forward operator while remaining consistent with the continuous PAT adjoint, enabling efficient least-squares solvers within the sampling procedure. We investigate the influence of temporal discretization, artificial boundary conditions, and adjoint consistency on posterior uncertainty and identify discretization strategies that avoid numerical artifacts. The framework is validated against exact posterior statistics, existing Bayesian PAT methods, and Hamiltonian Monte Carlo using the No-U-Turn Sampler (NUTS), and is demonstrated on a three-dimensional problem with approximately $2\times 10^5$ unknowns on a general finite-element domain. To the best of our knowledge, this is the first large-scale Bayesian PAT study on general three-dimensional finite-element geometries, and the methodology extends naturally to a broad class of linear PDE-constrained inverse problems.

Entanglement distillation is a fundamental task in quantum information theory. In this work, we prove that Werner states in arbitrary dimension are $2$-copy distillable if and only if they are $1$-copy distillable. This answers the longstanding open question of the $2$-copy distillability of Werner states. This is an important step on determining whether every non-positive partial transpose (NPT) state is distillable, which remains one of the central open problems in the field of entanglement distillation.

We prove a weighted sum formula for double Eisenstein series. Its corresponding identity for the generating series of multiple divisor sums was conjectured by the author in his master's thesis. The double Eisenstein series identity follows from the restricted double-shuffle relations proved by the author and Tasaka, while the proof of the divisor-sum identity is combinatorial and uses generating series.

We study a free boundary problem of minimising a functional containing a non-local term rewarding depth into the zero phase: for $u\geqslant 0$ on a bounded, open set $\Omega\subset\mathbb R^d$, we minimise $$J(u) = \int_\Omega \left(\frac12|\nabla u|^2 - fu\right) \;-\; \int_{\{u=0\}} F\big(\mathrm{dist}(x,\partial \{u=0\})\big)\;\mathrm{d}x. $$ This kind of functional arises, for example, from a two-membranes problem with an adhesive contact energy. We first address a well-definedness issue caused by the non-local, boundary-sensitive nature of the functional and prove existence of minimisers, establishing along the way a weak lower semicontinuity result for the non-local term. We then derive stationarity conditions for minimisers, including a variational (Euler–Lagrange type) inequality, a PDE on the positivity set, and, under a mild non-degeneracy assumption, a free boundary condition obtained via inner variations and a Danskin-type differentiation of the distance function.

We analyze the training dynamics of variational physics-informed neural networks applied to linear coupled parabolic convection–diffusion–reaction systems, called heterogeneous when only a subset of the components undergoes convective transport. In the neural tangent kernel regime, the gradient flow on the space-time variational residuals reduces to a linear differential system whose operator is a Gram matrix built from the space-time symbol of the system operator and the matrix tangent kernel. The main result is a double screening theorem. Under dominant convection, the Schur complement of this matrix relative to the block of convective components converges to an expression that involves only the diffusive block of the symbol, with no coupling term, together with the Schur complement of the tangent kernel. From this we derive four consequences, namely an exact identity quantifying the screened coupling energy, a degradation law for the training rate governed by the canonical correlations of the kernel, a bound on the condition number in terms of the Péclet number, and the non-participation of temporal frequencies in the screening mechanism. The growth of the condition number slows down shared-step gradient descent, whereas the continuous flow suffers no slowdown, which makes the training difficulty attributable to the optimizer rather than to the approximation. We finally show that the Adam optimizer, through its adaptive scaling, mitigates this difficulty provided that the architecture separates the parameters associated with the convective and diffusive components, confirming the architectural prescription that follows from the second screening. The predictions are validated numerically down to machine precision, on a two-dimensional exchanger, and by the full training of finite-width networks.

This short note provides two examples of aperiodic frieze patterns of the plane, supported on the rhombic Penrose tiling and the Godrèche–Lançon–Billard tiling. That is, we provide a decoration of their vertices with positive integers which satisfy the diamond rule, in analogy to the usual (in)finite frieze patterns as defined by Conway and Coxeter.

Let \(\cL_D^\theta\) be a Brownian loop soup of intensity \(0<\theta<1/2\) in a bounded smooth planar domain. We estimate the probability that one cluster of intersecting loop traces meets a fixed number of separated shrinking bulk discs. For every fixed \(q\ge3\), this probability is, up to an arbitrary loss in the logarithmic exponents, the product of the corresponding one-arm probabilities. The upper bound is not obtained by multiplying pairwise estimates or by a formal application of BK. We condition on the loops not confined to one target collar, charge a local arm down to their penetration radius at every target, and control the joint penetration depths by a multi-target Brownian loop-measure estimate. The latter follows from a marked Schur-complement expansion on finite killed networks, uniform planar capacity estimates, and a random-walk-loop-soup coupling. The lower bound uses winding separators, one macroscopic bridge loop, and Poisson FKG. The one-window exponent and the two-window estimate are due to Jego, Lupu and Qian.

Let $\mathrm{IMS}_n(m)$ count the $n\times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$. We determine the Ehrhart series $F_n(q)=\sum_{m\geq0}\mathrm{IMS}_n(m)q^m$ as reduced rational functions for $n=7$ and $n=8$. Their numerator–denominator degrees are respectively $(366,373)$ and $(540,548)$. Both numerators have positive integer coefficients, are palindromic and strictly unimodal. The proofs share one finite architecture: a signed SimpCone decomposition is evaluated by quotient characters over finite fields, a certified common denominator and Ehrhart reciprocity reduce the rational identity to finitely many coefficients, an explicit counting bound lifts modular congruences to integer equalities, and exact gcd computations prove reducedness. For order eight, a face-index pole certificate gives a degree-$598$ common denominator without enumerating the full face lattice, leaving $296$ independent coefficients in degrees $0$ through $295$. Once the candidate rational function is known, the first six production primes certify this finite prefix by the same bounded-coefficient argument used for order seven.

Let $M^2\to\mathbb{S}^4$ be a closed minimal immersion, let $S$ be the squared norm of its second fundamental form, and let $\lambda_1\geq\lambda_2\geq0$ be the eigenvalues of Lu's fundamental matrix. We classify all such immersions for which $S+\lambda_2$ is constant. We prove that the constant can only be $0$ or $2$. In the first case the image is a totally geodesic $2$-sphere; in the second case it is either a Clifford torus in a totally geodesic $\mathbb{S}^3$ or the Veronese surface in $\mathbb{S}^4$. In particular, there is no closed minimal surface in $\mathbb{S}^4$ with constant $S+\lambda_2>2$. Consequently, Lu's second-gap conjecture holds for minimal surfaces in codimension two. Together with the hypersurface result of Peng–Terng and the counterexamples of Li–Zhao in every codimension $m\geq3$, this completes the codimension picture for minimal surfaces.

The Ozsváth-Szabó contact invariant $c^+(\xi)\in\mathrm{HF}^+(-Y)$ of the link of a normal surface singularity equipped with its canonical contact structure $(Y,\xi)$ was transposed to lattice homology theory by Bodnár-Plamenevskaya. When considering a transverse algebraic knot $L$ in the link, the chain complex computing $\mathrm{HF}^+(-Y)$ can be equipped with an Alexander grading, and we can define an element $\mathcal{L}(L)$ in the bigraded theory $\mathrm{HFK}^+(-Y,L)$, which maps to the contact element by forgetting the filtration. We show that the Alexander grading (as defined by Ozsváth-Stipsicz-Szabó) of this element is invariant under all blow-ups of the underlying plumbing graph. Furthermore, we utilize the fact that for specific types of blow-ups, the resulting lattice chain complexes are filtered chain homotopic and the chains maps map this element in one chain complex to the other, thereby providing a partial combinatorial description of the Legendrian invariant.

We introduce and study discrete analogs of Scott's and Murillo-Wu's topological complexity of maps. We prove that these discrete analogs are contiguity invariants and are, in fact, equivalent. Furthermore, we establish the fundamental theoretical properties and computational aspects of the discrete topological complexity of simplicial maps.

We prove that given any sufficiently small and regular solution $B$ of the stationary Euler equations there exists an infinite dimensional family of solutions $(u,b)$ of the non-resistive magnetohydrodynamics equations (MHD) that relax to $(0, B)$. More precisely, $(u,b) \to (0, B)$ exponentially fast as $t \to +\infty$. This family may be viewed as lying in the stable manifold of the non-resistive MHD equations around the equilibrium state $(0, B)$. The problem whether it actually coincides with the stable manifold remains open. As a byproduct of our result, we provide a large class of global regular solutions of the non-resistive MHD equations. Another consequence is that any sufficiently small and regular solution of the stationary Euler equation is (non-trivially) topologically accessible via MHD from a large class of magnetic fields according to the definition of Moffatt and, in this scenario, the topology of the magnetic lines is (entirely) preserved in the limit $t \to + \infty$.

A logic is Post complete if it is consistent but has no consistent proper extensions. In this article, we systematically investigate the Post complete extensions of certain basic conditional logics. We identify all of the finitely many regular and normal Post complete conditional logics, and prove analogues of Makinson's embedding theorems. We also show that certain basic conditional logics have uncountably many Post complete extensions for which closure under some, but not necessarily all, rules peculiar to the conditional are relaxed. We reflect on what our results tell us about the structure of certain lattices of conditional logics and also draw some morals for multimodal logic.

We prove that there are infinitely many closed contact 3-manifolds supported by planar open books, each admitting infinitely many pairwise non-homeomorphic Stein fillings. This answers K3 Problem 4.105. As a corollary, there are contact 3-manifolds that admit infinitely many Stein fillings but do not admit arbitrarily large ones.

The paper studies a dynamical system in the Wasserstein space, where the evolution is governed by a Fokker-Planck-Kolmogorov equation with coefficients chosen from a prescribed set that depends at each point on that point and on the current measure. Under the assumption that the multivalued mapping determining the constraints is convex-valued, upper semicontinuous and satisfies certain growth conditions, we prove an existence theorem and establish the compactness of the solution set. Moreover, we study an optimal control problem for such dynamical systems.

We present an unfitted boundary algebraic equation method for the two-dimensional exterior/interior Stokes equations on a staggered MAC grid. By constructing an explicit free-space pair of velocity and pressure lattice Green's functions (LGFs) from free-space Laplace LGFs, we represent homogeneous fields using sources supported exclusively on thin staggered boundary layers. This formulation imposes physical Dirichlet data at cut points via local interpolation, while sampled-normal rank updates remove hydrostatic null modes associated with single or multiple obstacles. The workflow parallels that of classical boundary integral formulations and requires no artificial boundary conditions for exterior flows, but follows a discretize-then-represent route and does not require singular/near-singular quadrature. The resulting dense boundary system is solved via GMRES, utilizing a componentwise discrete Calderón preconditioner built from the scalar Laplace kernel and padded FFTs for fast volume convolutions. Extensive numerical validation, including multiply connected domains, narrow gaps, and Moffatt eddies, confirms discrete incompressibility to solver accuracy and recovers the expected Moffatt eddy scaling. We achieve second-order velocity and pressure convergence and bound maximum discrete divergence within numerical accuracy. The discrete Calderón preconditioner reduces the condition number by orders of magnitude and yields nearly mesh-independent conditioning in exterior configurations, while remaining effective—though more demanding—for narrow-gap and fine-grid interior problems.

We study the critical-exponent, or equivalently entropy, spectrum arising from free type-preserving actions on the biregular tree $\mathcal T_{r+1,s+1}$. For an action with a nonempty finite quotient core, the critical exponent agrees with both the volume entropy of the universal cover of the core and the topological entropy of the associated non-backtracking edge shift. The unrestricted spectrum is $[0,\frac12\log(rs)]$, whereas the finitely generated spectrum is a countable dense subset of this interval obtained by taking logarithms of Hashimoto spectral radii. We stratify this finite-state spectrum by the circuit rank of the quotient core. At each fixed rank, finitely many typed kernels parametrize all values. Moreover, every positive entropy value has only finitely many kernel–length realizations, up to type-preserving isomorphism, and every positive accumulation point of a fixed-rank spectrum belongs to a lower-rank stratum. At rank two, the corresponding exponential rates admit a complete inverse classification in terms of three explicit polynomial families, together with a finite exact membership test for algebraic-integer inputs.

The solution of the Cauchy problem for homogeneous abstract hyperbolic equations, together with its derivative, admits a vector representation in terms of a unitary operator group associated with a rotation matrix. A rational approximation of this unitary group is constructed and shown to possess optimal fourth-order convergence. The order of convergence is determined in accordance with the smoothness scale. Based on this rational approximation, a two-layer semi-discrete scheme is constructed for the approximate solution of Cauchy problems for nonhomogeneous abstract hyperbolic equations in both the linear and semilinear settings. The convergence properties of the scheme are examined in relation to the regularity of the solution.

Scattered data approximation with discontinuities is challenging due to the Gibbs phenomenon, which significantly reduces accuracy near interfaces. The recently introduced Non-Linear Partition of Unity Method (NL-PUM) addresses this by combining Radial Basis Function (RBF) interpolation with a non-linear Weighted Essentially Non-Oscillatory (WENO) strategy. While effective, NL-PUM's performance relies heavily on two fixed hyperparameters: the RBF shape parameter and the patch radius. This work extends NL-PUM by adapting both hyperparameters locally using Leave-One-Out Cross-Validation (LOOCV) minimized via Global Optimization with Optimistic Improvement (GOOI). Our main innovation is a smoothness indicator linking a discontinuity-aware shrinkage process to LOOCV-based radius selection: patches in smooth regions remain unchanged, while those near discontinuities automatically shrink to avoid crossing the interface. The resulting method, LOOCV-NL-PUM-GOOI, requires no prior knowledge of interface geometry and introduces no extra cost beyond standard adaptive shape parameter selection. Numerical experiments on synthetic test functions with jump discontinuities and a real-data application to Norwegian Fjords elevation data confirm that this approach substantially reduces approximation error near discontinuities while preserving full accuracy in smooth regions.

In this paper, we consider the Navier-Stokes initial boundary value problem in exterior domains with initial data in the Lebesgue space $L^p$, $p\in (2,3)$, and show the existence of a global weak solution. For the quoted solution, we also furnish a structure theorem. In particular, we see that the weak solution becomes regular after a certain instant of time and it is also regular a. e. in time. Although a general $L^p$-theory for local strong/mild solutions is well-established in the literature, a corresponding theory for global weak solutions in exterior domains seems to be new. Of course, our results hold in the particular cases of the Cauchy problem and the initial boundary value problem in a half-space.

For a self-adjoint operator $A$ on a Hilbert space, the dyadic resolvent representation of Castillo, Costin and Costin expresses the resolvent $R_A(i\lambda)=(A-i\lambda)^{-1}$ as a series in the unitary group $U_t=e^{-itA}$ sampled at the dyadic times $\{2^{-k}\}_{k\ge 0}$. We develop this representation structurally and through several spectral applications. We first exhibit the underlying operators as a one-parameter scale family obeying a Landen doubling recursion whose telescoping recovers the representation, and record its readings as a series of Zak transforms and as a dyadic filter bank. As a worked instance of the Zak-transform reading we obtain dyadic-Bessel series for the lattice Green functions of $\mathbb{Z}^d$, with closed-form special values: the lemniscatic constant $\Gamma(1/4)$ in two dimensions and Watson's integral in three. For the Laplacian $A=-\Delta$ on $\mathbb{R}^n$ we expand $R_A(i\lambda)\varphi$, for $\varphi \in L^1(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)$, as a series of convolutions with the free Schrödinger propagator, and derive explicit dyadic representations of the fundamental solutions of the Laplace and Poisson equations in $\mathbb{R}^3$ and of the one-dimensional heat equation. Finally, we reconstruct spectral data of $A$ from the dyadic representation: the spectral measures on $\sigma(A)$, including, through the limiting absorption principle, the absolutely continuous spectral density of $-\Delta+V$, together with the density of states and the spectral zeta function, the last reducing to the Riemann zeta function for $-\Delta$ on the circle and yielding the functional determinant of $-\Delta+m^2$ there. We close by showing that the dyadic samples determine $A$ uniquely, an exact anti-aliasing of the propagator, so that all of this spectral data is a function of the dyadic samples alone.

We construct a covariant representation of the causal logic for the Dirac system and the electron, based on the conserved Dirac probability current and a general current-to-localization procedure for achronal regions. A polynomial decay estimate, derived by a non-stationary phase argument, verifies the conditions required to define covariant achronal localization for the full Dirac system. Extending this localization to complete spacetime regions yields the corresponding causal-logic representation; restriction to the positive-energy invariant subspace gives the analogous construction for the electron. We establish several structural properties. On Euclidean space, Dirac localization agrees with canonical projection-valued localization, while its position operator differs from the Newton–Wigner operator by an explicitly determined bounded self-adjoint correction. Full Dirac achronal localization is projection valued and represents achronal separateness by orthogonality, thus satisfying microscopic causality. After compression to the electron subspace, it becomes positive-operator valued while preserving causality. We further extend known results from spatial to general achronal regions, prove separation and norm-one criteria, and analyze the high-boost limit, obtaining Lorentz contraction in a precise probabilistic sense. Finally, we discuss the electron–positron decomposition and the state transformation induced by position measurements, showing that measurement-induced positron production is universal and independent of the specific measuring device.

We study the spectrum of the Orr–Sommerfeld operator associated with the incompressible plane Poiseuille flow in the high-Reynolds-number regime. In the Tollmien–Schlichting eigenvalue region, we prove the existence and uniqueness of the lower and upper branches of the neutral curve. For each sufficiently small fixed wavenumber $\alpha$, the viscosities corresponding to the lower and upper neutral branches satisfy $\nu\sim |\alpha|^7$ and $\nu\sim |\alpha|^{11}$, respectively. Equivalently, for each sufficiently small viscosity $\nu$, the corresponding lower and upper neutral wavenumbers satisfy $\alpha^2\sim \nu^{2/7}$ and $\alpha^2\sim \nu^{2/11}$, respectively. We also establish the simplicity of the neutral eigenvalues and verify the transversal crossing condition on both neutral branches. The proof is based on a boundary-adapted version of the Rayleigh–Airy iteration scheme, together with precise expansions and refined estimates for the correction terms and their parameter derivatives. These spectral results verify the assumptions required in the classical Hopf bifurcation framework of Joseph–Sattinger \cite{JS1972} and Iooss \cite{Iooss1972}, and hence yield traveling-wave solutions bifurcating from the plane Poiseuille flow at the neutral points.

Let $\mathscr{B}$ be the Bruhat–Tits building of $\mathrm{PGL}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathscr{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincide. This allows us to use the local combinatorial definition on quotients $\Gamma\backslash\mathscr{B}$, without referring to the universal covering. When $\Gamma$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $\epsilon$-twisted variants $Z_k^\epsilon$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(\Gamma\backslash \mathrm{PGL}_n(F))$: $(1-u^n)^{\chi(\Gamma\backslash\mathscr{B})}L(\Gamma,q^{(n-1)/2}u)=\prod_{k=1}^{n-1} Z_k^\epsilon(\Gamma\backslash\mathscr{B},u)^{(-1)^{k+1}}$. This gives a uniform Ihara-type identity for all $\mathrm{PGL}_n$. We also extend the construction and the identity to $\mathrm{PGL}_n(D)$, where $D$ is a central division algebra over $F$; in that setting the residue parameter is $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.

We study the Chebyshev extremal problem on a compact Riemann surface $X$ of genus $g>0$. As an analog to monic polynomials, we consider admissible meromorphic functions having no poles away from a marked point $P_{\infty}$ (with adequate normalisation). For a nonpolar compact set $E\subset X\setminus\{P_{\infty}\}$, we show that the $n$-th root of the Chebyshev constant $t_n(E)$ converges to the capacity of $E$, and we establish the corresponding Bernstein-Walsh inequality. In the second part of the paper, using a Cauchy kernel adapted to Riemann surfaces, we study the refined Szegő–Widom asymptotics for the extremals as well as the Widom factors $$ W_n(E)=\frac{t_n(E)}{\operatorname{cap}(E)^n},$$ assuming that $E$ is a finite union of $p$ closed discs with analytic boundaries. The geometry of the Schottky double, which has genus $2g+p-1$, enters explicitly into these asymptotics. We find that there is no analogue of Faber-type asymptotics even for one single boundary curve. We conclude by constructing explicit examples in genus $1$ using the Weierstrass-$\wp$ function.