Papers

We establish large deviation principles both for the CRT in the Gromov-Hausdorff topology and for the CRT weighted by its mass measure in the Gromov-Hausdorff-Prohorov topology. In the first case, the rate function of a tree $\mathcal{T}$ is half the square of the length of $\mathcal{T}$. In the second case, the rate function of a pair $(\mathcal{T},\mu)$ is half the integral with respect to the length measure of $\mathcal{T}$ of the inverse of the absolutely continuous part of $\mu$.

Spatial transcriptomics measures gene expression together with spatial coordinates, but many existing analysis methods represent each spot primarily by a single feature vector. We propose GrassST, a subspace method for spatial transcriptomics analysis and cross-slice alignment. For each spatial spot, GrassST constructs a neighborhood from tissue coordinates, fits a low-dimensional subspace to the embedded expression profiles in that neighborhood, and represents the spot by the resulting subspace. These representations are points on a Grassmann manifold and can be compared using distances between subspaces. GrassST selects the neighborhood size and subspace rank from the spectral energy of the data, allowing these parameters to vary across datasets rather than being fixed globally. Experiments on four spatial transcriptomics datasets show that GrassST achieves competitive clustering and cross-slice integration performance under a unified evaluation pipeline.

We study weakly interacting bosons on the three-dimensional torus at temperatures proportional to the critical temperature for Bose–Einstein condensation. We show that, as the expected particle number tends to infinity, the grand canonical Gibbs state is asymptotically described by the coherent state quantization of a random field that converges to a novel Bogoliubov random field. Moreover, the point process associated with the Gibbs state can be approximated by a Cox process. These results are based on and extend the approximation of the Gibbs state recently obtained by the first two authors and Nam. Using these approximations, we compute limiting distributions of the number of particles outside the condensate, the joint occupation statistics of finitely many momentum modes, and particle-number fluctuations in macroscopic subsets of the torus, all governed by the Bogoliubov field. We further prove convergence of the empirical measure of the associated point process to the uniform measure and convergence of the microscopic particle-number statistics to the boson point process. The latter result establishes the universality of particle-number statistics on microscopic length scales.

Let $X:M^n\to\R^{n+1}$ be an $n$-dimensional complete self-shrinker. We obtain a complete classification of complete self-shrinkers with positive constant scalar curvature. More precisely, we prove that the round sphere $S^n(\sqrt n)$ and the standard generalized cylinder $S^k(\sqrt k)\times \R^{n-k}$ for $2\leq k\leq n-1$ are the only complete self-shrinkers with positive constant scalar curvature. The key difficulty is to characterize the residual case in Cheng-Li-Wei \cite{CLW}: $R>0$, $S<1$, and $\sup_M S=1$, where $R$ and $S$ denote the scalar curvature and the squared norm of the second fundamental form, respectively. For this case, it seems a hard task that the generalized maximum principle yields a useful information. In order to overcome this substantial difficulty, our key ingredient is to get a uniform positive lower bound for the Bakry-Émery Ricci curvature so that we can make use of the comparison theorem of Wei-Wylie \cite{WeiWylie} to conclude that the Gaussian volume is finite. Furthermore, the gap theorems on $S$ are given.

Let $P=(P_j)$ be a random mass partition and let $(X_j)$ be iid real marks, independent of $P$. We construct a fixed mark distribution for which the law of the real random variable $\sum_j P_j X_j$ determines the law of $P$ on the Kingman simplex. The observation map is an affine topological embedding. One construction uses an infinite convolution of stable laws with rationally independent indices; for every prescribed $q\in(0,\infty)$, a determining mark can instead be chosen symmetric, centered, compound Poisson, and in $L^q$. At any known positive time, the law of the ranked block frequencies of a $\Xi$-coalescent started from singletons$\unicode{x2014}$equivalently, the exchangeable partition probability functions of all its finite restrictions at that time$\unicode{x2014}$determines the finite collision measure $\Xi$. Composing the two results therefore identifies $\Xi$ from the law of one real marked sum. Within the $\Lambda$ subclass, an asymmetric Bernoulli coloring determines the normalized collision measure, as does any centered nonconstant mark whose absolute moment of some order $a\in(1,4)$ is finite while every higher absolute moment is infinite. We also prove that, for every $d\ge 2$, a pair of distinct laws supported on strictly decreasing $(d+1)$-tuples of positive masses with a fixed total mass remains indistinguishable for every mark supported on at most $d$ points. For every fixed number $d\ge 2$ of types, this obstruction yields distinct normalized collision measures with the same mutation-free neutral $d$-type $\Xi$-Fleming$\unicode{x2013}$Viot transition semigroup.

We extend Gauss composition and Gauss genus theory over $\mathbf{Z}$ to $\mathbb{F}_{2^n}[T]$, a polynomial ring over a finite field $\mathbb{F}_{2^n}$ of characteristic 2. We find new invariants of binary quadratic forms over $\mathbb{F}_{2^n}[T]$ by using Arf invariant and introduce new definitions of proper equivalence and direct composition, and prove that the direct composition makes the set of proper equivalence classes of binary quadratic forms with the same invariants into a finite Abelian group, which is isomorphic to a Picard group of a corresponding extension ring of $\mathbb{F}_{2^n}[T]$. Building on this, we develop genus theory in characteristic 2 and prove that the kernel of the generalized Gauss's map is the subgroup of all squares in the class group.

Aeroelastic ?utter is normally a failure mode to be designed against, yet the limit-cycle oscillations (LCOs) that follow it convert flow energy into sustained structural motion that a piezoelectric transducer can turn into electrical power. A transducer is embedded in a three-degree-of-freedom pitch-plunge aerofoil with a finite-mass trailing-edge flap and unsteady strip-theory aerodynamics, giving a fifteen-state electro-aeroelastic system with a cubic hardening pitch spring, and the system is reduced by biorthonormal projection of the Taylor-expanded residual onto eigenvectors of the coupled Jacobian. The testbed is deliberately low order, so that every reduced prediction can be checked against the full-order system it replaces. The degree of freedom that carries the transducer is a first-order design variable, since it sets both the sign of the shift in the flutter boundary and the magnitude of the harvested power. The error of the reduced model is dominated not by the size of the retained basis but by how the reduced operator is made to depend on low speed. Expanding the retained eigenvalues leaves the reduced operator block-diagonal, with no coupling between the retained modes at any order of the expansion, whereas re-projecting the exact Jacobian onto the same frozen basis supplies that coupling and restores the structural response without enlarging the basis. It is found that the error that remains lies in the harvested voltage, which the equilibrium eigenvectors carry in a fixed proportion to the motion, and this is not removed by adding the electrical modes to the basis, whereas the flap detection, which the same basis also misses, is recovered by retaining the flap modes. The accuracy of a harvester reduced-order model must therefore be reported on the harvested quantity itself.

Classical convergence guarantees for stochastic gradient methods typically assume Lipschitz-smooth objectives and finite-variance gradient noise, both frequently violated in practice. In contrast, we study nonconvex stochastic optimization under the joint relaxation of these assumptions: objectives with $(L,s)$-Hölder continuous gradients, $s\in(0,1]$, and gradient noise satisfying only a bounded $\alpha$-th moment condition for $\alpha\in(1,2]$. We establish three convergence results. Firstly, that standard SGD converges at rate $O(T^{-s/(1+s)})$ whenever $\alpha\ge1+s$, extending the classical nonconvex SGD rate to heavy-tailed noise and Hölder smoothness simultaneously. Secondly, we analyze $\delta$-regularized gradient clipping ($\delta$-GClip), a provable trainer of wide and deep nets, and establish a stationarity rate of $O(T^{-2s(\alpha-1)/[(1+s)(2\alpha-1)]})$ under the same condition. Thirdly, we analyze standard gradient clipping (G-Clip) and show that it recovers the above rate for $\alpha\ge1+s$ while in the very heavy-tailed regime $\alpha<1+s$, it has a convergence rate $O(T^{-2s(\alpha-1)/[(\alpha-1)+s(2\alpha-1)]})$ — the first convergence guarantee in this regime for any stochastic gradient based method.

In 2015, Liu, Qu (Commun. Math. Phys., Vol. 338) show that the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian admits a positive lower bound independent of the coupling constant lambda: dim_H sigma(H_{tm,lambda}) ge (log 2) / (140log 2.1). In this paper, we prove that the spectrum of the Thue–Morse Hamiltonian has full Hausdorff dimension for any lambda in mathbb{R}: dim_H sigma(H_{tm,lambda})=1.

We construct a permutation of the positive integers containing no four-term arithmetic progression as a subsequence.

For integers $m\geq3$ and $n\geq1$, an $m$-gonal form is called tight $\mathcal{T}(n)$-universal if it represents exactly the positive integers $\mathcal{T}(n)=\{ n, n+1, n+2, \ldots \}$. In this paper, we study the minimal criterion set $\mathrm{CS}(m,n)$ for tight $\mathcal{T}(n)$-universality. Our main result determines $\mathrm{CS}(m,n)$ for all $n \geq8$, $3 \leq m \leq \left\lfloor \frac{3n+1}{2} \right\rfloor$. More precisely, \[ \mathrm{CS}(m,n)= \begin{cases} \{ n, n+1, \ldots, 2n-1 \}, & m=5,\newline \{ n, n+1, \ldots, 2n \}, & m\neq5. \end{cases} \] We also establish the corresponding tight $\mathcal{T}(n)$-universality results and show that the upper bound on $m$ is optimal.

Higher-order regularization has shown improved results over first-order methods such as total variation. In this paper, convergence of a piecewise linear finite element discretization is shown for the second-order total generalized variation functional. Additionally, this convergence result is extended to a spatiotemporal setting after the spatiotemporal function space is comprehensively introduced. In both cases, discrete minimizers converge to the continuous minimizer with rate $h^{1/4}$, matching the best known rate for finite element discretizations of total variation in the same general setting. Numerical experiments confirm the convergence rates and indicate faster convergence in practice. Comparisons with related spatial image reconstruction algorithms show comparable reconstruction results. The spatiotemporal case is discussed using the example of the inverse problem in electrocardiographic imaging, where the proposed functional improves the state of the art.

We introduce a class of finite- and infinite-horizon discrete-time control problems for studying macroscale systems in discrete time, which we refer to as mean-field terminal value problems (MFTVPs). An MFTVP takes a terminal \(Q\)-function as input and, starting from this terminal \(Q\)-function, seeks an indefinite backward evolution of optimal policies and state measures, as required for an infinite-horizon non-stationary mean-field equilibrium (MFE) of a discrete-time mean-field game (MFG). We study the existence and uniqueness properties of this problem. As an application, using the uniqueness properties of MFTVPs, we show that, in the average-cost setting (and in the discounted setting for sufficiently small discount factors), MFTVPs can be used as an intermediate step to develop approximation schemes between finite-horizon MFEs and average-cost MFEs (respectively, stationary MFEs). In contrast to approaches that use infinite-horizon MFGs as intermediate problems, MFTVPs can be employed when the discount factor is sufficiently small (or, in the average-cost setting, when the drift factor is sufficiently small), together with a regularizer, to approximate a stationary MFE. Assuming uniqueness of the solution of a MFTVP, the restrictions we impose on the Lipschitz parameters of the system are weaker than the contraction conditions available for regularized stationary MFGs, and our approximation scheme remains applicable even in the presence of multiple stationary MFEs.

Stochastic gradient descent (SGD) is the primary workhorse for large-scale optimization. While the average behavior of its iterates, typically characterized by mean-squared error bounds, is well-understood, obtaining high-probability guarantees for the last iterate remains challenging. Prior approaches to this problem have either imposed restrictive assumptions (such as bounded domains or gradients) or relied on complex proofs involving auxiliary sequences. In this work, we propose Batched SGD, a simple variant that partitions online samples into epochs and performs a single update per epoch using a refined, low-variance gradient estimate. Our main contribution demonstrates that this batching mechanism enables a surprisingly simple high-probability analysis that avoids both restrictive assumptions and auxiliary sequences. Under standard smoothness and norm-sub-Gaussian noise assumptions, we establish near-optimal rates for both strongly convex and non-convex objectives. Furthermore, we show that our batching idea extends naturally to federated learning (FL). We provide the first high-probability guarantees for FL, achieving logarithmic communication complexity, linear speedup in the number of agents, and resilience to data heterogeneity.

We establish an asymptotic formula for the first moment of the central values of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble $\mathcal{H}_{2g+2}$ over $\mathbb{F}_q[x]$, for every fixed odd prime power $q$. Working with the completed $L$-function yields an exact two-term central-value formula with truncation levels $g$ and $g-1$; combining the two truncations before the square-dual contour shifts leaves the three cubic points $z^3 = q^{-4}$ as the only secondary poles, and they contribute the secondary term $q^{2g/3+2}(a_g g + b_g)$ with an error $O_\varepsilon(q^{g(1+\varepsilon)/2})$. The coefficients are real and $m \mapsto (a_m, b_m)$ has minimal period exactly three, so the coefficient of $g$ at the order $q^{2g/3}$ depends on $g$ modulo 3, in contrast with the earlier even-degree formulas; the difference is confirmed numerically.

Rohn's sign-accord algorithm is a classical single-coordinate sign-flip method for $Ax-|x|=b$, selecting the least-index mismatched coordinate at each step. We construct rational instances on which this rule requires exactly $2^{n-1}-1$ flips from natural initialization, even when both $\|A^{-1}\|_1$ and $\|A^{-1}\|_\infty$ are arbitrarily small. We propose two alternative selection rules, Gauss–Southwell and steepest single flipping, which require only one flip on the same instances. A discrete potential framework with positive polarity factors identifies solutions with global maximizers. Under positive polarity with ratio $\kappa$ and $r=\|A^{-1}\|_\infty<1$, both rules terminate from any initial signature after at most $\lfloor s\kappa(1+r)/(1-r)\rfloor$ flips, where $s$ is the number of nonzero solution coordinates. For fixed $\eta_1<1/2$ and $\eta_2<1$, both rules have worst-case flip complexity $O(n)$ on the class $\|A^{-1}\|_1\le\eta_1$ and $\|A^{-1}\|_\infty\le\eta_2$. An $n-1$ lower bound from natural initialization shows that this linear order is sharp. Under stronger, explicit sign-correction conditions, each rule makes exactly as many flips as there are initially incorrect signs among the nonzero solution components. Thus, each rule makes at most \(n\) flips, and this bound is optimal. A five-dimensional example requiring six greedy flips shows that the bound of $n$ does not extend to the entire common small-norm class. Building on Radons–Rump sign certificates, we further develop three classes of sign-flip algorithms using elimination guidance, cyclic scanning, and randomized $m$-flipping. Under $\rho(|A^{-1}|)<1/2$, we prove at most $n$ flips for elimination guidance, $n^2$ inspections for cyclic scanning, and $O(n^2/m)$ expected batch updates for randomized $m$-flipping. Exact computations and retained paired experiments distinguish flip counts from computational cost.

A large $p$-subgroup of a group $G$ is a self-centralizing $p$-subgroup $Q \le G$ whose normalizer controls the normalizers of all the non-trivial, central subgroups of $Q$. In 2016 Meierfrankenfeld, Stellmacher and Stroth produced a result describing the $p$-local structure of a finite group having a large $p$-subgroup (the main examples arising from finite groups of Lie type in defining characteristic $p$). This result is a major success within the wider framework of studying groups of local characteristic $p$. We attempt to produce a result analogous to that of Meierfrankenfeld, Stellmacher and Stroth, but for fusion systems and localities. Previous work of Ellen Henke and the author shows that reasonable generalizations can be formulated and solved equivalently either within the realm of fusion systems or in the world of localities. In particular, in the present paper we set the stage for our analysis, showing how working within a locality grants a clear advantage: it allows to follow the same lines of reasoning as for a group. We therefore produce analogous reduction results and case subdivision as those in the 2016 result of Meierfrankenfeld et al. and, proceeding according to the analogy, we deal with occurrences of certain natural orthogonal modules and with the first of the cases that are to be studied, the so-called symmetric case. The remaining cases will appear in future publications.

Given a coherent sheaf on a smooth projective hypersurface X, we prove an explicit formula for its Chern character as a Cech cocycle in terms of the free resolution of the associated module and calculate its image in the Jacobian ring under the Griffiths residue map. The formula is a geometric analogue of the Kapustin-Li formula for Landau-Ginzburg models, but proven directly using Hodge-theoretic techniques. This yields an effective method to compute the primitive part of the Chern character of any coherent sheaf using commutative algebra. We finish by proving the Hodge conjecture for the degree 33 Fermat fourfold.

Let $0<\alpha<2$ and $p=2/(2-\alpha)$. We study critical nonlinear convolution inequalities for kernels $K:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^m$ and functions $\Phi:\mathbb{R}^m\to\mathbb{R}$ positively homogeneous of degrees $\alpha-2$ and $p$. We assume angular Lipschitz regularity for $\Phi$ and angular $C^{0,\gamma}$ regularity for $K$, where $0<\gamma\le1$ and $\gamma=1$ if $p<2$. On bounded finitely cornered planar domains with a common Dini tangent modulus on the boundary branches, we characterize $|\int_\Omega\Phi(K*f) dx|\le C\|f\|_{L^1}^p$ for real densities $f\in L^1(\mathbb{R}^2)\cap L^s_{\mathrm{loc}}(\mathbb{R}^2)$, $1<s\le\infty$, by signed angular cancellation on the plane, tangent half-planes, and full vertex tangent cones. An analogous criterion holds on infinite sectors for compactly supported mean-zero densities. For the Newton kernel and each nonzero angularly Lipschitz, positively two-homogeneous $\Phi:\mathbb{R}^2\to\mathbb{R}$ with vanishing half-circle moments, we construct a bounded convex $C^1$ domain $\Omega_\Phi$, independent of $\varepsilon$, and smooth tests $u_\varepsilon$ satisfying $\|\Delta u_\varepsilon\|_{L^1(\Omega_\Phi)}+\|\partial_nu_\varepsilon\|_{L^1(\partial\Omega_\Phi)}\le2$, while, despite tangent cancellation, their nonlinear integrals grow in magnitude like $\log\log(1/\varepsilon)$. On the Dini class, a constant-preserving linear extension with compact support modulo the domain mean and controlled measure-valued Laplacian yields the same necessary-and-sufficient criterion for the Newton-kernel PDE inequality, without exact corner charts. We also construct infinite-dimensional cancellation spaces containing nonquadratic functions and classify the quadratic locus in polygon moduli.

This is an extended version of arXiv:2511.22129. We develop a general construction of the superstrip algebra $\mathbf{sStr}_{\mathscr C}(\mathcal M)$, the fermionic counterpart of the strip algebra, for a gapped $(1+1)$-dimensional system with a fermionic superfusion category $\mathscr C$ and the vacuum structure described by the supermodule category $\mathcal M$. As illustrations, we work out the superstrip algebra for two simple examples: the fermionic variant of the Fibonacci fusion category and the q-type $\mathbb{Z}_2$ symmetry. We then apply the framework to the gapped IR phases of the $\mathcal N=2$ and $\mathcal N=1$ superconformal minimal models deformed by their least relevant operator and compute the Witten indices. We explain how the particle and soliton spectra, together with their fermion parity, are organized into representations of the corresponding superstrip algebras. A mixed 't Hooft anomaly between the spontaneously broken symmetry and the unbroken fermion parity $(-1)^F$ further forces each $\mathcal N =2$ soliton to carry a fractional fermion number. Finally, we set up the boundary SymTFT for fermionic systems and reproduce the results in several examples studied in this paper.

Motivated by the questions posed by W. V. Li and A. Wei and the conjecture of E. Lundberg and A. Thomack, we study the expected number of zeros of random harmonic polynomials $H_{n,m}(z)=p_n(z)+\overline{q_m(z)}$ with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is $O(n)$ when $\deg p=\alpha\deg q$, where $0\leq\alpha<1$. This result extends the previous estimates when $m$ is a fixed constant or $m=n$ to more general case.

We bound the eventual order types of Skolem functions below finite exponential towers. Writing $E_0(u)=u$, $E_{n+1}(u)=2^{E_n(u)}$, and $\omega_0=1$, $\omega_{k+1}=\omega^{\omega_k}$, the argument gives \[ |\Sk_{<E_n(x^m)}|<\omega_{r_n},\qquad r_n=2+\frac{n(n+3)}2\quad(n\ge1,\ m\ge2\text{ fixed}). \] For triple towers we obtain the sharper bound $|\Sk_{<E_3(x^m)}|<\omega_{10}$ for every fixed $m\ge1$. The proof combines comparisons of asymptotic expansions with finite recursive decompositions and ordinal estimates for ordered sums and products. The analytic part is developed in the classical field of logarithmic-exponential series. We prove the required discreteness, support and truncation statements in this setting. The external inputs are the series construction of van den Dries–Macintyre–Marker and the order and ordinal estimates recalled from Berarducci–Mamino. These bounds imply $|\Sk|=\varepsilon_0$.

We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials $P_n$, where $n$ denotes the degree and is assumed to be sufficiently large. For each fixed $B$ above an absolute threshold, let $I_n=[-\sqrt n+B\sqrt{\log n}, \sqrt n-B\sqrt{\log n}]$. Uniformly over deterministic compact intervals $I\subseteq I_n$ whose length $\ell$ is sufficiently large and depends on $n$, the distribution of the standardized number of real roots in $I$ has Kolmogorov distance at most $C\log\ell/\sqrt\ell$ from the standard Gaussian distribution. Consequently, every such interval sequence with $\ell\to\infty$ satisfies a central limit theorem. In particular, taking $I=I_n$ gives the bound $C\log n/n^{1/4}$. The same bound holds for the distribution of the standardized number of real roots on $\mathbb R$. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.

For every positive integer $k$, we prove that the homogeneous cut semigroup of every $K_5$-minor-free graph is saturated at height $k$ if and only if every planar $k$-graph is $k$-edge-colourable. Here a $k$-graph is a loopless $k$-regular multigraph in which every odd vertex cut has size at least $k$. A triangle expansion of a cubic plane dual converts cut decompositions into perfect-matching decompositions. The converse uses symmetric difference with a fixed perfect matching. The equivalence identifies the normality conjecture for $K_5$-minor-free cut polytopes with Seymour's planar edge-colouring conjecture. In particular, the known cases $k\leq 8$ give saturation through height eight.

In this article, we compute the Hilbert series of the generalized binomial edge ideal associated with the join of two graphs. Consequently, we compute the Hilbert series of complete $r$-partite graphs. Further, we show that if a vertex satisfies a certain degree condition, then some Hilbert coefficients remain unchanged upon its removal. This provides a reduction technique for computing them. As an application, we compute the dimension and multiplicity of several classes of graphs.

We study the zeros of the quasimodular forms $G_{\{2\}^n}$ defined by iterated sums. We first show that, for every $n>0$, $G_{\{2\}^n}$ has exactly $n$ simple zeros on each of the vertical half-lines $\Real(\tau)=0$ and $\Real(\tau)=1/2$, and that the zeros for consecutive values of $n$ satisfy an interlacing property. The proof is based on an expression of $G_{\{2\}^n}$ in terms of the $n$-th derivative of $\eta^3$ and on the theory of bell-shaped functions, rather than on Rankin–Swinnerton-Dyer method. We also determine the asymptotic behavior of these zeros as $n\to\infty$. In addition, we prove that all zeros of $G_{\{2\}^n}$ are simple and that $G_{\{2\}^n}$ has infinitely many $SL_2(\ZZ)$-inequivalent zeros. We further show that quasimodular forms of maximal depth have no zeros at CM points. In particular, none of the zeros of $G_{\{2\}^n}$ are CM points. Finally, in the special case $G_{2,2}$, we show that each Ford circle contains exactly two distinct simple zeros.

The complementary prism $G\overline{G}$ of a graph $G$ is obtained from the disjoint union of $G$ and its complement $\overline{G}$ by adding an edge between each vertex $a$ in $G$ and its copy $a'$ in $\overline{G}.$ This paper explores a general framework for studying fractional revival with respect to real symmetric matrices with a block structure. The framework is then used to show that, for a fixed state $\mathbf{u}$ in $G$ orthogonal to the all-one vector, the complementary prism $G\overline{G}$ exhibits fractional revival from the state $[\mathbf{u},\mathbf{0}]^T$ with respect to the adjacency, Laplacian, and signless Laplacian matrices. We further characterize perfect pair state transfer in the complementary prism of a complete graph and establish the existence of perfect pair and plus state transfer in the complementary prism of a complete bipartite graph.

We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley $g$-functions in the spirit of Stein's work for compact Lie groups and prove their $L^{p}$-boundedness for $1<p\leq 2$ and for some of them also for $1<p<\infty$. As an application, we define associated Riesz transforms and imaginary powers and prove their $L^{p}$-continuity for $1<p<\infty$. Passing to the average with respect to the action of the associated reflection group, we obtain $L^{p}$-boundedness of $g$-functions for the symmetric Poisson semigroup for all $1<p<\infty$. In particular, our framework covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases.

This paper is a direct continuation of papers arxiv:2401.00053 and arxiv:2507.09332. By this reason, neither the introductory part of the paper nor the bibliography are duplicated here. However, for the reader's convenience, the formulas from the previous papers that are referenced are given in a special addendum at the end of the paper under their original numbers. In this paper, we investigate new local foliations: major and medium pockets and several bilinearity regions. The appearance of such local foliations is illustrated by further analysis of examples with third-degree polynomials. At the end of the paper, a complete overview of possible foliations is given when the boundary values are third-degree polynomials.

The asymptotic behavior of the coefficients of a sequence of polynomials is given under the conditions that all the zeros are real and positive and the zeros have an asymptotic zero distribution on $(0,\infty)$. We show how the limit for the coefficients is related to the Stieltjes transform of the asymptotic zero distribution and make a connection with the $S$-transform in free probability. We illustrate with some examples involving orthogonal and multiple orthogonal polynomials.