Papers

The facial distance of a polytope, equivalently its pyramidal width, is a geometric condition measure that appears in the convergence bounds for Frank–Wolfe algorithms. Peña and Wirth developed calculus rules for this quantity and conjectured that the ratio of the facial distance to the diameter multiplied by the square root of the dimension is bounded by a constant. We show that this quantity is uniquely maximized by a regular simplex in each dimension, which implies that the conjecture holds and that $\sqrt{2}$ is the smallest constant. We also compute the facial distance of the Birkhoff polytope.

The dominated coloring of a graph $G$ is a proper vertex coloring such that each color class is dominated by a vertex. The minimum number of colors required for a dominated coloring of a graph $G$ is called the dominated chromatic number of the graph $G$ and is denoted by $\chi_{dom}(G)$. A graph $G$ is said to be chromatic dominated if $\chi(G)=\chi_{dom}(G)$. In this paper, we characterized the perfect graphs, which are chromatic dominated. Also, we studied the dominated chromatic number of some classes of unicyclic graphs.

In this paper, we prove that for $\ell>5$, the number of $\ell$-regular conjugacy classes of a finite group is at least that of the normalizer of a Sylow $\ell$-subgroup.

We construct Laurent phenomenon structures on reductive branching algebras. For $A_{2n-1}\downarrow C_{n}$, $n\ge 2$, and the exceptional inclusions $D_4\downarrow G_2$, $E_6\downarrow F_4$, and $F_4\downarrow B_4$, we identify the branching algebras with upper Laurent phenomenon algebras, keeping the frozen coefficients polynomial. Degree-fibred categories of projective presentations give a common construction. It also gives cluster seeds of type $A_1$ for $B_3 \downarrow G_2$ and $G_2\downarrow A_2$. We give uniform sufficient conditions identifying a specialized Keel–Yu mirror algebra with an upper Laurent phenomenon algebra admitting a suitable binomial seed. The resulting theta basis is simultaneously adapted to the frozen boundary valuations. For $A_{2n-1}\downarrow C_{n}$ with $2\le n\le5$ and for the other exceptional inclusions, we obtain homogeneous theta bases parametrized by lattice points of explicit rational polyhedral cones, whose weight fibres compute every branching multiplicity. We also prove mutation invariance of finite upper bounds for LP patterns over a UFD.

We prove that the strong spherical maximal operator in $\mathbb{R}^n$ is bounded on $L^p$ for $n\geq4$ and $p>(n+1)/(n-1)$. This establishes the conjecture of Hickman and Zahl in the remaining dimensions. The main ingredient is a local $L^2$ estimate for the pieces obtained by a dyadic decomposition in the normal variable. After squaring one input coordinate, we reduce this estimate to a $TT^*$ argument for diagonal quadratic phases.

This paper is concerned with the maximal regularity theory for non-autonomous stochastic evolution equations with a generalized fractional derivative in UMD spaces. The generalized time-fractional derivative provides a unified framework covering both the classical Riemann-Liouville and Caputo fractional derivatives, which accommodates a wider class of anomalous diffusion processes with intermediate memory effects. Based on the singularities of the initial term and the stochastic convolution kernel, a time-weighted space and a regular-singular decomposition are used to obtain the well-posedness, space-time regularity, and stochastic maximal $L^p$-regularity results. Our results are applied to non-autonomous stochastic diffusion equation and stochastic fractional reaction-diffusion SIR model.

Bayesian state estimation for high-dimensional nonlinear dynamical systems entails a fundamental tension between statistical fidelity and computational tractability, as particle weights can collapse, while Gaussian ensemble updates can miss non-Gaussian posterior structure. Score-based diffusion filters offer a sampling-based alternative, but existing training-free score filters often rely on heuristic likelihood corrections, which can compromise posterior accuracy by neglecting uncertainty about the system state associated with each noisy reverse particle. To address these issues, we propose AECSF, a training-free adaptive ensemble conditional score filter. AECSF constructs an analytically tractable score estimator from the conditional Tweedie identity, which recasts noisy posterior score estimation as estimating the conditional mean of the system state given a noisy reverse particle and the observation. To estimate these conditional means efficiently, AECSF employs a shared adaptive weighted proposal ensemble, while particle-specific conditional weights yield an estimate for each noisy reverse particle without separate proposal sampling. The proposal ensemble is updated using reverse-particle information within the same reverse-diffusion run to improve conditional-mean estimation. Theoretically, we characterize when a fixed weighted proposal measure yields the exact noisy posterior score. Under stated assumptions, we establish a bound relating conditional-mean estimation errors to reverse-sampling endpoint error. Numerical experiments demonstrate that AECSF improves the accuracy of posterior sampling and nonlinear filtering in high-dimensional problems with limited forecast ensembles.

Finite-difference discretizations of Laplace operators yield discrete Laplacians whose spectral properties are closely tied to the stability, convergence, and physical fidelity of numerical solvers. While symmetric discretizations are well understood, many high-order finite-difference schemes produce nonsymmetric matrices for which rigorous spectral analysis is overlooked. Surprisingly, we prove that two fourth-order schemes with Dirichlet boundary conditions yield nonsymmetric discrete Laplacians whose spectra are purely real and strictly positive. Computer-assisted computations show that this property does not hold for higher-order extensions of these schemes in general.

We develop a structure-preserving locally optimal block preconditioned conjugate gradient(LOBPCG) method for computing several of the largest positive generalized singular values of a matrix pair \((A, B)\), where \(B\) has full column rank, together with the associated generalized singular vectors. The method is derived from the Hermitian-definite Jordan–Wielandt pencil and therefore avoids explicitly forming the normal-equation matrices. By exploiting the block structure and spectral symmetry of this pencil, we construct separate search subspaces for the left and right generalized singular vectors. The resulting Rayleigh–Ritz procedure requires only a singular value decomposition of a small projected cross matrix. We adapt the improved Hetmaniuk–Lehoucq (IHL) trick to construct stable conjugate search directions in the two component spaces. We also establish an explicit correspondence between the coefficient matrices produced by the componentwise and augmented Rayleigh–Ritz procedures. This correspondence permits the componentwise IHL construction to be used in an augmented, structure-preserving LOBPCG formulation without explicitly assembling the augmented basis. Numerical experiments demonstrate the computational advantages of the proposed method over unstructured and Gram-matrix-based LOBPCG implementations.

We construct a Lipschitz, non-$C^1$ map $u\colon \mathbb{R}^2\to \mathbb{R}^6$ which is the unique global minimizer of a smooth, strongly polyconvex functional. Such an example is necessarily non-homogeneous.

A mixed graph $M=(V,E\cup A)$ is acyclic if its directed part $(V,A)$ is an acyclic digraph. In this note we study the so-called orientation completion problem for the class of acyclic mixed graphs. That is, given an acyclic mixed graph $M$ and a property ${\cal P}$; can we orient the edges of $M$ so that the resulting digraph is acyclic and has property ${\cal P}$. We prove that one can decide in polynomial time whether $M$ can be completed to an acyclic digraph with an out-branching from a prescibed vertex $s$, while it is NP-complete to decide whether $M$ has an acyclic orientation which has both an out-branching and an in-branching (a bipolar orientation). We show that it is NP-complete to decide whether $M$ can be oriented so that it contains a directed path between two prescribed vertices. Finally we describe a polynomial algorithm for deciding whether an acyclic digraph $D$ has an out-branching $B^+_s$ such that the digraph $D-A(B^+_s)$ is connected (in the underlying sense). Based on this we pose as an open problem the complexity of deciding whether the edges of an acyclic mixed graph can be oriented so that the result is an acyclic digraph with a non-separating out-branching.

Given a plurisubharmonic function $\varphi>-\infty$ on the unit ball $\mathbb{B}^n$ and a constant $0<p\leq2$, Guan-Zhou's optimal $L^p$ extension theorem yields a sharp lower bound for the weighted $p$-Bergman kernel. We show that the equality holds at some point if and only if $\varphi$ is pluriharmonic on $\mathbb{B}^n$. This provides an analogue of the equality part of Suita's conjecture. As an application, we prove a Forelli-type result. Let $\varphi$ be a real-valued function on $\mathbb{B}^n$. If every slice of $\varphi$ through the origin is harmonic and if $\varphi$ is plurisubharmonic in a neighborhood of the origin, then $\varphi$ is pluriharmonic on $\mathbb{B}^n$.

Plug-and-Play (PnP) methods solve imaging inverse problems by incorporating deep denoisers into iterative optimization algorithms. Although practical implementations often decrease the denoiser noise level $\sigma$ along iterations, most existing convergence analyses assume a fixed denoiser. In this work, we establish convergence guarantees for a broad family of Plug-and-Play algorithms with annealed noise level, spanning deterministic methods (RED–GD and PnP–PGD) and stochastic methods (SNORE, equivariant RED, and a variant of PnP–Flow). For each method, we identify an explicit, nonconvex objective associated with the terminal denoising level and prove asymptotic stationarity of the iterates with respect to this objective. Our analysis does not prescribe any decay rate for the noise schedule, and our assumptions cover both learned gradient-step denoisers and exact MMSE denoisers. Overall, our theoretical results bridge the gap between existing PnP convergence theory and the decreasing-denoising practices used by state-of-the-art image restoration methods. We empirically demonstrate the benefits of such schedules and illustrate the predicted convergence behavior on several imaging inverse problems, including inpainting, super-resolution, demosaicing and tomography.

Let \(0<\alpha\leq1\), and let \(g_\alpha\) be the Littlewood-Paley function associated with the fractional Poisson kernel, which reduces to the classical \(g\)-function when \(\alpha=1\). We establish dimension-free \(L^p\) bounds for \(g_\alpha\) for every \(1<p<\infty\) and identify its exact \(L^2\) norm as \(\sqrt{\alpha/2}\). We further determine the precise asymptotic behavior of the weak-type \((1,1)\) constant as \(n\to\infty\), yielding quantitative upper and lower bounds in terms of the dimension.

We construct a sequence of AE Kähler metrics on C^2 with nonnegative and integrable scalar curvature whose ADM mass tends to zero but which, for a suitable choice of basepoints, fails to converge to Euclidean space in the pointed Gromov-Hausdorff sense. This shows that, without any additional assumptions, the excision in Klemmensen's stability result for the positive mass theorem for Kähler manifolds is crucial, which aligns with the findings in the (3-dimensional) Riemannian case.

The present study explores integer and rational matrix solutions for the Diophantine equation $X^{4}+Y^{4}=2Z^{4}$, establishing that infinitely many solutions exist over different matrix rings, including $M_2(\mathbb{Z})$, $M_3(\mathbb{Z})$, $M_4(\mathbb{Z}).$ Furthermore, expanding the scope beyond the integer domain, the analysis demonstrates that an infinite number of matrix solutions can be found in degree-4 and degree-16 field extensions, specifically within $M_4(\mathbb{Q}(\sqrt[4]{l}))$ and $M_4(\mathbb{Q}(\sqrt[4]{l_1}, \sqrt[4]{l_2}))$ when the rational numbers involved lack perfect fourth powers and are not perfect squares.

Geometrically parametrised PDEs arise in shape optimisation, where solving a PDE over many deformed domains at manageable cost is a central challenge. Free-form Deformation (FFD) parametrises shape variations through a lattice of control points, but the resulting parameter space is typically high-dimensional, with most directions barely affecting the quantity of interest (QoI). We address this with a two-stage reduction pipeline. An Active Subspace (AS)-guided structured control-point selection identifies the low-dimensional subspace of FFD weights driving the QoI. A Sinkhorn Autoencoder (SAE) is then trained on the reduced weights, learning their distribution in a compact latent space by minimising the Wasserstein distance between aggregated posterior and prior. We compare three non-intrusive Reduced-order Models (ROMs), mapping full FFD weights, AS-reduced weights, or SAE latent codes to the QoI, each using Random Forests, Gaussian Processes, and K-Nearest Neighbours. We test the methodology on a non-linear heat equation with Gaussian random Robin boundary data on a deformed Stanford Bunny, and on drag prediction for the DTC Hull bulb under FFD deformations. In both cases, the generative model captures the bulk of the QoI distribution over a comparable support, thinning the shoulder opposite the design objective, while its extremes in the design-relevant direction match the FFD sample within numerical and sampling uncertainty. ROMs learned in the SAE latent space yield smaller test errors than those on full FFD weights in most regressor-metric combinations, whereas AS-reduced weights give no comparable gain. Only the latent representation attains a positive predictive coefficient $Q^2$.

Let $A$ be a commutative associative unital algebra over a field $K$ of characteristic zero, and let $(G,\rho)$ be a Lie–Rinehart algebra over $A$. We revisit the Gerstenhaber structure carried by the exterior algebra $\bigwedge_A G$ from the point of view of graded derivations. Particular attention is paid to the well-definedness of the Schouten–Nijenhuis extension over the coefficient algebra $A$: the Lie–Rinehart identity is shown to provide exactly the correction terms needed for the bracket to descend to the $A$-balanced exterior algebra. After establishing graded antisymmetry, the Leibniz rule and the graded Jacobi identity without circularity, we study the inner graded derivations $\adSN(P)=[P,\cdot]_{SN}$ and prove \[ [\adSN(P),\adSN(Q)]=\adSN([P,Q]_{SN}). \] Finally, we give a complete proof of the converse construction: Gerstenhaber brackets on $\bigwedge_A G$ are in one-to-one correspondence with Lie–Rinehart structures on $G$.

Let $(\G,\rho,\omega)$ be a symplectic Lie–Rinehart–Jacobi algebra over a commutative algebra $A$, equivalently a Lie–Rinehart algebra $(\G,\bar\rho)$ endowed with a $1$-cocycle $\chi$ and a nondegenerate $2$-form closed for the twisted differential $d_\rho=d_{\bar\rho}+\chi\wedge$. Starting from the standard extension by the $2$-cocycle $\omega$, we study the strict lifting problem for its canonical $1$-form $\eta$ in the LRJ setting. We prove that the strict symmetries of $\eta$ are precisely the elements $X_f+fe$ and that $f\mapsto X_f+fe$ identifies the Jacobi Lie algebra $(A,\{\,,\})$ with this Lie algebra of strict symmetries. For locally Hamiltonian elements, the obstruction to a strict lift is the class $[i_x\omega]\in H^1_\rho(\G,A)$, yielding the exact sequence $0\to\ham\to\hamloc\to H^1_\rho(\G,A)\to0$. We also relate Okassa's symplectic curl to the variation of $d\eta$ and establish a conformal triviality criterion. In the smooth case the framework reduces to locally conformally symplectic geometry, and on a closed surface of genus $g\geq2$ we obtain structures that are not conformally trivial and whose obstruction space has dimension $2g-2$.

Let \begin{equation*} f_p(x)=\frac{\mathcal{K}(\sqrt{x})}{p-\ln\sqrt{1-x}}, \end{equation*} where $\mathcal{K}$ denotes the complete elliptic integral of the first kind, and set $g_p=1/f_p$. Tian and Yang conjectured that, at $p=\ln4$, both $-f_{\ln4}^{\prime\prime\prime}$ and $g_{\ln4}$ are absolutely monotonic on $(0,1)$. Writing \begin{equation*} \frac{2}{\pi}f_{\ln4}(x)=\sum_{n=0}^{\infty}a_n(\ln4)x^n,\quad \frac{\pi}{2}g_{\ln4}(x)=\sum_{n=0}^{\infty}b_n(\ln4)x^n, \end{equation*} we prove \begin{equation*} a_n(\ln4)<0\quad(n\geq3),\quad b_n(\ln4)>0\quad(n\geq0), \end{equation*} thereby settling both conjectures. The proof of the first inequality is based on the representation \begin{equation*} a_n(\ln4)=\frac{(-1)^nC}{15^{n+1}}-M_n, \end{equation*} where $C>0$ and $(M_n)$ is a positive moment sequence. Its strict log-convexity, together with estimates for the initial coefficients, determines the sign of every $a_n$. The second inequality follows from a factorization of the quadratic truncation of the normalized Taylor series and a reciprocal-series argument.

We consider the quadratic NLS system \begin{equation*} \begin{cases} i\partial_tu+\Delta u=v\bar{u},\newline i\partial_tv+\kappa\Delta v=u^2, \end{cases} \qquad (t,x)\in\mathbb{R}\times\mathbb{R}^5, \end{equation*} where $\kappa>0$. If $\kappa=1/2$, which is called mass resonance condition, then scattering below the ground state is shown by Hamano. Moreover, when $\kappa \neq 1/2$, scattering of radial solutions is proved in Hamano–Inui–Nishimura (2021). In the present paper, we prove scattering below the ground state for $\kappa\neq1/2$ without radial symmetry. Our proof is based on the concentration compactness and rigidity method by Kenig–Merle (2006). For the rigidity argument, following Pausader (2010), we use a localized virial argument in the direction orthogonal to the momentum.

We study matrix semirings over the three-element flat additively idempotent semiring with elements one, a, and infinity, where the square of a is infinity. We determine the associated matrix-variety chain completely. The scalar, two-by-two, and three-by-three cases generate three distinct varieties, while every matrix dimension at least three generates the same variety. The proof shows that every failure of an identity in arbitrary dimension is already witnessed on three indices, and it yields a coordinate criterion for all identities in the stable variety. We also realize every graph semiring arising from a directed graph of in-degree and out-degree at most one as a divisor of a direct power of the two-by-two matrix semiring. Consequently, the variety generated by all three-nilpotent flat semirings is contained in the two-by-two matrix variety. Directed cycles and independent reversal identities then embed the power-set lattice of the odd primes into each interval between the base variety and a nontrivial matrix variety. Thus every such interval has continuum cardinality and contains continuum-sized chains and antichains. Finally, we determine the last three powers of the multiplicative subsemiring obtained by deleting the constant all-one matrix, and we develop general matrix operators on the lattice of additively idempotent semiring varieties, including stable closures, stable cores, and propagation of equality along matrix-dimension chains.

Solution of convection-dominated problems using stabilized methods often requires to specify stabilization parameters whose optimal choice is not known but which considerably influence the quality of the approximate solution. In a recent work of the authors, an approach for computing these parameters by machine learning was proposed and applied to the streamline upwind/Petrov-Galerkin (SUPG) method for convection-diffusion equations. In the present paper, this approach is studied numerically for unstructured meshes and extended to meshes obtained by anisotropic mesh adaptation.

An explicit finite identity basis is given for the eight-element semiring of upper triangular Boolean \(2\times2\) matrices in the signature \((+,\cdot)\). The basis consists of a known multiplicative basis, the ai-semiring laws, and 30 mixed identities, each using at most eight variables. The proof combines a finite basis for the multiplicative reduct with finite rules for shuffling words, duplicating a marked occurrence, and interchanging adjacent occurrences while adding prescribed witnesses. This converts the one-letter gap criterion into a derivation of every valid semiring identity. We also study the band subvariety, a distinguished 156-element interval of the subvariety lattice, congruences, flat members, and finite representations of free algebras. Every \(m\)-letter word has an equivalent subword of length at most \(m^2\) for \(m\geq2\), and the free algebras have doubly exponential rank growth. For arbitrary partially ordered sets, we determine the equational theory of Boolean relation semirings and of finite-support incidence semirings over nontrivial bounded distributive lattices: finite height \(h\) gives the theory of \(T_h\), while unbounded height gives precisely the identities of all additively idempotent semirings.

In the setting of a hyperbolic (polynomial) system $(\mathcal{V}, p, e)$ of degree $n$, an $n$-tuple $\mathbf{A} = \big[ a_1, a_2, \dots, a_n \big]$ is said to be an $e$-doubly stochastic $\mathcal{V}$-matrix if each $a_i$ belongs to the hyperbolicity cone, has trace $1$, and $a_1 + a_2 + \dots + a_n = e$. In this article, we characterize linear preservers of such $\mathcal{V}$-matrices, describe some connections between $e$-doubly stochasticity and majorization, and study extreme points of the set of all $e$-doubly stochastic $\mathcal{V}$-matrices. We show, for example, that $(i)$ when $n>1$, positive, unital, and trace-preserving transformations are (the only) linear transformations on $\mathcal{V}$ that preserve $e$-doubly stochasticity; $(ii)$ when $\mathbf{A}$ is $e$-doubly stochastic, the eigenvalue vector of the linear combination $\sum_{i=1}^{n} r_ia_i$ is majorized by the coefficient vector $(r_1, r_2, \dots, r_n)^{T} \in \mathbb{R}^n$; and $(iii)$ when $p$ is complete, $e$-doubly stochastic $\mathcal{V}$-matrices induced by (generalized) Jordan frames are extreme points of the compact convex set of all $e$-doubly stochastic $\mathcal{V}$-matrices.

We study equilibrium joining strategies in an M/M/1-type queueing system with strategic customers operating in a two-phase random environment described as a continuous-time Markov process. Strategic customers, upon arrival, choose whether to join or to balk based on available information and anticipated utility, considering the trade-off between reward from service and waiting cost. Four observation scenarios are analysed: fully observable (both queue length and environment phase are disclosed to a customer upon arrival), queue-only observable, environment-only observable, and fully unobservable. In each case, equilibrium joining strategies are analysed. In the environment-only observable and fully unobservable cases, explicit solutions and equilibrium conditions are derived under rapid oscillations and under very slow transitions between environment phases.

Motivated by recent work of Aguilera–Bagaria–Lücke and Aguilera–Bagaria–Goldberg–Lücke (ABGL), we analyze various effects of set-theoretic forcing upon the classes of exacting and ultraexacting cardinals. Using Magidor support products of Prikry forcings, we prove that an I2-embedding yields a transitive, set-sized model of ZFC with a proper class of exacting cardinals. We show that under Woodin's HOD Conjecture, no exacting cardinal can be a limit of exacting cardinals. In contrast, we prove that models of ZF containing large cardinals beyond choice have forcing extensions that are models of ZFC in which a regular cardinal is a stationary limit of ultraexacting cardinals. We also show that, assuming the HOD Conjecture, an analogue of the classical Levy–Solovay theorem holds for exacting and ultraexacting cardinals. Finally, we analyze the large cardinal properties possessed by exacting cardinals in HOD. We prove that if $\lambda$ is exacting and $V_\lambda$ satisfies the HOD Hypothesis, then $\lambda$ has strong large cardinal properties in HOD. Carrying forward forcing constructions from ABGL, we start with a model of ZFC with an I2-embedding and produce a model of ZF in which the successor of an exacting cardinal $\lambda$ is extendible in all models of the form $\mathrm{HOD}_x$ for $x\subseteq \lambda$.

We show that every left-invariant elliptic structure of minimal rank on an odd-dimensional compact Lie group admits an algebraic decomposition analogous to those obtained by Pittie in the complex case and by Charbonnel and Khalgui for left-invariant CR structures of hypersurface type. We then use this decomposition to prove that the cohomology associated with such an elliptic structure can be reduced to a computation on a suitable maximal torus.

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and every separable Banach space $V$, we use an oracle-relative two-sorted finite-extension construction, whose oracle records the local rational structure of $V$, to construct a separable Banach space $X_V$ such that $$ \operatorname{Cal}(X_V)=\mathcal{B}(X_V)/\mathcal{K}(X_V)\simeq \begin{pmatrix} \mathbb{K}&0\\ V^*&\mathbb{K} \end{pmatrix} $$ as Banach algebras. The identification of $V^*$ with the Jacobson radical is isometric. Consequently, every nonzero dual Banach space with a separable predual admits, after an equivalent renorming, a unital Banach-algebra structure isomorphic to the Calkin algebra of a separable Banach space. Moreover, there is a separable Banach space $X$ such that every separable Banach space is isometric to $\mathcal{J}/\mathcal{K}(X)$ for a closed two-sided ideal $\mathcal{J}$ of $\mathcal{B}(X)$, and the subspace–ideal correspondence preserves the canonical order.

For a unital Banach algebra $A$ with $\|1_A\|=1$ and a decreasing sequence $(I_r)$ of proper closed two-sided ideals of finite codimension, we introduce the bounded detection envelope $$\widehat A:=\operatorname{Env}^{b}_{(I_r)}(A)=\varprojlim^{b}(A/I_r,\pi_r^s),\qquad \|(a_r)_r\|=\sup_r\|a_r\|.$$ Combining vector-valued Bourgain–Delbaen extensions, oracle Turing-machine recursion, and Argyros–Haydon operator analysis, we construct a separable Banach space $X$ such that, isometrically as unital Banach algebras, $$\operatorname{Cal}(X)=\mathcal{B}(X)/\mathcal{K}(X)\simeq\widehat A. $$ We also characterize these envelopes as the unital Banach algebras with norm-one units admitting an isometric predual and countably many jointly faithful weak-star-to-norm continuous finite-dimensional unital representations. Applications include arbitrary finite-dimensional unital Banach algebras with norm-one units and their countable $\ell_\infty$-sums, as well as unitized coordinatewise algebras over normalized $1$-unconditional boundedly complete Schauder bases. These include incidence algebras, $\ell_\infty$, and the unitization of Tsirelson's original space. Further applications concern analytic operator algebras, Fourier–Stieltjes and Fourier multiplier algebras of countable discrete groups, and classical and quantum convolution algebras. In particular, we realize $H^\infty(\mathbb D)$, $\operatorname{Mult}(H_2^2)$, $\mathcal L_2$, $M_{\mathrm{cb}}A(\mathbb F_2)$, and $C^u(O_3^+)^*$ as Calkin algebras, with their canonical norms and products.