Papers

A graph G with q edges and p vertices is edge-graceful if some bijection f from E(G) onto {1,...,q} makes the induced vertex sums f^+(v), the sum of f(e) over the edges e incident to v, distinct modulo p. Lee conjectured in 1989 that every tree of odd order is edge-graceful; the broadest general result we have located, due in equivalent form to Kaplan, Lev and Roditty, covers trees of odd order with at most one vertex of degree two. We prove that every tree of odd order with at most two vertices of degree two is edge-graceful. The proof combines zero-sum block partitions of Z_n with perfect and hooked Langford sequences; the residual case analysis is verified symbolically for all odd n <= 5001 and holds uniformly beyond, and the construction was executed and independently re-checked on all 2,245,070 trees of odd order at most 25 with exactly two vertices of degree two. Since an edge-graceful graph is antimagic, the theorem also enlarges the family of trees known to be antimagic.

This monograph investigates the Lebesgue boundedness of planar directional maximal operators $D_{\Omega}$. These are maximal averages of functions over line segments in $\mathbb R^2$ whose slopes lie in a specified set $\Omega\subseteq\mathbb R$. A large body of work has identified a geometric property of $\Omega$, called finite-order lacunarity, as a key factor in ensuring that $D_{\Omega}$ is Lebesgue bounded. While several variations of this notion exist, they all centre on the distribution of gaps in $\Omega$. Building on earlier work, an article of Bateman(2009) asserted a dichotomy for such operators. Namely, $D_{\Omega}$ is bounded on $L^p$ for all $p\in (1,\infty)$ precisely when the slope set $\Omega$ is finite-order lacunary, or equivalently, when $\Omega$ does not admit Kakeya-type sets. Conversely, sublacunary direction sets $\Omega$ admit Kakeya-like phenomena, implying that $D_{\Omega}$ is unbounded on $L^p$ for all $p\in [1,\infty)$. Recent work of Hagelstein, Radillo-Murguia, and Stokolos(2024) identified a gap in the proof of this assertion and produced counterexamples for which the separation mechanism underlying that proof fails, demonstrating the need for a corrected framework. We establish the corrected characterization by introducing a new notion of admissible finite-order lacunarity that faithfully reflects the combinatorial structure of the direction set. This leads to a tree-theoretic characterization in terms of finite splitting number and provides the foundation for new geometric and probabilistic constructions establishing the equivalence between finite-order lacunarity, the absence of Kakeya-type sets, and the boundedness of directional maximal operators. The resulting framework not only resolves the gap in the earlier proof, but also identifies admissible finite-order lacunarity as the structural invariant governing these phenomena.

We study the cofree functor on the category of group actions and examine its categorical and combinatorial properties. Motivated by this construction, we introduce a family of functors associated with subgroups and develop their theory through a corresponding coreflective subcategory of group actions. Finally, we investigate the comonad in the category of sets induced by the cofree adjunction and prove that it classifies the groups.

Sport leagues assign a fixed score-point to each match outcome and rank the players (or teams) by their accumulated scores. Since the score-points are mandated without a probabilistic model in mind, we investigate models that make the score the right summary of the data. We find the score to be a sufficient statistic if and only if the points rule is constant-sum, so that every match distributes a fixed number of points between its participants, and the outcomes follow the adjacent categories model whose slope parameters are the score-points themselves. The constant-sum condition fails in the National Hockey League and in football, which awards three points for a win instead of the two required by sufficiency. We also establish that when two players meet every other opponent equally often, and each plays as often at home as away, they are ordered identically by the score and by the estimated skills; a venue-balanced round-robin satisfies the condition for all pairs. The Elo ranking then produces the same order. In nine leagues from football, ice hockey and volleyball, the two orders differ only where the schedule condition fails, and the uniform rule awarding one point per outcome level agrees with the estimated skills more closely than each league's own rule, unless that rule is already the uniform one.

We construct a bilinear pairing for $p$-adic families of cohomology classes associated to locally symmetric spaces of a certain level at $p$ and obtain a comparison between the specialisations of constructed pairing and the natural duality pairings on cohomology of automorphic sheaves under specialisation maps. We show that the considered locally symmetric spaces with the chosen level admit an automorphism which naturally generalises the Atkin–Lehner involution for modular curves and interchanges the ordinary and anti-ordinary parts of the cohomology thereby making it possible to pair two anti-ordinary classes. Our construction is motivated by the pairing constructed by Ohta for modular curves and recovers it.

We study the generalized covering radii of binary primitive BCH codes, which measure how many parity-check columns suffice to span several prescribed syndromes. For the four-error-correcting family of length $2^m-1$, the second radius is exactly $11$ for $m\geq55$, and it is either $11$ or $12$ for $m\geq16$. For every fixed error parameter at least two, we give an explicit stable bound with two adjacent possible values for the second radius and an arithmetic criterion for exactness. At every higher order, we obtain explicit lower and upper bounds whose additive gap is bounded independently of the order for each fixed error parameter, once explicit field-size conditions hold. The bounds are therefore asymptotically tight as the order grows. The higher-order upper bound retains the common-core support count established by Xiong, Yip, and Zullo; our refinement reduces its sufficient field-size threshold at large order. The refinement combines componentwise mixed degrees with the ordinary degree of a multicone. The upper bounds come from constructing syndrome representations with a common set of parity-check columns.

In this report, we recall Christiansen's [2011, Num. Math.] discrete elasticity complex that uses spaces of polynomials and distributions. Using the degrees of freedom of those spaces, we write a complex using vector-valued chains and co-chains that is isomorphic to Christiansen's complex. The operators of the new complex are algebraic, but use the geometry of the simplicial complex. Finally, we define a de Rham-like map from the smooth elasticity complex to this discrete complex.

The ultra-weak variational formulation (UWVF) is known to be the Trefftz discontinuous Galerkin scheme obtained from the standard PWDG flux family by setting \(\alpha=\beta=\delta=1/2\). We give a short structural explanation of this fact. On every interior mesh edge, the two PWDG jump terms at these parameter values are exactly one half of the sum of the squared norms of the two oriented impedance-transmission defects used by the UWVF. Thus the Trefftz-DG norm is the natural Riesz norm of the global transmission defect. In PWDG coordinates its Gramian is the anti-Hermitian part of the discrete matrix, and Riesz normalization puts the operator in the normal form \(K+\ii I\), with \(K\) Hermitian. The same argument applied directly to the discrete UWVF yields the complementary normal form \(\tfrac12 I+\ii K_{\rm U}\). This identifies the classical equivalence as a relation between broken-field and impedance-trace coordinates for the same edge-coupling problem, rather than only as a particular choice of penalties.

Timely access to automated external defibrillators (AEDs) remains a major limitation of public-access defibrillation systems because stationary AED networks cannot adapt to spatial and temporal variation in out-of-hospital cardiac arrest (OHCA) demand. We study a platform-enabled mobile AED model in which AEDs are carried by existing urban ride-hailing fleets that can be dispatched to OHCA locations. Rather than assuming additional AED capacity, we hold the total AED supply fixed and compare alternative allocations between stationary and mobile deployment, including hybrid policies that retain both forms of coverage. Our analysis leverages real-world urban data from New York City and Toronto, integrating georeferenced public AED inventories, EMS-reported OHCA incident data, and ride-hailing mobility data. The results show that mobile AEDs can substantially improve both response times and reliability, but the benefits depend on how AED capacity is divided between stationary and mobile deployment. In New York City, average response-time gains exceed 2.5 minutes and peak when approximately 55 to 70 percent of AED capacity is mobile; beyond this range, both gains and reliability decline as stationary coverage is reduced. In Toronto, average gains approach 4 minutes and then plateau as the system moves toward a fully mobile configuration. These contrasting patterns show that mobile AEDs can materially improve access, but the best mix of stationary and mobile AEDs must be tailored to local coverage and mobility conditions.

We give an equivalent definition of a non-pointed torsion theory, also called a pretorsion theory in the literature. This uses a new, more general, notion of a near-torsion theory, and does not use non-pointed exactness. Furthermore, a general method of associating the `largest' (non-pointed) torsion theory to any given near-torsion theory is described. We also describe a near-torsion theory formed by the class of cokernel diagrams and the class of kernel diagrams in a pointed category with kernels and cokernels, and its associated torsion theory.

Given an ideal $\mathcal I$ on $\omega$, a subgroup $H$ of the unit circle $\mathbb T$ is said to be $\mathcal I$-characterized if there exists a sequence of integers $(a_n:n\in\omega)$ such that $$ H= \left\{ x\in\mathbb T: \mathcal I\text{-}\lim_{n\to\infty}a_nx=0 \right\}. $$ We investigate the descriptive complexity and Polishability of these subgroups in terms of the structural and topological properties of the ideal $\mathcal I$. Our main structural result shows that $\mathcal{I}$ is an analytic $P$-ideal if and only if all $\mathcal I$-characterized subgroups are Polishable. In such case, we explicitly describe a compatible finer Polish group topology. Using results on Polishable subgroups, we obtain a trichotomy for their possible Borel complexities. If $\mathcal{I}$ is a generalized density ideal, we show the sharper dichotomy that every proper $\mathcal I$-characterized subgroup is either countable or $F_{\sigma\delta}$-complete. We also prove that this fails for general analytic $P$-ideals by constructing a subgroup characterized by a summable ideal which is neither $F_\sigma$ nor $F_{\sigma\delta}$-complete. Finally, we give explicit descriptions of the subgroups associated with the sequences of powers, the Fibonacci sequence, and the sequence of factorials. We conclude with several open questions.

We provide proofs of several variants for spectra of the original Adams Conjecture. Our approach uses F-spaces (also known as Segal Gamma-spaces), resulting in establishing both unstable and stable homotopy equivalences between maps of spectra associated to the J-map and the Adams operations. We employ a rigid version of Artin-Mazur etale homotopy theory applied to commutative diagrams of simplicial schemes defined over the Witt vectors of fields of positive characteristic. Much of this text is devoting to developing a theory of X-fibrations and l-complete X-fibrations over the F-spaces we consider.

We give a one-parameter functional equation for the real Rogers dilogarithm, with arguments in $\mathbb{Q}\bigl(u,\sqrt{4-3u^{2}}\bigr)$, and prove it by an explicit array of ten instances of Rogers' five-term relation: the fifty arguments so contributed cancel down to the five of the identity, which admit no shorter relation among themselves. The equation comes from a pair of integrals whose equality is elementary, and we show that the underlying integrand is essentially forced. Specialising the parameter gives identities over $\mathbb{Q}(\sqrt{33})$ and $\mathbb{Q}(\sqrt{17})$, a relation between $\mathbb{Q}(\sqrt{13})$, $\mathbb{Q}(\sqrt{3})$, and $\operatorname{Cl}_2(\pi/3)$ with a new analogue for Catalan's constant, and a pair of dilogarithm ladders of quartic base over $\mathbb{Q}(\sqrt{33})$. The base equations of these ladders, and of four conjectural ones located by integer-relation search, are irreducible quartics with four real roots. All previously recorded ladders of degree four known to us have base equations with two real roots, so these appear to be the first totally real ladders of degree exceeding three.

We prove a parabolic Bogomolov–Gieseker inequality for stable regular parabolic Higgs sheaves on a compact Kähler manifold \((X,\omega)\) equipped with a simple normal crossing divisor \(D\). We also prove that a polystable regular parabolic Higgs sheaf admits a pluriharmonic metric on \(X\setminus D\), provided that each stable summand has parabolic degree zero and vanishing integrated second parabolic Chern character.

We develop a Monte Carlo lattice Boltzmann method to compute statistical solutions to the three-dimensional incompressible Navier–Stokes and Euler equations. Entropic space-time adaptive relaxation of the higher order kinetic moments yields stable numerical solutions with decreasing viscosity. We provide a convergence analysis that is conditional on four explicitly stated assumptions regarding the discrete dynamics. Under diffusive scaling, the laws of the discrete ensemble converge along a subsequence to a limit satisfying the Foias–Temam Liouville formulation of the Navier–Stokes equations. Consequently, provided the structure function scaling holds uniformly, the vanishing viscosity limit of these measures satisfies the multi-point statistical Euler hierarchy of Fjordholm, Mishra, and Weber. The limit measures inherit the known weak-strong uniqueness principle on the interval of existence of a strong Euler solution. Under explicit scaling assumptions, a Kuznetsov-type argument yields a fractional 1-Wasserstein convergence rate. We present three-dimensional computations of time-dependent statistical solutions along the inviscid limit of the incompressible Navier–Stokes equations together with convergence measurements in the Wasserstein metric. Numerical experiments on a randomized Taylor–Green vortex with 24-dimensional initial uncertainty recover Kolmogorov's K41 scaling for energy spectra and structure functions, exhibit the failure of pathwise strong convergence, and yield Wasserstein convergence rates of about $0.5$ at the onset of turbulence. Finally, error measurements with respect to spectral hyperviscosity computations indicate that the computed limit measure is independent of the numerical regularization.

It was proven by Yoshioka that given a complex abelian surface $A$ of Picard rank $1$ whose primitive polarization is non-principal of type $(1,d)$, there is an isomorphism $\Psi:\mathrm{Hilb}^d_A\times\hat{A}\to M_{\hat{A}}(0,\hat{l},-1)$ where $\mathrm{Hilb}^d_A$ is the Hilbert scheme of lenth-$d$ subschemes of $A$ and $M_{\hat{A}}(0,\hat{l},-1)$ is a moduli space of Gieseker-stable sheaves on the dual abelian surface. Specifically, $M_{\hat{A}}(0,\hat{l},-1)$ parametrizes rank $1$ torsion-free sheaves with Euler characteristic $-1$ that are supported on a curve whose Néron–Severi class is dual to that of the polarization on $A$. The Fourier–Mukai transform is a crucial component of $\Psi$. In this paper, we use the isomorphism $\Psi$ to deduce information about curves on abelian surfaces. In the case that $Z\in \mathrm{Hilb}^d_A$ is symmetric, i.e., fixed by the inverse map $\iota$ on $A$, we use quadratic forms to compute information about the sheaf $\Psi(Z)$ and its supporting curve. It was recently shown by Knutsen and Lelli-Chiesa that any singularity on a curve of geometric genus $2$ contained in a general $(d_1,d_2)$-polarized abelian surface must have multiplicity at most $6$, among other constraints. In contrast, we demonstrate there are curves with singularities of arbitrarily high multiplicity contained in general $(1,d)$-polarized abelian surfaces for sufficiently large $d$. Furthermore, we identify the isolated fixed points of $\iota$ in acting on the variety of Kummer type $K_{\hat{A}}(0,\hat{l},-1)$ when $d=4$. Along the way, we prove some structural results on symmetric line bundles, showing that if $d$ is even, the dual of an odd line bundle is odd.

We prove a Tachibana-type result for Kähler-Einstein manifolds isolating complex projective space based on its Calabi curvature operator. The proof uses a new Bochner formula expressing the Lichnerowicz curvature term via the Calabi curvature operator. This resolves one of the problems posed at the AIM workshop "The Bochner Technique” (May 2026).

A classical idea in analysis going back at least to work of Simon (1997) is that Liouville theorems for solutions to elliptic or parabolic PDEs are equivalent to Schauder type regularity estimates. The goal of this course is to describe some recent developments of this idea concerning the regularity of the complex Monge-Ampère equation with respect to singular reference metrics. We will start with a quick look at the classical $C^2$ and $C^3$ estimates of Calabi-Aubin-Yau and then present a new proof of the Evans-Krylov $C^{2,\alpha}$ estimate on a Euclidean ball. Based on this we will consider the case of singular backgrounds such as cylinders and cones, discussing some recent work by Hein, Tosatti, Lee and Klemmensen. Our discussion is far from complete and knowledge of Kähler geometry including the Aubin-Yau theorems is assumed. The course includes five exercises with solutions and a problem list.

We develop a general procedure for constructing time-dependent integrable chiral field theories from autonomous unitary difference-form $S$-matrices satisfying the Yang-Baxter equation. Spectral parameters are transported along the free right- and left-moving characteristics, defining a map from physical spacetime to spectral space on which the two-body scattering data are evaluated. Requiring spatially homogeneous right-left scattering forces the characteristic map to be affine. The inverse Cayley transform then determines the local contact interaction, while evaluation along the affine spectral trajectory fixes its time dependence. Thus, the nonautonomous interaction is determined by the autonomous scattering data together with chiral kinematics and spatial homogeneity, rather than being introduced independently. The Yang–Baxter equation supplies the factorized many-body transport, and on a spatial circle periodicity leads to quantum Knizhnik–Zamolodchikov (qKZ) equations whose compatibility defines a flat discrete transport in spectral space. Pulling the corresponding spectral-space amplitudes back to physical coordinates gives the time-dependent many-body wavefunctions. Rational $SU(N)$, trigonometric $U_q(\widehat{\mathfrak{sl}}_2)$, and rational $O(N)$ scattering illustrate how the same mechanism generates distinct nonautonomous interactions. The resulting framework gives a geometric route from autonomous factorized scattering data to time-dependent integrable field theories.

We analyze the convergence rate of certain sequence of the duality gaps in the generalized Frank-Wolfe method, under certain regime of the weak growth condition proposed in Peña (2023). Our analysis leverages a recursion lemma that may be of independent interest.

Iteratively reweighted least squares (IRLS) methods constitute a natural approach to nuclear norm minimization, but their convergence rates and the role of the weight operator have remained poorly understood. This paper establishes sharp convergence rates for IRLS methods for constrained nuclear norm minimization in low-rank recovery. A central ingredient is a new majorization analysis for the smoothed nuclear norm: we prove that the harmonic-mean weight operator defines a valid global quadratic majorizer. Furthermore, we show that this weight operator is optimal within the family of power-mean weights, clarifying why it improves over classical one-sided reweighting schemes that use only row- or column-space information. Under a Schatten-1 null space property, we prove global linear convergence of IRLS algorithms using a variety of weight operators, including the harmonic-mean weights. For IRLS with harmonic-mean weights, we prove a dimension-independent, locally linear convergence rate. We provide a counterexample showing that this dimension-independent local rate cannot in general be obtained for IRLS algorithms using one-sided weight operators, which predominate in the literature. Numerical experiments corroborate the theoretical results and illustrate the practical advantage of harmonic-mean reweighting across square, rectangular, and adversarially initialized recovery problems.

Point clouds arising in image collections, samples from Markov chain Monte Carlo, or states of a random walk, often have a simple underlying geometry which is obscured by noise, high ambient dimension, and the failure of Euclidean distance to reflect similarity. Methods such as UMAP and t-SNE condense such data into usable form, but rely on heuristic choices and provide no guarantee that the output reflects the topology of the input. We introduce a condensation method that comes with such a guarantee. Encoding the data as a positive $m\times n$ stochastic matrix $Q=(q_{ij})$, for instance the transition matrix of a random walk on the point cloud, we define a potential function $\psi(p)=\log \sum_{i} \exp(-KL(p,q_{i\bullet}))$ on the probability simplex $\Delta_n$, where $KL(p,q)$ is the Kullback-Leibler divergence, and prove that $\psi$ is $c$-convex in the sense of Optimal Transport Theory for the cost function $c(p,q)=KL(p,q)$. The associated transport map collapses noisy directions while provably preserving topology: the super-level sets of $\psi$ are homotopy equivalent to those of a $c$-conjugate function, whose image is a condensed, resampleable family of topological spaces which can be interpreted as a continuous analog of an alpha shape. We demonstrate the method by recovering the circle of camera angles from the COIL image dataset, where a standard PCA pipeline produces spurious homology, and the quotient $SO(3)/A_4$ from $45{,}000$ views of a tetrahedron in the SYMSOL pose-estimation benchmark.

We consider a many-server queue where tasks with general patience and service times arrive as a renewal process. Contrary to the standard assumptions, we allow for preemption and abandonment during service, relevant in applications such as electric vehicle charging and anytime algorithms in cloud computing. Moreover, we do not assume independence of patience and service times, which has been a critical assumption in the literature. We describe the state of the system using a discrete measure on a two-dimensional orthant, such that the coordinates of its atoms represent the attained service and time in the system of tasks. Under mild assumptions, we derive the fluid limit as the number of servers approaches infinity and the arrival rate of tasks grows proportionally. The limit is given by a measure-valued integral transport equation that we solve explicitly when the initial condition is the null measure. We also prove that all solutions, regardless of the initial condition, converge to the same fixed point, which admits a closed-form expression. Our results focus on the preemptive Last-Come-First-Served (LCFS) policy, which has been identified as practically appealing in recent work.

We investigate Bell-CHSH correlations for a chiral massless Majorana field localized in a double cone. The conformal modular flow of the associated light-ray interval is converted into translations by a rapidity-like coordinate, and the one-particle scalar product is derived in modular momentum space. Particular attention is paid to the fact that this scalar product carries a Fermi-Dirac weight. Requiring the modular conjugation to be antiunitary with respect to that weighted scalar product fixes the Fourier-space realization of the modular objects. The resulting Tomita-Takesaki operator and its adjoint generate Alice and twisted-dual Bob directions for which all four crossed CAR orthogonality relations follow automatically from modular localization. The Bell problem then reduces to a simple one-function functional. For a Gaussian modular-momentum profile $h_\sigma(k)=\exp[-k^2/(2\sigma^2)]$ the Bell parameter is larger than the classical bound for a broad range of widths and approaches the Tsirelson value analytically, \[ \lim_{\sigma\to 0^+}{\it B}(\sigma)=2\sqrt2 \] Thus near-maximal Bell violation is obtained from an explicit sequence of smooth rapidly decreasing profiles concentrated around zero modular momentum, corresponding to the modular spectral value $\lambda=1$.

We classify regular polynomial endomorphisms of $\mathbb C^k$ with the same equilibrium measure under some mild condition on the dynamics at infinity. In particular, for $f$ in a Zariski dense open subset of the parameter space and any $g$ of the same degree, $\mu_g=\mu_f$ if and only if $f=g$. We also establish finiteness results for maps with a prescribed equilibrium measure. Our approach rests on the stable manifold structure near the hyperplane at infinity. Applications include a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.

For a $n$-dimensional compact Kähler manifold $X$ with a pair of Kähler classes $(\alpha,\beta)$, we show that the algebraically defined $J$-null locus is equal to the analytically defined $J$-non-ample locus under the assumption of $J$-bigness(which is automatic for semistable pair up to dimension $3$ \cite{3d}), and boundary cone condition $c_{\alpha,\beta}\omega^{n-1}-(n-1)\omega^{n-2}\wedge\chi\geq 0$ for some Kähler form $\omega\in \alpha,\chi\in \beta$. The key tool is the generalized Khovanskii-Teissier inequality associated to $J$ equation formulated by Collins \cite{CT21} and its extension to singular Kähler space.

We formulate split Galois descent for fusion categories through framed equivalences, reducing the existence of a split form to a group-theoretic splitting problem. The associated degree-two obstruction combines categorical coherence with stable descent of the simple objects, while fusion spaces impose index and parity restrictions. For quantum groups at roots of unity, we use the quasi-$R$-matrix to make the bar involution tensor-compatible and prove that the underlying fusion category $\mathcal C(\mathfrak g,k)$, associated with the simply connected root datum, has a split form over the maximal totally real subfield of the cyclotomic field generated by the quantum parameter. The same holds for all its cyclotomic Galois conjugates; in particular, every $\mathcal C(\mathfrak g,k)$ has a split real form. By contrast, no split real form exists for the standard braiding except on $\mathcal C(E_8,1)$ and $\mathcal C(D_{4m},1)$, $m \geq 1$. We further show that associative zesting can destroy real descent of $\mathcal C(\mathfrak g,k)$, while every braided zesting of $\mathcal C(\mathfrak g,k)$ still admits a split real form as an underlying fusion category. Applications to pointed and finite-group representation categories give criteria for real descent in terms of cohomology and Frobenius–Schur indicators.

Recent work gives an all-$\beta$, all-order stochastic-zeta representation of the correlation functions of the $\Sine_\beta$ process and determines their leading Vandermonde asymptotics when several variables merge. We compute the first nontrivial correction throughout the regime $m\beta>1$. If $a_1,\ldots,a_m$ are distinct real numbers and \[ V(a)=\sum_{1\le i<j\le m}(a_i-a_j)^2, \] then, as $\varepsilon\to0$, \[ \rho^{(m)}_\beta(\varepsilon a_1,\ldots,\varepsilon a_m) =C^{(m)}_\beta |\varepsilon|^{\beta\binom m2}\prod_{i<j}|a_i-a_j|^\beta \left[1-\frac{\beta^2 V(a)}{8(m\beta-1)(2m+1)}\varepsilon^2+o(\varepsilon^2)\right]. \] For $m=2$ the normalized second-order coefficient is $-\beta^2/[40(2\beta-1)]$ for every $\beta>1/2$. The proof combines a finite-$N$ rotational Ward identity, exact Hua–Pickrell trace moments, compact moment bounds for the stochastic-zeta entire function and its derivatives, and a quantitative multivariate expectation–Taylor lemma. As a by-product we evaluate \[ \E_{\HP_{\beta,m\beta/2}}\sum_x x^{-2}=\frac{m\beta}{4(m\beta-1)(2m+1)}. \] The pole at $m\beta=1$ marks the boundary of the present second-moment argument and suggests a transition in the form of the next fusion correction.

Let $G$ be a quasi-reductive supergroup (so its underlying algebraic group $G_{\bar 0}$ is reductive). We consider two trivially intersecting classes of odd elements: neat elements and balanced elements. Neat elements are always $ad$-nilpotent and may be embedded into subalgebras that are isomorphic to $\mathfrak{osp}(1|2)$, a simple Lie superalgebra whose underlying Lie algebra is $\mathfrak{sl}_2$. Balanced odd elements, on the other hand, are a natural generalization of the notion of a self-commuting element (an element $x\in Lie(G)_{\bar 1}$ for which $[x,x]=0$). Balanced elements are used to define homology-type functors on the category of representations of $G$. We show that any element $x\in Lie(G)_{\bar 1}$ may be written as a sum of a neat and a balanced odd element which commute with each other. This theorem has a categorical application. Let $\mathfrak{g}^{(1|1)}$ be the $(1|1)$-dimensional Lie superalgebra generated by $x \in Lie(G)_{\bar 1}$. The semisimplification of the category of finite-dimensional super-representations of $\mathfrak{g}^{(1|1)}$ is a functor $S: Rep(\mathfrak{g}^{(1|1)}) \to Rep(SOSp(1|2))$. Any $x\in Lie(G)_{\bar 1}$ induces a homomorphism $ i_x:\mathfrak{g}^{(1|1)}\to Lie(G)$. Let $$\Phi_x=S\circ (-)\downarrow_{i_x}:Rep(G)\to Rep(SOSp(1|2))$$ be the composition of the restriction functor $(-)\downarrow_{i_x}$ and the functor $S $. We show that the functor $\Phi_x$ may be described explicitly using the homology-type functor $\Phi_{x_{bal}}$ corresponding to the balanced part of $x$ in the above decomposition. These homology-type functors are known as Duflo-Serganova functors. Finally, we provide a full classification of distinguished odd elements in simple quasi-reductive Lie superalgebras and show that in all cases except $\mathfrak{spe}(n)$, such elements are either balanced or neat.

We prove a Fefferman–Stein good-$\lambda$ inequality for the Dunkl Poisson semigroup associated with a finite reflection group and a non-negative multiplicity function. For arbitrary complex-valued $f\in C_c^\infty(\mathbb R^N)$, with no $G$-invariance assumption, it compares the orbit-conical non-tangential maximal function $\mathcal N_P^\beta f$ with the area function $\mathcal S_Pf$ formed from the full space-time Dunkl carré du champ, including its reflection-difference energy. The main obstruction is that a general cut-off creates wall differences not controlled by the local Euclidean-gradient product identity. The good set $E=\{x:\mathcal N_P^\beta f(x)\le\lambda\}$ is $G$-invariant; by the equivariance of the Poisson semigroup, so is $a=\varphi(P_t \mathbf 1_E)$, and hence all reflection differences of the cut-off vanish. Poisson maximal and tail estimates, together with the $L^2$ Littlewood–Paley estimate for $P_t \mathbf 1_{E^c}$, then yield the desired distribution inequality. Its integrated form gives maximal-to-area estimates for every $0<p<2$ and endpoint $H^1$-to-$L^1$ bounds for the orbit-conical and Euclidean-conical intrinsic area functions. For chamber lifts of globally smooth data, the inequality has an equivalent formulation on a fundamental chamber, where orbit cones become Euclidean cones and the reflection energy becomes a finite wall coupling. Combined with the known semigroup square-function characterization, these bounds characterize the Dunkl Poisson maximal Hardy space among $L^1(d\omega)$ data.