Papers

In this paper, we establish the boundedness of the bilinear Bochner-Riesz means $\mathcal{B}^{\alpha}_R$ on the complex sphere $\mathbb{S}$. More precisely, we prove that $\mathcal{B}^{\alpha}_R$ is bounded from $L^{p_1}(\mathbb{S}) \times L^{p_2}(\mathbb{S}) \to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\leq p_1, p_2 \leq \infty$, for an admissible range of exponents, with the required smoothness parameter $\alpha$ described in terms of the topological dimension of $\mathbb{S}$. To facilitate our proof, we establish several analytic estimates, including restriction-type estimates, weighted Plancherel estimates with large power of weights and bilinear weighted Plancherel estimates, which are derived from the ground up in our setting and may be considered of independent interest.

We prove mode stability for the Klein-Gordon equation on Kerr-de Sitter black hole spacetimes in the full subextremal range and for the range $m_{\rm KG}^2\in[0,\Lambda\sqrt{2}]$ of squared scalar field masses that includes, in particular, the minimally coupled case $m_{\rm KG}=0$ and the conformally coupled case $m_{\rm KG}^2=\frac{2}{3}\Lambda$ (i.e., the Teukolsky equation for spin $0$). In combination with recent work by Petersen-Vasy, this proves, unconditionally, that sufficiently regular solutions of the scalar wave equation decay exponentially fast to constants, and in fact to zero for nonzero $m_{\rm KG}$.

Böhm metrics on $S^{k+1}\times S^l$ and $S^{k+l+1}$ ($k,l\geqslant 2$, $k+l\leqslant 8$) occur in sequences of Einstein metrics that converge to a cone. We prove that, along such a sequence, the number of negative eigenvalues of the Lichnerowicz Laplacian acting on transverse-traceless tensors tends to infinity: these metrics become increasingly unstable. This result gives a partial answer to a conjecture of Gibbons, Hartnoll and Pope concerning the instability of the generalised black hole spacetimes built from Böhm metrics.

The huge amount of literature about the divergence equation in bounded Lipschitz domains of $\R^n$ ($n\ge2$) is fairly disconnected and even apparently simple problems remain unsolved. We go several steps further in the knowledge of this equation and of some related inequalities. We prove that among its infinitely many solutions there exists a special one obeying elliptic regularity theory. This solution simplifies the definition of the Bogovskii constant $C_B$ and allows us to prove its attainment in smooth domains. We then obtain a universal lower bound for $C_B$ in any Lipschitz domain as well as a non-minimality criterion. As expected, balls are minimisers as the domain varies, although no symmetrisation technique is used. We also analyse ellipsoids and annuli: for the first we improve the (so far) best asymptotic inequality for thinning domains. Finally, we introduce higher-order Bogovskii constants, which lead to a polyharmonic Stokes problem. Not only regularity theory applies without smoothness of the domain when the source has some vanishing traces, but we also prove that balls are again minimisers among Lipschitz domains with the very same Bogovskii constant. Three main challenging open problems are suggested.

We study steady solutions to the two-dimensional incompressible Euler equations in an infinite channel, whose far-field limits are uniformly non-stagnant shear flows. In the smooth category,for a broad class of prescribed far-field shear profiles, non-shear steady states exist via the construction of two-dimensional solutions of the semilinear elliptic equations of stream function by the min–max method. In the analytic category, we establish a comparison principle for the analytic steady states and we show for a dense family of analytic uniformly non-stagnant shear profiles, every analytic steady state with the prescribed far field must itself be a shear flow. In particular, there are far-field shear profiles which exhibit flexibility in the smooth category but rigidity in the analytic category. Furthermore, the dense rigidity is sharp in the sense that there exists analytic shear profile which admits flexibility in the analytic category.

For a finite associative unital ring $R$, the null ideal of $R$ is the collection of polynomials with coefficients from $R$ that send each element of $R$ to zero under evaluation. It was conjectured that the null ideal of $R$ is always a two-sided ideal of its overlying polynomial ring. The conjecture was proved to be false with the construction of a subring of $4 \times 4$ upper triangular matrices over $\mathbb{F}_2$ for which the null ideal is not two-sided. The Jacobson radical of this counterexample ring has nilpotency 4. We prove that if the Jacobson radical of $R$ has nilpotency at most 3, then the null ideal of $R$ is two-sided. By extending the known counterexample ring, for each $n \geq 5$ we present a ring for which the null ideal is not two-sided, and the Jacobson radical has nilpotency $n$.

We develop a neural-network-accelerated multiscale hybridizable discontinuous Galerkin method for elliptic problems with heterogeneous coefficients. The method preserves the standard MsHDG local-to-global structure: fine-scale HDG problems on coarse blocks define discrete Dirichlet-to-Neumann operators, which are assembled through the standard MsHDG global skeleton equations. To reduce the cost of constructing these local operators, we train a neural network on coefficient fields defined on a reference block and use the predicted operators in place of repeated fine-scale local solves. The numerical experiments assess both the accuracy and online efficiency of the resulting NN-MsHDG method. For two-dimensional binary permeability fields with moderate contrast, the neural method reproduces the standard MsHDG solution with relative modeling errors of a few percent while reducing the total online computational cost by factors of approximately 5 to 16, depending on the coarse trace dimension. In the high-contrast regime, the chosen polynomial coarse trace spaces already yield substantial discretization errors in the standard MsHDG method, indicating the need for more effective coarse spaces, such as coefficient-adapted spectral trace spaces. In addition, the learned local operators introduce modeling errors that become severe as the trace space is enriched. These results demonstrate the potential of neural surrogates for accelerating multiscale HDG computations while also highlighting the need for improved operator representations and a better understanding of error amplification in high-contrast problems.

We apply the adaptive Nesterov momentum (ANM) method [30] to electrical impedance tomography under the complete electrode model. The forward problem is formulated in the variational CEM setting, accounting for finite electrode size, contact impedance, insulating gaps, and the mean-free voltage gauge. The resulting nonlinear inverse problem is treated within a unified dual-to-primal framework using three classes of strongly convex structural penalties: an L2-type penalty, an L1-type penalty promoting sparse deviations from a calibrated homogeneous background, and a TV-type penalty favoring approximately piecewise-constant conductivities with sharp interfaces. The TV class is implemented using both smoothed TV and Huber-TV formulations. The data- misfit gradient is computed through CEM adjoint equations and stabilized by Sobolev smoothing. The method is evaluated on the publicly available KIT4 tank measurement data after calibration of the background conductivity and contact impedance from no-object measurements. The experiments include single, multiple, mixed-conductivity, and geometrically challenging phantom configurations. The L2-type penalty generally produces smooth but diffuse reconstructions, whereas the L1-type penalty yields cleaner backgrounds with occasional geometric distortion. The smoothed TV and Huber-TV penalties provide more spatially coherent localization and exhibit similar reconstruction behavior across most tested configurations. These results demonstrate the practical applicability of the adaptive Nesterov framework to measured CEM-EIT data.

For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erdős and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)) \frac{n^2}{16}$. Balogh and Liu (JCTB, 2014) disproved this conjecture and determined the asymptotic value of $f_4(n,\mathrm{ex}(n,K_3)+1)$. He, Ma, Ma and Ye (JCTB, 2023) later determined $f_{p+1}(n,\mathrm{ex}(n,K_p)+1)$ asymptotically for every $p\ge 3$, and asked for the value of $f_{p+1}(n,m)$ for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. In this paper, we answer their question asymptotically for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. We also determine the exact value of $f_3(n,m)$ for all $0\le m\le \mathrm{ex}(n,K_3)$ by a different method.

Adapting ideas of \cite{akutagawa2007perelman}, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in \cite{wei2009comparison}. Using this we show that low dimensional GRS are diffeomorphic to $S^3 / \Gamma$ or $S^3 \times S^1 / \Gamma$. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion $G_2$, strong torsion $\mathrm{Spin}(7)$-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered $5$-manifolds over complex orbifolds \cite{kollar2006circle}.

The motion of a slender vortex in ideal incompressible fluid is a classic problem in hydrodynamics. Formulae for the velocity of a vortex ring go back to work of Kelvin, Helmholtz, Hicks and Dyson in the nineteenth century, while more recently a number of models have been created for simulating the motion of slender tubes of vorticity of general shape. For a vortex tube or filament localised near to a curve C(t), the relevant small parameter to measure slenderness is the tube radius divided by the radius of curvature of C. The present paper revisits this range of classic problems by introducing a coordinate system closely linked to the geometry of vortex surfaces in a slender vortex. The motion of fluid elements in this coordinate system has an action–angle form, and with this the vorticity equation simplifies radically. At the same time, however, most aspects of the shape or evolution of a vortex are thrown into the description of the coordinates, and in particular the corresponding metric and volume form. As the coordinate system is non-orthogonal and time-dependent, tools of differential geometry are most easily used to describe the structure of both vorticity and coordinate system, and a general mathematical framework is set out. This resulting system of equations is taken as far as possible with only the assumption of vortex slenderness in place, but allowing arbitrary motions and distortions of the vortex core and of the curve C(t). The modelling is applied to calculate the motion of a slender vortex ring with arbitrary axial flow, solving directly for the shape of perturbed vorticity surfaces and giving results in agreement with earlier studies. The general framework set up in this paper is suitable for the development of simplified equations for vortex motion and interaction in future studies.

We study a fast-forwarded quantum algorithm for solving weakly nonlinear dissipative ordinary differential equations. Our approach is a combination of the Carleman embedding technique and the linear combination of Hamiltonian simulation algorithm for linearized systems with fast-forwarded scaling. The complexity of our algorithm does not explicitly depend on the evolution time $T$, thus greatly improving the previous state-of-the-art $\widetilde{\mathcal{O}}(\sqrt{T})$ to $\mathcal{O}(1)$, and any remaining time dependence enters through the output norm and forcing parameters. We rigorously analyze the performance of this approach by convergence guarantees of the Carleman embedding for time-dependent coefficient matrices and detailed complexity estimates, and improve the realization of the Carleman-embedding-based algorithms by simplifying the post-selection step. In addition, we perform a numerical study on differential equations beyond the weakly nonlinear case, and identify possibility of achieving fast-forwarding scaling for systems with stronger nonlinearity or linear non-resonant effect.

Let $\alpha$ be irrational and let $q_j$ be the denominators of its continued-fraction convergents. We study the function \[ h(x)=\sum_{j\geq1}\frac{\cos(2\pi q_jx)}{q_j} \] and the skew product \[f(x,y)=(x+\alpha,y+h(x))\quad\mathrm{mod}\quad\mathbb{Z}^2.\] The function $h$ is Hölder continuous of every exponent below one. A Fourier argument shows that $h$ is not Lipschitz. The map $f$ is a toral pseudo-rotation with rotation vector $(\alpha,0)$, but it has neither bounded mean motion nor $C^0$-rigidity. Suppose $\alpha$ satisfies the Diophantine condition $\mathrm{DC}(\tau)$. Then, $f$ has $(C,1-1/\tau)$-deviation when $\tau>1$; and it has $(C_\delta,\delta)$-deviation for every $0<\delta<1$, but not for $\delta=0$ when $\tau=1$.

We prove an asymptotic formula with a power-saving error term for the second moment $$ \sum_{q \in \mathcal{Q}} \; \; \sideset{}{^\flat}\sum_{\chi \bmod q} \left| L\big( \tfrac{1}{2} , \chi \big) L\big(\tfrac{1}{2},f\otimes \chi)\right|^2 $$ of the twisted ${\rm GL}(2)$ {$L$}-function and the Dirichlet {$L$}-function at the central point under the assumption of Selberg's eigenvalue conjecture. Here $f$ is a fixed Hecke holomorphic cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and the sum over $\chi$ runs over all primitive even Dirichlet characters modulo $q$, and $$\mathcal{Q}=\left\{q=q_1 q_2\asymp Q: q_1 \leq Q_1, q_2 \leq Q_2,\ Q_2\asymp Q^{\delta_1},(q,6)=1,(q_1,q_2)=1 \right\}$$ with $0<\delta_1<0.0004$.

Monotone rearrangements provide a simple way to enforce shape constraints of an estimator, but existing distributional theory does not cover flat regions, where the target induces no local ordering. We study rearranged estimators in two canonical flat settings. First, for a histogram estimator of the uniform density, we establish functional weak convergence of its non-decreasing rearrangement at the parametric rate on compact subsets of the interval $(0,1)$ after an additional deterministic centering. This result is strikingly different from what is known for strictly monotone densities. Second, we consider two rearranged estimators of rearranged copulas under independence, based on empirical-copula increments and on a checkerboard approximation. After appropriate centering and rescaling, both estimators converge weakly on $[0,1]^2$ to an integrated Gaussian process which was not known before. We further use these results to prove asymptotic normality for a broad class of rearranged copula-based dependence measures, which were recently discussed in Strothmann et al. (2024).

Packing-optimal $(M,n)$ spherical codes attain the largest achievable minimum distance and hence the optimal error exponent at high SNR. Among these codebooks held fixed as SNR grows, the smallest leading coefficient, $K_{M,n}^{\mathrm{fix}}$, equals the smallest number of ordered closest pairs. We show that reoptimizing the codebook at every SNR can yield a smaller leading coefficient $K_{M,n}^\ast$. Specifically, we show that the SNR-wise minimum exact maximum-likelihood error probability is $P_e^\ast=B_{\gamma}^\ast(K_{M,n}^\ast+o(1))$, while the minimum exact error over packing-optimal codebooks is $P_e^{\mathrm{fix}}=B_{\gamma}^\ast(K_{M,n}^{\mathrm{fix}}+o(1))$, where $B_{\gamma}^\ast$ is a common factor. We prove that $K_{M,n}^\ast\leq K_{M,n}^{\mathrm{fix}}$. Strict inequality, $K_{M,n}^\ast < K_{M,n}^{\mathrm{fix}}$ therefore gives a smaller exact error at all sufficiently high SNRs, i.e., $P_e^\ast<P_e^{\mathrm{fix}}$. We also characterize $K_{M,n}^\ast$ as the limit of the minimum Gaussian soft-packing energy. An orthoplex-bound construction explains strict inequality: an SNR-dependent perturbation makes one class of limiting closest pairs slightly closer, without changing the optimal exponent or their unit contributions to the constructed family's leading coefficient, while moving the remaining closest pairs farther apart on a larger but still vanishing scale, causing their contributions to vanish. For $2\leq M\leq n+1$, we prove $K_{M,n}^\ast=K_{M,n}^{\mathrm{fix}}=M(M-1)$. For $M=n+k$, $2\leq k\leq n$, we prove $K_{n+k,n}^{\mathrm{fix}}=4n(k-1)$ and $K_{n+k,n}^\ast\leq 4k(k-1)$. In particular, $K_{n+2,n}^\ast=8<4n=K_{n+2,n}^{\mathrm{fix}}$ for $n\geq3$, so SNR-wise optimization improves the asymptotic error by the factor $n/2$. At $k=n$, both coefficients equal $4n(n-1)$. For $3\leq k\leq n-1$, we conjecture $K_{n+k,n}^\ast=4k(k-1)$.

There is an isomorphism between the frame of radical thick tensor ideals of an essentially small tt-category and the frame of open sets of the Hochster dual of its Balmer spectrum. This isomorphism allows us to compare topological properties of the latter with tt-geometric properties of the former. We use this to give an intrinsic definition of those tt-categories which satisfy a variant of the second De Morgan law, and furthermore give a construction for producing such tt-categories.

We study a class of nonlocal evolutionary partial differential equations with weakly singular temporal kernel and spatially variable diffusion coefficient. The model is posed on $\Omega \subset \mathbb{R}$, and involves a left-sided Riemann–Liouville fractional derivative in space multiplied by a variable coefficient $a(x)$. The temporal derivative is approximated by an $L1$ type scheme, while the spatial operator is discretized by finite difference techniques, resulting in large scale all at once linear systems with a twolevel Toeplitz like structure. We develop and analyze a block lower triangular strategy that mimics the structure of the coefficient matrix while simplifying its components for computational efficiency. The analysis is carried out at the level of matrix sequences by means of generalized locally Toeplitz (GLT) theory. Within this framework, we characterize the asymptotic spectral distribution of the discretized operators and use the associated GLT symbol to guide the construction of the structured approximation. Numerical experiments using the GMRES solver demonstrate that the proposed preconditioning strategy significantly improves convergence rates, robustness, and scalability for large-scale problems. Open problems and possible extensions are briefly discussed at the end of the present work.

We prove that Pyber's upper bound $2^{O(n\log(n))}$ for the number of minimally transitive subgroups of $S_n$ is best possible along the powers of every fixed prime, even when the groups are counted up to permutational isomorphism. As a byproduct, our construction shows that, along the powers of every fixed prime, the maximum order of a minimally transitive permutation group of degree $n$ is $2^{\Theta(n)}$. For completeness, we also present Pyber's previously unpublished proof of his upper bound. We further deduce that the numbers of labelled vertex-transitive graphs and digraphs of order $n$ are both $2^{\Theta(n\log(n))}$, and discuss the implications of our results for approaches to the McKay–Praeger conjecture.

We define a reformulation of the Collatz $3n+1$ map which allows us to express it as an operator on the set of sequences of Boolean polynomials. We express this operator in terms of certain carry sequences that arise from addition in base-2, and then we simplify them and find explicit formulas for them.

Berge's maximum theorem ensures the continuity of the value function and the upper semicontinuity of the solution mapping in parametric optimization problems. This theorem plays a central role in optimization theory, game theory, and dynamic programming. Motivated by the inherent inaccuracies in optimization data, this paper investigates the stability of such problems under sequential perturbations of both the objective function and the feasible mapping. The analysis focuses on the convergence of sequences of value functions and solution mappings via variational approximations of the data. To this end, we employ lower and upper continuous, epi- and hypo-convergence notions for functions, together with lower and upper continuous and graphical convergence notions for multifunctions. In addition, we study some relationships among these types of convergence and provide examples and counterexamples associated with the corresponding notions. Our results extend and complement existing stability results in the literature. We provide applications to generalized Nash equilibrium problems, where stability is obtained via a direct approach, as well as to finite-horizon dynamic programming models under novel perturbation assumptions.

Boson sampling is one of the most prominent methods to verify quantum advantage, harnessing the computational complexity of the permanent of a network matrix. Similar quantum sampling problems utilize other expressions for the output probabilities, such as the fermion-based analog applying determinants. In this work, we study sampling setups of interacting light-matter systems, combining fermion and boson sampling. The mixed network input consists of fermionic and bosonic excitations. Sampling the output probability of the modes then yields contributions that are neither purely determinant nor permanent but general immanants. Specifically, we prove that these immanant contributions are linearly independent from determinants and permanents, rendering our mixed sampling network an ideal candidate for light-matter quantum application beyond pure fermion and boson sampling. Furthermore, quantum sampling with statistics that are neither bosonic nor fermionic results in expressions which even exceed immanants, further extending the functionality of quantum samplers.

We obtain a uniform pointwise estimate for short Weyl sums of degree $n\geq3$ in the intermediate rational-approximation range $$ \frac{1}{qx^{n-2}y}\ll|\lambda| \ll\frac{1}{qy^{n-1}}. $$ This range arises from a second application of Dirichlet's rational approximation theorem in estimating the residual integral that occurs in the derivation of an asymptotic formula for Waring's problem with almost proportional summands. For $r=2^n+1$ and fixed positive numbers $\mu_1,\ldots,\mu_r$ satisfying $$ \mu_1+\cdots+\mu_r=1, $$ we derive an asymptotic formula for the number of representations $$ x_1^n+\cdots+x_r^n=N, \qquad |x_i^n-\mu_iN|\leq H,\qquad 1\le i \le r, $$ valid for $$ N^{1-\theta(n,r)+\varepsilon}\le H \le \frac{N}{\ln N}, \quad \theta(n,r)= \frac{2}{n\bigl((r-1)(n-1)+2\bigr)}. $$ The resulting admissible lower bound for $H$ improves the previously known bound for every $n\geq3$.

We study weighted periodic-orbit zeta functions for an infinitely cusped tree lattice $Y_q$, where $q\ge2$ is even and the quotient is a one-sided comb. The global Euler product fails coefficientwise because infinitely many primitive cycles have length four. A first-return determinant at a finite directed-edge set nevertheless exists, and stationary Schur elimination gives an algebraic formula for the root local zeta and its dominant poles. For the two-step multiplicity potential we compute the Gurevich pressure $P_G=2\log(q+1)$ and pressure at infinity $P_\infty=\log(4q)$, yielding strong positive recurrence and exponential local-orbit asymptotics. The height-damped transition operator is trace class. After subtraction of an explicit integrated-pressure counterterm, the inverse Fredholm determinant has a locally uniform finite part, expressed by a convergent dilogarithmic product and covariant under changes of height.

We study proper actions on homogeneous spaces of the group $\mathrm{B}(n)$ of invertible upper triangular matrices from a coarse-geometric viewpoint. We obtain sufficient conditions for the properness of the natural $L$-action on $\mathrm{B}(n)/H$, where $L,H$ are closed subgroups of $\mathrm{B}(n)$. We also give an explicit sufficient condition for properness of the $L$-action on a homogeneous space naturally identified with $\mathrm{B}(n)/H\cong \mathbb{R}^{n-1}$. In addition, we discuss extensions to the case where the ambient group $G$ is a subgroup of $\mathrm{B}(n)$ and $L,H$ are closed subgroups of $G$, with particular attention to the unipotent subgroup $\mathrm{N}(n)$.

We study a class of discrete-time interacting particle systems arising from mean-field coupled maps. We establish a Dobrushin-type estimate, a relative entropy estimate and Central Limit Theorem (CLT) type results for this class of systems. First we control the growth of the 1-Wasserstein distance between the empirical measures and the limiting distributions. Then we use the relative entropy method to show the propagation of chaos. Finally we consider the asymptotic behavior of the fluctuations for the empirical measures and prove that the sequence of fluctuation processes converges in distribution to some Gaussian process, where we establish both qualitative and quantitative results.

Epidemic models often pool air travel, commuting, and other mobility layers into a single weighted network. Pooling keeps the total imported infections into a region but discards the transport mode and route that delivered them, the information a mode-specific intervention needs. We make this precise for directed multilayer flows with heavy-tailed variation. Pooling acts as an asymmetric filter. The magnitude of an extreme importation, its radial tail index, is an exact invariant of sampling and nonnegative aggregation, whereas its composition across layers and routes, the angular part, is reweighted by layer-specific sampling and can be irrecoverably merged. We give a necessary and sufficient condition for recovering route composition, attach a surveillance decision cost to the loss, and show that the first established route need not be the busiest. A controlled metapopulation study calibrates the cost, and two contrasting reconstructions show where it bites. In US pandemic influenza, air adds fitted information beyond commuting, and a layer-resolved ranking targets more air-import burden than a pooled one. In the early Italian COVID-19 wave, commuting improves onset reconstruction while air attribution stays unresolved. Pooling can therefore support total-importation surveillance but cannot in general identify the mode or route behind an importation.

A frequency domain bootstrap procedure for functional time series is proposed and applied to the class of spectral mean operators. The procedure works by first generating independent pseudo periodogram operators across the positive Fourier frequencies using an estimator of the spectral density operator involved. Functional replicates of the spectral mean operators of interest are then generated. Through an additive, projection-based decomposition of the bootstrapped spectral mean operator, its leading $m$-dimensional part is properly complemented to also capture the relevant fourth order characteristics of the process. The complementation is achieved by means of a resampling procedure based on convolved periodogram operators of subsamples. The resulting bootstrap spectral mean operator consistently estimates the entire second order as well as the $m$-dimensional fourth order structure of the distribution of spectral mean operators. By allowing for the decomposition parameter $m$ to increase to infinity as the sample size increases to infinity, consistency in estimating the entire fourth order structure of the process also is achieved. The asymptotic theory developed investigates properties of the procedure for fixed and for increasing $m$ and establishes validity of the frequency domain bootstrap proposal under rather weak conditions on the underlying functional process class.

We propose a rigorous computational method that certifies the separator required by Temple's inequality and its Lehmann–Goerisch refinements, and with it unconditional ground-state bounds for $H=-\Delta+V$ on $R^3$ with attractive Coulomb singularities. For H$_2^+$ the method certifies $\lambda_2^{\rm eee}\ge-0.2035$ in place of the analytic $-\tfrac14$, and Temple's inequality with a global trial then gives an enclosure of width below $6.5\times10^{-51}$. For linear H$_3^{2+}$, whose central nucleus defeats the analytic separator, the certified widths are $1.3\times10^{-12}$ and $3.1\times10^{-13}$ at two nuclear separations. Such a separator, a lower bound on the second eigenvalue, is supplied by theory only for special configurations; elsewhere it is estimated and the bound is conditional. Here every hypothesis is verified in interval arithmetic at run time. Three constructions make this possible on an unbounded domain with a potential of unbounded $L^\infty$ norm: a two-sided Neumann/Dirichlet truncation with a computable remainder; a localized Hardy inequality and sharp form bound with explicit constants, restoring coercivity; and a projection stage that certifies lower bounds on the first two eigenvalues, the second serving as separator. On a box, a Lehmann–Goerisch step places $\lambda_1\in[-0.553291,-0.550740]$ for H$_2^+$. The coercivity constant, we show, must bound its own optimum from above. All drivers and certificates are released.

We study multiple orthogonal polynomials with orthogonality defined by multiple positive discrete measures on the real line. Focusing on type~I and type~II multiple orthogonal polynomials and their step-line recurrence relations, we consider the reconstruction of the associated banded upper Hessenberg recurrence matrix from given nodes and weights of the measures. We propose efficient updating and downdating procedures that modify an existing recurrence matrix when nodes are added to or removed from the discrete measures. The updating strategy is based on solving an inverse eigenvalue problem, while the downdating procedure relies on an eigenvalue deflation technique inspired by a QR-type of algorithm. Numerical experiments confirm the stability and efficiency of the proposed approaches.