Papers

We characterize essential ideals in polynomial rings $R[(X_\lambda)_{\lambda\in\Lambda}]$ over a commutative ring $R$ with $1$. We present several characterizations, letting $R$ vary among notable classes of rings. The idea is to detect essentiality by checking the intersections with principal ideals generated by polynomials in a test class. In the general commutative case, we apply McCoy's zero-divisor criterion to construct an efficient test class of polynomials in terms of annihilators of content ideals. We then refine the test class and strengthen the result in several ways, assuming further properties on the coefficients ring $R$. Assuming that $R$ is Noetherian, we reduce the test class using the associated primes of $R$. When $R$ satisfies Serre's condition $(S_1)$, it suffices to use minimal primes. We also study the cases of $R$ Artinian, $R=\mathbb{Z}/n\mathbb{Z}$ and $R$ equal to the ideal-adic completion of an excellent ring, among others.

We employ a simplified form of Tod's ansatz—which is built on the LeBrun–Tod ansatz—to construct one-sided type-D Ricci-flat multi-centre metrics. These metrics are Hermitian non-Kähler and conformally Kähler, and they are determined by a pair of generating potentials: an axisymmetric harmonic function (different from the one originally proposed by Tod) in an auxiliary 3D flat space and its harmonic conjugate. We carry out a systematic study of these multi-centre metrics, including their rod structure, asymptotic structure and recovering from them various known closed-form examples. In particular, we show that, when fitted into the scheme of multi-soliton solutions on flat space constructed by the present author using the inverse-scattering method, these metrics, with number of centres $n\le 3$, have a free-soliton number 2 or 3 greater than their phantom-soliton number; we conjecture that this is true for general $n$.

The alternating direction method of multipliers (ADMM), as a landmark algorithm, has attracted tremendous research attention and extensive practical applications over the past two decades. It is well known that, although the two-block ADMM enjoys well-established theoretical convergence guarantees, its direct extension to the three-block case may fail to converge, as demonstrated by existing counterexamples [5]. However, to the best of our knowledge, the case in which the third constraint block is the identity remains unresolved: the existing literature gives neither a general convergence proof nor a counterexample for this subclass. In this paper, we give a negative answer: direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics. Using Codex with GPT-5.6 Sol, we construct an explicit rational counterexample candidate and verify it along a piecewise-affine reduction path; exact checks show that direct three-block ADMM on this instance produces a bounded nonconvergent orbit of period 66. Within the same Codex workflow, we further guide a study of multiplier relaxation and clarify when convergence can be restored at the fixed-instance and class levels: a problem-dependent small dual step can restore convergence, whereas no positive relative step works uniformly over the whole class. Furthermore, we also test the recent Kimi Code with Kimi K3 model without the Codex candidate or project-specific route guidance; along a different path it produces an exact locally attracting period-23 certificate, convertible to an equivalent all-identity instance. The comparison suggests that different research-harness configurations can shape the mathematical objects explored and the certificates pursued.

We study purely atomic measures whose range is a central Cantor set and characterize those central Cantor sets that are the range of exactly one such measure. Next, we extend the characterization to the case of symmetric Cantor sets using a different method of proof. Finally, we make a few initial observations and remarks on recovering a measure whose range is a Cantorval.

Let $f$ and $g$ be nonzero functions in $L^2([0,1])$. The woven weighted exponential system (associated with $f$ and $g$) is defined by $$\Wc(f,g)=\bigset{\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}\,|\,J\subset\Z}.$$ We say that $\Wc(f,g)$ is wovenly complete, (resp. wovenly minimal, a woven frame) if the weaving $\set{fe^{2\pi i nt}}_{n\in J} \cup \set{ge^{2\pi i nt}}_{n\in J^c}$ is complete, (resp. minimal, a frame) for all $J\subseteq \Z.$ In this paper, we study conditions that imply certain approximation properties of $\Wc(f,g)$, such as completeness, minimality and the frame property. We first provide a complete characterization of the woven weighted exponential systems that are wovenly complete. We also show that $\Wc(f,g)$ is a woven frame if $f/g$ is strictly positive or strictly negative over $[0,1].$ Additionally, several counterexamples are provided to show that certain seemingly correct conditions do not imply the desired approximation properties of $\Wc(f,g).$ All results presented in this paper apply equivalently to systems of regular translates and Gabor systems at critical density in $L^2(\R)$.

This paper studies periodic inventory oscillations in delayed supply-chain networks of identical warehouses arranged in a ring. After reducing the inventory–replenishment system to a second-order delay equation, we formulate the periodic problem as an equivariant compact perturbation of the identity, with symmetries generated by the ring structure, the odd nonlinearity, and time translations. Using equivariant degree theory, we characterize critical parameter pairs, compute the associated crossing numbers, and construct a local bifurcation invariant whose nonvanishing guarantees nonconstant periodic solutions and provides information about their spatial and spatio-temporal symmetry types. By restricting the two-parameter problem to an admissible transversal curve, we further establish a global continuation alternative for the bifurcating branches. Examples with \(D_7\)- and \(D_8\)-symmetry illustrate the computation of critical parameters, bifurcation invariants, and admissible orbit types. Finally, Floquet theory is applied to small-amplitude periodic branches, yielding a leading-order stability criterion for nonresonant perturbation modes. The results provide a unified framework for describing the existence, symmetry, global continuation, and leading-order stability of periodic inventory oscillations in delayed ring supply-chain networks.

We introduce a procedure to obtain lump solutions via the formation of a kink-antikink pair, consisting of the superposition of kinks whose distance from the origin is controlled by a single parameter $a$. For large values of $a$, a wide plateau appears in the solution, which we call a large lump. The procedure involves the use of a first-order equation that allows the construction of the potential associated with the lump solution. We then investigate several known scalar field models where the parent kinks are capable of giving rise to novel lumps. The lump inherits the tails of the parent kink, allowing for either short-range exponential profiles or long-range profiles characterized by distinct power-law decays. We also show how to verify if an arbitrary lump solution can be obtained via our method and illustrate this possibility with a novel vacuumless lump.

A classical class number relation of Hurwitz expresses the Fourier coefficients of the product of a unary theta function and the class number generating function. Here, we establish an infinite family of analogous class number relations obtained by replacing the unary quadratic form $m^2$ by positive-definite binary quadratic forms. These identities involve Cohen's generalized class numbers and depend only on the genus of the underlying quadratic form. For this, we construct a genus-dependent level-lowering operator.

We consider the notion of shattering introduced by Ben Arous and Jagannath for spherical pure $p$-spin glasses with overlap $q$. For every $p\geq 3$ and $0<\beta\leq\beta_{\mathrm{sh}}(p)$, we rule out shattering whenever $q\leq2^{-1/2}$ or $q>\sqrt{(p-2)/(p-1)}$. The proof combines a deterministic $N+1$ bound for disjoint bands in the first range with a general-$p$ sign law showing that their total marked weight has subdominant free energy in the second. A spherical-code bound and Hölder's inequality give an additional $q$-dependent obstruction; in particular, they rule out every fixed overlap for $0<\beta\leq\sqrt{\log2}$. For $p=3$, the first two ranges already exhaust every fixed $q\in(0,1)$, so the landscape is not shattered at any $T\geq T_{\mathrm{sh}}$. For $p\geq4$, the cases not covered by our criteria are confined to $2^{-1/2}<q\leq\sqrt{(p-2)/(p-1)}$ and $\sqrt{\log2}<\beta\leq\beta_{\mathrm{sh}}(p)$. In particular, this paper partially resolves Conjecture 1 of the paper above and also suggests new methods to show non-shattering.

This work provides explicit expressions, in all sectors of the Fock space: soliton, antisoliton and breather, for the fundamental building blocks -the form factors- of the generalised integral kernels which represent the exponential of the field operator in the 1+1 dimensional Sine-Gordon quantum field theory. The obtained form factors reproduce the various results obtained earlier in the literature using, in particular, the free field approach.

A geometric model for string algebras was recently established in \cite{BC24}. Building upon this framework, we characterize the class of string algebras whose geometric models are unique up to equivalence of labelled tiled surfaces. Moreover, we provide a necessary and sufficient condition for all geometric models of a string algebra to be entirely free of red punctures, and further give a combinatorial description of string algebras with a common red puncture across all geometric models. In addition, we derive a sufficient condition for a string algebra guaranteeing the presence of red punctures in all geometric models.

We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Ampère equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.

For a finite connected simplicial complex $X$, the strand map $\iota_\ast$, from $\mathbb{P}_n(X)$ to $\prod_{i=1}^n\pi_1(X,x_i^0)$, sends a pure braid to the homotopy classes of its strands. A theorem of Goldberg (1973) computes its kernel when $X$ is a closed surface other than $S^2$ and $\mathbb{RP}^2$: the kernel is the normal closure of the pure braids supported in an embedded disc. We extend this picture to arbitrary finite connected simplicial complexes. Call $X$ $\textit{weakly Goldberg}$ if some contractible subcomplex $X_0\subseteq X$ realises Goldberg's description, $\ker\iota_\ast=\left\langle \operatorname{im}(\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)) \right\rangle$, and $\textit{Goldberg}$ if $X_0$ can moreover be chosen so that $\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)$ is injective. We prove that the strand map is surjective if and only if $X\not\cong S^1$; that $X$ is weakly Goldberg if and only if its free part is a forest; and that $X$ is Goldberg if and only if it admits an $\textit{admissible tree}$ – a maximal tree of a scaffold, compatible with the boundary and interior types of the attachments of the free part to the thick components. We also classify the complexes for which the kernel is trivial, settle the exceptional surfaces $S^2$ and $\mathbb{RP}^2$, and obtain complete answers for manifolds and for graphs. The main tools are a graph-of-spaces decomposition of the configuration space at a point of $X$ and a resolution procedure reducing an arbitrary complex to a simple model.

One can distinguish four major formulations of the Bloch-mode problem in linear periodic media. First, there are two common complementary ways of slicing the Bloch variety (Bloch wave vectors paired with the corresponding frequencies): frequency vs. wave vector or, conversely, wave vector vs. frequency. This paper deals exclusively with the latter option, directly applicable to lossy media, evanescent waves, and complex band structures. A separate crossroads is the full field (FF) vs. the lattice-periodic Bloch factor (PF) formulations. These are fully equivalent on the continuous level, but common discretization techniques may break this equivalence not just approximately but qualitatively. In the FF problem, the primary unknown eigenvalue is the Bloch phase factor (not the wavenumber), which enters only through opposite-boundary coupling. The PF formulation, on the other hand, injects the Bloch wavenumber into the differential operator and leads to a volume quadratic pencil. Consequently, standard PF discretizations of the type considered here, in contrast with the corresponding FF discretizations, violate the reciprocal-lattice (Brillouin-zone-shift) covariance – which, as the theory and numerical examples in the paper show, may lead to inaccurate or even nonphysical modes. From the mathematical perspective, the paper highlights the role of structure-preserving algorithms. The practical importance of the results and recommendations stems from the wide adoption of PF formulations in optics, photonics, and beyond – such as the computation of propagating and evanescent Bloch modes in photonic and acoustic structures, topological photonics, effective-mass and topological insulator band theories.

We present new inference tools for change point detection in high-dimensional time series. We discuss two distinct statistical applications: First, sequential change point testing in an incoming data-stream. Second, retrospective localization of multiple changes, with confidence intervals at a globally controlled error level. Test statistics are built on the maximum norm to generate power against sparse and asynchronous changes. Both problems are tackled by related multiscale statistics that search for changes in the data at many different levels of resolution. For fixed dimension, our statistical approaches can be validated using traditional Hölderian invariance principles. In this paper, we present the high-dimensional analogue: Hölder-Gauss-approximations, which can be (roughly) interpreted as the Gaussian approximation for a Hölder-norm of the high-dimensional partial sum process. Such approximations are of interest beyond change point detection and can be used for other problems such as for stationarity testing in high dimensions. We evaluate finite-sample performance in a simulation study and give an application to air contamination due to wildfires in California, which occurs asynchronously across a panel of measuring stations.

We give a deterministic proof of Bourgain's slicing theorem. The proof uses neither stochastic localization nor stochastic calculus, does not rely on Guan's covariance bound, and avoids $M$-ellipsoids, Pisier's regularization machinery, and complex interpolation. Instead, the argument is based on a deterministic one-parameter quadratic perturbation of the measure, a covariance-survival principle, and a geometric analysis of $L_p$-centroid bodies.

We introduce and study the category of unipotent character sheaves on the non-degenerate locus of the Vinberg semigroup associated to a reductive group of adjoint type. We show that appropriate perverse objects in this category can be glued from perverse parabolic character sheaves on the strata. As an application, we study the intermediate extensions of simple unipotent character sheaves from the group to its wonderful compactification, proving a conjecture of He. We also disprove a conjecture of Lusztig regarding the intermediate extension of the Steinberg character sheaf.

We study minimum degree conditions for tight Hamiltonian cycles in uniformly dense $3$-uniform hypergraphs. We prove that for every $d,\alpha>0$, every sufficiently large $(\rho,d)$-dense $3$-graph on $n$ vertices with minimum codegree at least $(1/3+\alpha)n$ contains a tight Hamiltonian cycle. This resolves a problem of Aigner-Horev and Levy in a stronger form, and the constant $1/3$ is asymptotically best possible. We also show that uniform density does not lower the asymptotic vertex-degree threshold: there are $(\rho,d)$-dense $3$-graphs with minimum vertex degree $(5/9-o(1))\binom{n}{2}$ and no tight Hamiltonian cycle. Finally, we construct $(\rho,2-\sqrt{3})$-dense examples with minimum codegree $(2-\sqrt{3}-o(1))n$ and no tight Hamiltonian cycle, answering negatively a question of Araújo, Piga and Schacht.

We study the action of the rational multiplication by $N$ map on the Chow groups of degenerations of principally polarized abelian varieties of torus rank at most one, as well as its interaction with the Fourier transform. As applications, we compute the class of the unit section, prove a generalized weight decomposition of the relative Chow motive of the universal family of such degenerations, and completely determine its tautological ring.

Wasserstein distributionally robust optimization (DRO) is a versatile and widely adopted framework for decision-making under uncertainty, yet its standard deterministic reformulations generally contain non-convex inner subproblems that are challenging to solve. To address this issue, we propose a shrinkage path heuristic that reduces the solution of a DRO problem to a one-dimensional search over the line segment connecting the (typically benign) sample average approximation (SAA) and the (more demanding but practically solvable) classical robust optimization solution. We derive a priori suboptimality bounds in stylized settings and, for the general case, a posteriori bounds obtained by applying a similar heuristic to a dual formulation. Numerical experiments on a multi-item newsvendor and an appointment scheduling problem show that the shrinkage path heuristic attains 85-110% (resp. 45-70%) of the out-of-sample performance improvements of Wasserstein DRO over SAA, at a fraction of the computational cost.

We study optimal control problems for incompressible fluids in two-dimensional domains with a cold boundary and internal heat sources and sinks. Given a kinetic energy budget, measured by the square of a nondimensional Péclet number $\mathrm{Pe}$, the goal is to maximize a cooling efficiency inversely proportional to the mean square gradient of the fluid's temperature. Using Lagrange duality, we formulate a well-posed dual problem whose solution yields an upper bound on the maximum cooling efficiency $\mathcal{E}(\mathrm{Pe})$. We then numerically approximate the dual problem using a convergent hierarchy of semidefinite programs obtained via discretization. We illustrate this computational approach on optimal cooling problems in a square and in an annulus, explaining also how problem symmetries can be exploited to reduce computational complexity. Finally, we construct admissible points for the dual problem to obtain new analytical upper bounds on the optimal cooling efficiency $\mathcal{E}(\mathrm{Pe})$. Specifically, we prove that $\mathcal{E}(\mathrm{Pe}) \lesssim \mathrm{Pe}^{2}$ for arbitrary domains and heat distributions, and that $\mathcal{E}(\mathrm{Pe}) \lesssim \mathrm{Pe}^{2}/ \ln^2\mathrm{Pe}$ for cooling flows in disks and annuli with heat source/sink distributions with a uniformly positive azimuthal average. These results generalize and improve known efficiency bounds for energy-constrained cooling flows in a disk.

Solving parametric partial differential equations (PDEs) repeatedly for many parameter values,as needed in optimal control, uncertainty quantification, and inverse problems, is often prohibitive with classical discretizations. This chapter introduces Reduced Order Modeling (ROM), which builds compressed yet accurate representations of parametric solution sets for fast online evaluation. We formulate the approximation of parametric PDEs as a supervised learning task and review linear approximation, very effective for elliptic and parabolic problems, before turning to nonlinear methods needed when solutions exhibit discontinuities or steep gradients. We then show how to use reduced-order models to efficiently recover the state of a physical system from limited measurements, which is an inverse problem known as state estimation. Throughout, we favor results with theoretical guarantees while also discussing practical methods, and provide a companion Jupyter notebook implementing the main algorithms.

While infinite-dimensional inverse problems are traditionally analyzed assuming continuous data, practical applications rely on finitely many discrete measurements. Recent deterministic approaches establish that unknowns belonging to a $d$-dimensional model class can be stably recovered from finitely many measurements. However, for severely ill-posed problems, such as the Calderón problem and inverse scattering, the known constructions may require a number of measurements that is exponential in $d$. We show that random sampling reduces this count dramatically if one asks only for exact identifiability. By exploiting the analytic geometry of the forward maps, we prove that, whenever the infinite-data problem is injective on the model class, $2d+1$ random scalar measurements determine the unknown uniquely, almost surely. Applications are given to the Calderón problem, with both infinite- and finite-dimensional boundary sampling, and to inverse medium scattering from randomly sampled far-field values.

Many control problems are classically formulated using integral costs that capture the cumulative performance of a system. Peak or worst-case behavior is captured via supremum or $L^\infty$-costs in another variety of control problems. When both cumulative and peak performance are important, it is natural to consider objective functions that combine the two costs. Although each criterion is well studied, their combination has not been explored extensively and we precisely work on such control problems. Towards establishing the existence, we first consider the relaxed framework, where control is considered using probability distributions. Using the well-known compactness and convexity properties of such control spaces, we establish the existence of an optimal relaxed control—we eventually establish the existence of an $\epsilon$-optimal pure (or classical) control, for every $\epsilon > 0.$ Despite these existence results, computing optimal policies remains challenging due to the non-smoothness introduced by the supremum term, and it is not clear whether the dynamic programming principle holds for our combined problem. To address this, we introduce a family of smooth approximations that yield standard control problems with well-defined optimal (pure) solutions. Using Maximum Theorem, we establish that the solutions of the smooth problems among pure controls form $\epsilon$-optimal for the original problem, with $\epsilon$ tending to zero as the smoothing parameter converges to zero. Finally using the methods proposed in this paper, we study a queueing problem to illustrate (among others) that the required trade-off between peak congestion levels and cumulative performance can be achieved.

In this paper, we study the Cauchy problems for a semilinear damped wave equation with a time-dependent coefficient for the damping term belonging to the class of non-effective damping terms with critical decay rates involving iterated logarithmic factors. As nonlinearities we consider both $|u|^p$ and $|u_t|^p$. Assuming nonnegative and compactly supported initial data, we establish the blow-up for weak solutions in the sub-Strauss range $1 < p \leq p_{\mathrm{Str}}(n)$ for the power of the nonlinear term $|u|^p$. The proof in the sub-critical case relies on an iteration frame for the space average of the solution, obtained by employing a time-dependent multiplier related to the coefficient of the damping term. On the other hand, in the limit case $p=p_{\mathrm{Str}}(n)$, we work with solutions of the homogeneous equation with separated variables, and we investigate the properties of a fundamental system of solutions for the corresponding time-dependent ODE, in order to derive an iteration frame for a suitable weighted space average of the solution. Finally, for the derivative type nonlinearity $|u_t|^p$ we prove the blow-up of weak solutions in the sub-Glassey range $1 < p \leq \frac{n+1}{n-1}$ by using a comparison argument for a suitable time-dependent function associated with the corresponding local in time solution.

The immersed boundary (IB) method models fluid–structure interaction using the natural Lagrangian and Eulerian formulations of structural mechanics and fluid dynamics, respectively, but explicit time discretization of the IB force imposes a stiffness-dependent upper bound on the time-step size. Treating these forces implicitly removes this restriction, but the resulting coupled linear systems become increasingly difficult to solve as the structural stiffness increases. Algebraically eliminating the Lagrangian degrees of freedom yields a reduced Eulerian velocity–pressure IB system. Here we introduce a coupling-aware Vanka (CAV) smoothing strategy to enable effective multigrid preconditioning of this system. CAV patches are built as unions of standard pressure-centered Vanka patches, with the graph of the Eulerian elasticity matrix determining which patches are combined. Under grid refinement, CAV patch sizes remain bounded, and the computational cost of each multigrid cycle thereby grows linearly with the number of Eulerian degrees of freedom. Tests using target-point, membrane, and beam force laws show that CAV-preconditioned FGMRES reduces the relative residual by ten orders of magnitude in $9--15$ iterations, with little growth under grid refinement. In a nonlinear benchmark modeling flow past a flexible fiber, the average number of FGMRES iterations per Newton solve increases only from $8.6$ to $9.5$ as the Eulerian grid is refined from $32\times32$ to $256\times256$ cells. To our knowledge, CAV provides the first robust multigrid strategy for time-dependent implicit IB formulations.

In this paper, we investigate the eigenvalues of character degree graphs, with particular emphasis on the arithmetic properties of their spectra. First, we study \((n-2)\)-regular character degree graphs of solvable groups and derive an explicit formula for their characteristic polynomials. We show that all their eigenvalues are rational and, consequently, that their splitting field is \(\mathbb{Q}\). We then consider supergraphs obtained by adding edges to these graphs and prove that the corresponding splitting field is a quadratic extension of \(\mathbb{Q}\). Next, using their structural decomposition, we examine a general class of Lewis graphs. For this class, we establish bounds on both the number of irrational eigenvalues and the degree of the associated splitting fields. Finally, we investigate prime character degree graphs of diameter \(3\), focusing on the arithmetic nature of their eigenvalues and the degree of their splitting fields.

In the frequency domain, nonlinear acoustic and electromagnetic wave propagation can, in certain regimes, be modeled by second-harmonic generation systems consisting of two coupled nonlinear Helmholtz equations. We analyze a conforming finite element approximation of a scalar second-harmonic generation scattering problem truncated using an exact Dirichlet-to-Neumann boundary condition. The discretization uses continuous piecewise polynomial finite elements of degree $p$. Under small-data conditions involving the incident-field and nonlinear susceptibilities, we prove existence and uniqueness of the continuous and discrete nonlinear solutions in a prescribed small ball. In the same regime, we derive quasi-optimal a priori $H^1$-error estimates by a nonlinear Céa-type argument combining linear Galerkin quasi-optimality with small-data stability estimates for the coupled nonlinearities. We also analyze the convergence of the fixed-point iteration used to compute the discrete nonlinear solution and quantify the combined effects of finite element approximation and nonlinear iteration error. A manufactured-solution experiment illustrates the predicted finite element convergence rates, while additional PML-based scattering computations demonstrate the behavior of the nonlinear fixed-point solver in two and three dimensions.

Fix nonzero $r_0,r_1\in\mathbb{C}$. Let $\widetilde{\mathcal H}$ denote a nil-DAHA of type $(C_1^\vee,C_1)$ defined by generators $t_0,u_0,t_1,u_1$ and relations $(t_i-r_i)(t_i-r_i^{-1})=0$ for $i\in\{0,1\}$, $u_0^2=u_0$, $u_1^2=0$, and $u_0t_0t_1u_1=0=t_1u_1u_0t_0$. Set $A=u_0t_0$ and $B=t_1u_1$. A finite-dimensional $\widetilde{\mathcal H}$-module is called $(A,B)$-multiplicity-free, or simply multiplicity-free, if $A$ and $B$ are simultaneously diagonalizable and every nonzero common eigenspace is one-dimensional. We consider finite-dimensional irreducible multiplicity-free modules that have a certain ordered basis, which we call an adapted block basis. For $D\geq 1$, we construct a family of $2D$-dimensional $\widetilde{\mathcal H}$-modules $E_D$, and for $D\geq 0$, we construct a family of $(2D+1)$-dimensional $\widetilde{\mathcal H}$-modules $O_D$. We determine which members of these families are multiplicity-free and irreducible. We prove that every finite-dimensional irreducible $\widetilde{\mathcal H}$-module that is multiplicity-free and has an adapted block basis is isomorphic to a module of the form $E_D$ or $O_D$. We also determine when two members of the same family are isomorphic.

For a locally symmetric space $\Gamma\backslash G /K$ we study the spectrum of the invariant differential operators of $G/K$ on $L^2(\Gamma\backslash G/K)$. This gives a purely analytical proof of the recent theorem of arXiv:2402.02530 that the quasiregular representation $L^2(\Gamma \backslash G)$ is tempered if the limit cone $\mathcal{L}_\Gamma$ of $\Gamma$ is contained in the interior of the Weyl chamber except if $G$ has real rank one or is locally isomorphic to $\mathfrak{sl}_3(\mathbb K)$, $\mathbb K=\mathbb{R},\mathbb{C},\mathbb H$, or $\mathfrak{e}_6^{-26}$.