Papers

Let $F_2(C_n)$ be the $2$-token graph of the cycle $C_n$ and let $\rho$ denote the packing number. Gómez Soto and Ríos-Castro recently proved that $\rho(F_2(C_n))\ge a(n)$ for $n\ge 19$, where $a(n)$ is an explicit expression. In this note, we prove that \[ \rho(F_2(C_n))\ \ge\ \left\lfloor\frac{n(n-2)}{10}\right\rfloor+1\qquad\text{for every } n\ge 3, \] which improves $a(n)$ by one whenever $n\equiv 0,2\pmod{10}$.

We study finite element approximations for an optimal control problem with pointwise tracking governed by a semilinear elliptic PDE. The control variable enters the state equation as a reaction coefficient, resulting in a bilinear control, and the cost functional includes point evaluations of the state variable. We propose two discretization schemes: a semidiscrete scheme, where the control variable is not discretized, and a fully discrete scheme, where the control variable is discretized using piecewise constant functions. In two and three dimensions, for both schemes, we establish convergence of discrete solutions and prove a priori error bounds that behave as $\mathcal{O}(h|\log h|)$ in the $L^2(\Omega)$-norm for approximating a locally optimal control. In two dimensions, these bounds are improved to $\mathcal{O}(h)$ for the fully discrete scheme and $\mathcal{O}(h^{2}|\log h|^2)$ for the semidiscrete scheme. We conclude with numerical experiments that demonstrate the performance of the proposed schemes.

This is a continuation of the author's earlier paper Block-count constrained harmonic sums: spectral expansion and block-directed Euler–Maclaurin. We presented in that reference the modal expansion of block-Irwin sums $I(b,w,k)$ for $k\geq1$. The case $k=0$ was fully developed when $w$ is a single digit, but it was presented only as a very short result summary for $w$ a multi-digit block. Although the framework is already completely exposed in the earlier paper, the case $k=0$ requires adding a separate, dedicated discussion, which is provided here.

The phenomenon of the top subspace stabilization of the Hessian matrix is an surprising and critical aspect in study of the second-order information of neural network training. Prior work argues that the top subspace of the Hessian stabilizes by measuring the overlap between the top subspaces of the step-wise Hessian, and explains this stabilization with diminishing parameter change in the late phase of training. In this paper, we define a new instability metric for the subspace evolution, and use it to detect subspace stabilization that is independent of the magnitude of parameter change. In the meantime, we observe that the gradient covariance matrix has a similar property of its top subspace to the Hessian. By using a between-class and within-class decomposition of the gradient covariance matrix, we identify an explicit form that gives a near-perfect approximation of the top-$(C-1)$ subspace of the Hessian and the gradient covariance matrix. In the gradient flow set-up, we show that the slow evolution of the idenfied approximation is due to the separation between the outlier and the bulk eigenvalues of the Hessian matrix, thus providing an explanation to the phenomenon of the top subspace stabilization of the Hessian matrix.

We study the number of minimal transitive factorizations of the identity permutation in $S_n$ into transpositions supported on a quasi-threshold graph. We show that this number is always divisible by $(2n-2)!/n!$, which is the factorization count for a star graph, as shown by Irving and Rattan. To prove this, we give a combinatorial formula for the number of such factorizations as a weighted sum over a subset of the $n^{n-2}$ factorization trees, which are edge-weighted spanning trees satisfying certain flow constraints.

Motivated by the recent paper [Sharp endpoint extension inequalities for the moment curve on finite fields II: an extremal property of the uniform distribution, arXiv:2609.29882], which proves sharp extension inequalities in finite fields via a two-point symmetrization argument, we prove here a more general component-wise inequality for permutation matches that implies theirs.

Instead of having a single "yes" or "no" result from a test of the global null hypothesis that a function is increasing, we propose a multiple testing procedure of the function's increasingness at several points. If the global null is rejected, then multiple testing provides more information about why. If the global null is not rejected, then multiple testing can provide stronger evidence in favor of increasingness, by rejecting null hypotheses that the function is decreasing. Our approach uses high-level assumptions that apply to a broad class of causal and descriptive statistical models. By inverting the proposed multiple testing procedure that controls the familywise error rate, we also generate "inner" and "outer" confidence sets for the set of points at which the function is increasing. With high asymptotic probability, the inner confidence set is contained within the true set, whereas the outer confidence set contains the true set. We also improve power with stepdown and two-stage procedures. Simulation and empirical examples illustrate the new methodology, and all code is provided.

In this paper, we provide a construction of an acyclic discrete vector field that is compatible with a total order associated with a lower star filtration on a simplicial complex, called the colex vector field. We show that the colex vector field induces a filtered acyclic vector field, whose resulting Morse complex computes the persistent homology of the lower star filtration on the underlying simplicial complex. We show that the colex vector field can be recomputed quickly when the vertex function that induces the lower star filtration is modified via order-adjacent vertex swaps. We provide a framework for storing and computing the number of paths between cells in the simplicial complex, as well as a method to update these values quickly when the colex vector field changes. Finally, we provide publicly available proof-of-concept code for the ideas shown. When applying it to the Persistent Homology Transform, we show that its runtime is comparable to a more standard matrix reduction approach.

We study density of states (DOS) estimation for large sparse real symmetric matrices by fitting the cumulative density of states (CDOS) and differentiating the fit. We use a stochastic Lanczos method to supply Ritz values and weights, which we average across Lanczos runs to construct CDOS midpoint data. Monotone piecewise-cubic interpolation of these data yields a smooth CDOS approximation that can be easily differentiated to yield a nonnegative, normalized, piecewise-quadratic DOS without requiring a Gaussian smoothing with a fixed bandwidth in its construction. We also consider approximating CDOS by using Gaussian-process regression (GPR) with single- and double-Gaussian covariance kernels, yielding an approximation with uncertainty information. DOS approximation is then obtained by analytic derivative of the GPR-based CDOS. Experiments on matrices from diverse scientific applications are performed to compare these estimators with Gaussian-broadened stochastic Lanczos and Jackson-damped kernel polynomial approximations. At a common Gaussian validation resolution, we use error measures that assess local discrepancies, integrated errors, and agreement in the overall spectral shape. The results demonstrate that the midpoint spline can produce accurate approximations to the DOS with relatively few matrix–vector products, supporting the use of cumulative interpolation as a practical DOS estimation.

We show that, on the Fourier side, the sharp second order Heisenberg Uncertainty Principle (HUP) and its sharp stability for functions are nothing but the first order $L^{2}$-Caffarelli–Kohn–Nirenberg inequality and its sharp stability, applied to their Fourier transforms. This gives a proof in a few lines of the sharp stability estimate that we established recently by a much longer argument. The same observation then produces two one-parameter families of sharp fractional Heisenberg Uncertainty Principles, in which the gradient is replaced by a fractional power of the Laplacian: for each of them we compute the sharp constants, characterize all the optimizers, and establish the sharp stability estimates. The stability constant of the first family is equal to $1$ for every nonnegative order and every dimension. Negative orders, where the fractional Laplacian becomes a Riesz potential, are treated as well. Finally, we show that the deficit carries much more information than the distance to the optimizers alone: it controls an explicit chain of successive remainder terms whose constants are the successive spectral gaps of an explicit operator. For the classical Heisenberg Uncertainty Principle this gives three remainder terms with optimal explicit constants, and for the second order HUP a chain of four remainder terms measured against explicit confluent hypergeometric profiles and coupled through one single set of parameters. As applications, we also establish the chain of stability of the HUP for curl-free vector fields.

We consider a McKean-Vlasov equation incorporating spatial fast oscillatory behavior into both the energetic and kinematic components of the model. The former is given in terms of a confining potential while the latter by an underlying metric. The equation is formulated as a gradient flow in the space of probability measures endowed with a Wasserstein metric. We identify the limiting dynamics which is also described as a gradient flow in an effective media. We further prove the long time exponential convergence of solutions to the invariant measure. The rate of convergence is expressed in terms of a Logarithmic Sobolev constant which is uniform in terms of the oscillatory length scale. Our approach is based on variational and functional inequalities. The homogenization result of the current paper extends the authors' previous work [GaoYip] on a linear Fokker-Planck equation to a nonlinear and nonlocal setting.

Multi-objective Bayesian optimization (MOBO) is commonly approached through specialized acquisition functions or scalarization schemes designed to explicitly account for trade-offs among non-preferential objectives. In this work, we show that such complexity might be unnecessary. We propose a framework that extends standard single-objective acquisition functions directly to the multi-objective setting through a hypervolume-based transformation. We further extend hedge strategies for acquisition functions, which are typically used only in single-objective optimization, to the multi-objective regime. Our approach requires minimal modification to existing Bayesian optimization pipelines and avoids the need for bespoke multi-objective formulations. We demonstrate how a broad class of commonly used single-objective acquisition functions and hedge strategies can be adapted in a principled manner to handle multiple objectives, while preserving their intuitive interpretation and computational efficiency. Empirically, we evaluate the proposed methods across a range of synthetic and real-world multi-objective benchmarks. Despite their simplicity, our extensions consistently match or outperform more complex state-of-the-art MOBO methods in terms of optimization performance and sample efficiency. These results suggest that effective multi-objective Bayesian optimization can be achieved by reusing and carefully extending well-established single-objective acquisition strategies, offering a simpler and more flexible alternative to existing approaches.

We study the empirical eigenvalue distribution of a standardized up Laplacian of the Linial-Meshulam model $Y_{q+1}(n, p)$. We first give a definition of a Gaussian–semicircle law: $\mathcal{N}(0, \sigma^2)\boxplus \operatorname{SC}(s\sigma^2)$, and give a combinatorial formula of its moments in terms of pairing partitions. In addition to that, we also prove that the limiting empirical eigenvalue distribution of this standardized up Laplacian follows a Gaussian–semicircle law, in the sense of almost surely weak convergence.

We study the divisibilities of Tamagawa numbers $ c(E) $ of elliptic curves $ E $ over global function fields $ K $ in terms of their torsion subgroups $ E(K)_{\operatorname{tors}} $. In particular, for a non-isotrivial elliptic curve $ E / k(t) $, where $ k $ is a finite field of characteristic greater than $ 3 $, we prove that $ |E(k(t))_{\operatorname{tors}}|^2 $ divides $ c(E) $, except possibly in four exceptional torsion families. More specifically, we give a complete characterisation of divisibilities for $ c(E) $ in each torsion family, and provide explicit examples to prove that they are best possible. Over a general global function field, we also prove that a rational point of prime order $ 5 \le N \le 101 $ on $ E / K $ forces $ N^2 $ to divide $ c(E) $, which motivates our result. Finally, we formulate a conjecture on the leading coefficient of the $ L $-function of $ E / K $, motivated by the integrality of Birch–Swinnerton-Dyer quotients.

We study a fixed-feature solver, referred to as Spectral-ELM, in which Chebyshev–Gauss–Lobatto (CGL) differentiation matrices are applied to the nodal values of an extreme learning machine (ELM) trial function. Curved domains are treated using a mapped multi-patch formulation. Each curved quadrilateral patch is represented as the image of a reference square, physical derivatives are computed from the associated metric terms, and adjacent patches are coupled by enforcing continuity of the solution and its normal flux. Local QR orthogonalization is used to reduce near-linear dependence among the discrete features. The numerical study includes comparisons with direct CGL collocation, an analytically differentiated ELM, a global Random Feature Method, and a TransNet sampling strategy. It also includes a matched KdV test, a small three-dimensional example, and a Poisson problem on a curved domain containing a hole. For a smooth elliptic problem on a square, direct CGL collocation gives the smallest error. When applied to the same fixed-feature trial space, spectral and analytic differentiation produce comparable errors. For the curved-domain problem, mapped direct CGL collocation and mapped Spectral-ELM achieve similar accuracy with comparable numbers of unknowns. The present formulation is particularly well suited to smooth PDEs in one to three dimensions, where tensor-product grids enable accurate high-order discretizations, while sparse-grid and dimension-adaptive strategies offer promising directions for extension to higher-dimensional problems.

By definition, a dual reflection group is a finite group which grades an Artin–Schelter regular algebra $A$ such that the identity component $A_e$ is again regular. In this paper, we completely classify the nonabelian dual reflection groups for regular algebras of dimension $3$: we first classify all possible nonabelian gradings which refine the natural $\mathbb{N}$-grading of the algebra, and then determine which of these gradings yield dual reflection groups. We show that none of the cubic algebras admit dual reflection groups, while for quadratic algebras, the dual reflection groups arise either from Ore extensions of two-dimensional regular algebras, or from the Sklyanin algebras $S_{q,0,1}$. In the Sklyanin case, the dual reflection groups form a new infinite family of groups $\Delta_n$ of order $27n^3$, answering a question of Goetz, Kirkman, Moore, and Vashaw. We also extend the four-dimensional examples of the same authors to two new infinite families of dual reflection groups. Finally, we show that an AS regular algebra admitting a dual reflection group need not have binomial relations, answering another of their questions. However, we conjecture that the relations must be binomial when $A_e$ contains no elements of degree $1$, and we prove this for nonabelian gradings in dimension $3$.

We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu–Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid $\mathrm{C}^*$-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu–Renault systems are constructed, each modelling the Cuntz algebra $\mathcal{O}_2$. The $\mathrm{C}^*$-algebras obtained from orbit-breaking for such systems come with explicit embeddings into $\mathcal{O}_2$. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into $\mathcal O_2\otimes\mathcal K$. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a $^*$-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit $^*$-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their $K$-groups.

We show that there are infinitely many k for which Sp_{2k}(2) is not generated invariably by two elements of prime order.

We study how thermal equilibrium states of infinite fermionic lattices respond to perturbations of the Hamiltonian. Our main result concerns extensive perturbations of a chosen KMS state. We show that exponential decay of correlations and exponential local indistinguishability under the extensive perturbation yield stretched-exponential decay of correlations for the perturbed state. Neither the initial nor the perturbed state is required to be unique, allowing stability to be studied in the presence of phase coexistence. Our approach uses Quantum Belief Propagation (QBP), which we establish directly on the underlying $C^*$-algebra. Combining QBP with Lieb–Robinson bounds, we show implications between decay of correlations, the principle that local perturbations perturb locally, and local indistinguishability, closing the cycle for perturbations of the tracial state under the stated decay, support and uniformity assumptions. For extensive perturbations under uniform clustering conditions, we also obtain a differentiable path of local equilibrium expectations and bound their change linearly in the interaction norm of the perturbation.

We study the nodal structure of Neumann eigenfunctions on planar dumbbell domains as the width of the neck joining two fixed end domains tends to zero. Under simplicity and non-degeneracy assumptions, we show that sufficiently thin necks force an increasing number of nodal domains as the limiting eigenvalue grows. At the bottom of the spectrum this forcing can attain Courant's upper bound: in particular, the third Neumann eigenfunction has exactly three nodal domains and is Courant sharp. At the same time, nodal deficiency of an eigenfunction on an end domain can persist on the full dumbbell, giving eigenfunctions that are not Courant sharp. The main analytic difficulty is that the singular limit may vanish on one or both end domains even though these regions contribute nodal domains for every positive neck width. We overcome this by deriving refined asymptotics for the first nonzero profiles on the vanishing regions, expressed through Neumann Green's functions with poles at the neck attachment points and obtained by a two-dimensional matched asymptotic analysis with logarithmic terms.

The Gromov-Wasserstein (GW) distance measures the discrepancy between metric measure (mm) spaces and identifies optimal alignments between them based solely on their intrinsic structure. Since it identifies isomorphic mm spaces, it provides a natural notion of distance for heterogeneous datasets which may admit isomorphic representations. In order to accelerate computation of GW distances, many practitioners employ entropic regularization to obtain an Entropic GW (EGW) problem. The most popular EGW solver is the Mirror Descent (MD) algorithm, which reduces EGW computations to an iterative process where an entropic optimal transport (EOT) problem is solved at each iteration. Despite its widespread use, the convergence of MD for this problem has only been established for restricted classes of costs. On the other hand, a recently proposed dual gradient method is available for general costs, but requires a choice of step size which depends on the regularization parameter. To address these two issues, we introduce Averaged Mirror Descent (AMD), which averages consecutive MD steps, and prove its convergence for arbitrary costs. Then, we establish that the dual gradient method with a fixed step size also converges for arbitrary costs at the cost of a more complicated iteration. In both cases, we also account for inexact iterations which are inescapable in practice. We compare the empirical performance of these methods across various settings and, in particular, show that AMD and the dual gradient method both converge on an example where classical MD fails.

We study the stationary state of driven-dissipative activated random walk on an interval of length $n$ and prove that subintervals down to size $\log n$ have particle density near the critical value. We then deduce that the avalanche area is heavy-tailed and that the process forgets an initial stationary state after $o(n)$ steps. We conjecture that the density fluctuations, avalanche area, and remixing time from stationarity are governed by the same critical exponent.

This study addresses a theoretical model regarding equity and accessibility challenges in designing shared transportation systems (such as micro-mobility networks) by applying graph-vertex domination setting. The work focuses on identifying dominating forced vertices, that represent nodes belonging to every dominating set of a graph of the smallest possible cardinality, and which correspond to critical, non-negotiable station locations essential for maintaining system efficiency and coverage. From a theoretical perspective, in the paper it is first demonstrated that determining whether a given vertex is a dominating forced vertex is co-NP-hard, establishing the computational infeasibility of exact identification in large networks. To analyze graph structures, sharp theoretical bounds on the maximum number of dominating forced vertices are established, proving that their count is bounded above by one-third of the order of the graph, and provide complete structural characterizations for graphs achieving this bound, as well as, trees with no dominating forced vertices. To overcome computational limits in practical urban settings, the study uses an iterated greedy metaheuristic framework to generate minimal dominating sets and approximate critical forced nodes based on their appearance frequency across iterations. The methodology is validated on strong grid graphs and applied to real-world road network models of Algeciras and La Línea de la Concepción, two cities in the area of Campo de Gibraltar, Spain, which successfully pinpoints candidate location points for scooter-sharing stations across both cities.

The Koopman operator is a well-established framework for lifting nonlinear dynamical systems to an infinite-dimensional space where dynamics are linear. Extended Dynamic Mode Decomposition (EDMD) is a widely used method in data-driven dynamics as it provides a Galerkin approximation of the Koopman operator on the finite-dimensional span of a prescribed dictionary of observable functions. In particular, EDMD only requires samples from the dynamical system, describing the Koopman operator without knowledge of the underlying dynamics. While many studies analyze asymptotic convergence results for EDMD in terms of data, the question of more practical finite-data convergence results remains incomplete in general settings. In this work, we prove bounds for the finite-data EDMD Galerkin approximation of the Koopman operator by a rate proportional to the inverse of the root of the number of samples (the Monte Carlo rate). These results apply to multiple classes of problems, i.e., for discrete or continuous dynamics, including stochastic systems. In certain more structured setting, we demonstrate super-polynomial rates are achievable both theoretically and computationally. Finally, we derive recovery guarantees for an EDMD variant in the undersampled regime inspired by compressed sensing techniques.

What should a compressed representation preserve when the information of interest lies in relationships among elements rather than in the elements themselves? We formulate relational compression in the classical source-description-reconstruction sense, but with relational structure itself as the fidelity-bearing content. Each instance specifies the source relation, retained description, reconstructed or evaluated relation, fidelity criterion, and constrained resource. We use this interface to situate selected methods from graph summarization, spectral sparsification, similarity-preserving representation, and relational distillation within a common formulation while keeping their different reconstruction and resource assumptions explicit. We develop finite-codeword collision as one concrete realization. Same-codeword probability yields a relational geometry linking pair-specific alignment and separation to aggregate Rényi-2 occupancy and the spherical geometry of categorical assignments, with exact objective correspondences to squared-Euclidean centroid reconstruction and normalized graph association and cut. Graph and image studies illustrate complementary routes within the finite-codeword family: graph- and teacher-defined relational requirements act directly on equality or collision, while reconstruction acts through a joint decoder. Together, these results illustrate how distinct relational requirements can be formulated and tested within a common constrained-representation framework.

Csernák and Soukup introduced the stacking number of a graph: the least integer $t\ge2$ such that every configuration of $t$ pebbles can be transformed by pebbling moves into a configuration supported on one vertex. They conjectured an explicit formula for the stacking number of a tree in terms of the degrees of its vertices and their distances from a chosen root. We prove this conjecture for every finite tree with at least two vertices. We also show that whether a configuration on a tree can be stacked at a given vertex is decided exactly by the sign of an integer score built from recursive branch messages; consequently, stackability can be tested with a linear number of arithmetic operations. The proof combines this criterion with an explicit zero-score obstruction and a weighted cancellation argument for arbitrary nonstackable configurations. The main theorem and the stackability criterion are formalized in Lean 4, and the main theorem has passed Palomar mechanical verification; it is publicly registered as PALOMAR-2026-09-25-000010.

For birth–death processes with a strong Allee effect, long survival near a positive stable equilibrium does not by itself determine the principal spectral gap. We identify its large-population limit when the birth and death rates at population size $n$ are $n\widetilde\lambda(n/K)$ and $n\widetilde\mu(n/K)$, where $K$ is the population scale and $\widetilde\lambda,\widetilde\mu$ are the per-capita rates. Write $V(x)=x(\widetilde\lambda(x)-\widetilde\mu(x))$ for the deterministic drift, with unstable threshold $x_1$ and positive stable equilibrium $x_2$. For the two smallest eigenvalues $\rho_1(K)<\rho_2(K)$ of the negative generator killed at extinction, we prove \[ \lim_{K\to\infty}\bigl(\rho_2(K)-\rho_1(K)\bigr) =\min\{\widetilde\mu(0)-\widetilde\lambda(0),V'(x_1),-V'(x_2)\}, \] so the limit depends on the linearization rates at all three equilibria. Each term can be the unique minimum even when $x_1$ and $x_2$ are fixed. The proof combines a frozen boundary model with two local oscillator limits through discrete Ismagilov–Morgan–Simon (IMS) localization. A uniform comparison of the global and stable local ground states controls the orthogonality constraint in the variational lower bound; projected local trial vectors give the matching upper bound. The result determines the limiting $L^2$ spectral gap and optimal Poincaré constant of the $Q$-process, which describes conditioning on indefinite survival. Combined with principal-eigenvalue asymptotics from a companion work, it separates an exponentially growing quasi-stationary mean extinction time from a spectral relaxation time with a finite positive limit.

We study density-dependent birth–death processes with a strong Allee effect and absorbing extinction. The deterministic system is bistable: extinction and a positive equilibrium are locally asymptotically stable, separated by an unstable Allee threshold. Let $K$ be the population-size scaling parameter. As $K\to\infty$, we derive a sharp Eyring–Kramers-type asymptotic formula, including the leading prefactor, for the principal eigenvalue of the killed generator, and a uniform approximation of the corresponding positive eigenvector. Consequently, we obtain a sharp asymptotic formula for the mean extinction time under the quasi-stationary distribution (QSD). In the bistable regime, the initial population determines the competition between early extinction and positive metastability. After initial relaxation, the law is approximated in total variation by an initial-state-dependent convex combination of the QSD and the Dirac mass at extinction. The mixing coefficient is identified as the probability of reaching the positive stable region before absorption. We identify a probabilistic transition layer of width $O(\sqrt K)$ around the Allee threshold, where the mixing coefficient converges to an explicit Gaussian profile. Notably, a population starting at the threshold reaches the positive metastable region or becomes extinct with asymptotically equal probabilities. We establish existence and uniqueness of the QSD, quantitative convergence to the quasi-stationary regime, and a discrete Gaussian approximation of the QSD centered at the positive stable level. Our analysis combines a self-adjoint spectral realization of the killed generator, matched asymptotic analysis of the associated second-order recurrence, and discrete Laplace estimates for reversible weights.

We investigate orthogonality under the shift-invariant resolvent delta operator $\mathscr L=D(I-D)^{-1}$. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is $\mathscr L$-Appell if and only if it is a translated Laguerre sequence. We then move beyond the Appell setting and show that the diagonal family associated with $$\mathcal S_\beta=\frac{\beta}{\beta+1}\mathcal L^0+\frac{1}{\beta+1}\delta$$ satisfies the exact Hahn relation $$\mathscr L B_{n+1}(x;\beta)=(n+1)B_n(x;\beta+1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=\theta\,\mathscr L P_{n+1}(c)$ already forces the orthogonality functional to be a translated Laguerre functional or a translated member of the diagonal family. Connection and structure relations, a semiclassical Pearson equation, a differential characterization, and a generating function complete the description. Thus the resolvent operator provides an explicit setting in which delta-operator lowering and semiclassical orthogonality are linked by exact structural identities.

Let $Z_\alpha$ be a positive $\alpha$-stable random variable, where $0<\alpha<1$. We prove that $Z_\alpha^p$ is infinitely divisible for every $p>0$. Combining this result with Jedidi and Simon's characterization of infinite divisibility of negative powers of $Z_\alpha$, we obtain, for $p\ne0$, $$Z_\alpha^p\ \text{is infinitely divisible} \quad\Longleftrightarrow\quad p\in\left(-\infty,-\frac{\alpha}{1-\alpha}\right]\cup(0,\infty).$$ The proof relies on Kanter's factorization and a general product property: if $X$ is a mixture of exponential distributions, $W$ is a generalized gamma convolution, and $X$ and $W$ are independent, then $(X+c)W$ is infinitely divisible for every $c\ge 0$.