Papers

We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert–Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.

We show the convergence of a commonly used mesh-free discretization of a dynamic non-linear bond-based peridynamic model with irreversible damage implemented via a bond-breakage criterion for a fixed horizon $\delta >0$. The model chosen here is a mathematically well-posed regularization of a discontinuous peridynamic operator often found in the literature, which assumes that fracture results from strain in tension only. We explicitly include the one-point quadrature used in this scheme in the analysis. The convergence is with respect to the natural norm of the solution space given by $C^1([0,T],L^\infty(\Omega)^d)$, and therefore $L^\infty(\Omega)^d$ in the spatial coordinates. This result heavily constrains the spatial regularity of the continuous solution which we prove afterwards. More specifically, for locally $L$-Lipschitz continuous input data, local Lipschitz bounds are derived for the displacement, regardless of damage. This is consistent with interpreting the peridynamic model used here as an approximation to a truly discontinuous fracture model with a length-scale parameter given by the horizon $\delta>0$. We note that we do not deal with the limit of the solutions as $\delta \searrow 0$.

In this chapter, we discuss the interaction between low-dimensional dynamics, symbolic dynamics, and geometric topology, focusing on the algebraic characterization and asymptotic behaviour of stretch factors or equivalently pseudo-Anosov dilatations. We first review the algebraic foundations linking Perron–Frobenius theory, multi-variable Euclidean algorithms, and Lind's characterization of topological entropy for shifts of finite type, emphasizing recent constraints on the Perron–Frobenius degree of expansion constants. We then relate these dynamical invariants to the combinatorial world, examining train tracks for surfaces and automorphisms of free groups. Finally, we briefly review the relevance of fibered 3-manifolds, detailing how the Thurston norm, Fried's concavity theorem over fibered cones, and McMullen's Teichmüller polynomial come into play.

We consider second-order strictly hyperbolic operators whose coefficients are Lipschitz continuous in space and may oscillate near the initial time. Their first two time derivatives are bounded by the corresponding powers of $\frac{1}{t}(\log\frac{1}{t})^\alpha$ with $0\leqslant\alpha<1$, and the first time derivative satisfies the corresponding spatial Lipschitz bound. By combining Tarama's energy with frequency-dependent weights, we prove an energy estimate without loss for $\alpha=0$ and with a stretched logarithmic upper bound for the loss when $0<\alpha<1$, which implies an arbitrarily small loss in the Sobolev scale. These estimates extend the analysis of Colombini, Del Santo and Fanelli [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 26 (2025)] for spatially Lipschitz coefficients to time-derivative bounds with an additional logarithmic factor.

This work extends the study of mobile resolvability in connected graphs by introducing completely mobile resolving sets. As with mobile resolving sets, we consider stationing a collection of robots at the vertices in a resolving set of a graph. Robots may move one at a time to any adjacent vertex as long as set of occupied vertices remains a resolving set. In this new variant, we ask that every robot can traverse every vertex in the graph through a sequence of permitted moves. Graphs possessing at least one completely mobile resolving set are termed mobile-resolvable. We completely characterise the non-mobile-resolvable graphs as paths of order at least three and subdivided stars with three or more leaves and at most two subdivided edges. For trees, we explicitly determine the completely mobile metric dimension, showing that it exceeds the classical metric dimension by one for most mobile-resolvable trees, and by two for certain subdivided stars. Furthermore, we evaluate this parameter across fundamental graph classes-including cycles, complete bipartite graphs, grid graphs, and circulant graphs.

We study the connection problem for the quantum differential equation of the Fano fourfold $X=\PP_{Q^3}(\OO\oplus\OO(1))$. The problem is to identify its canonical asymptotic solutions with exceptional objects and compute the associated Stokes data. We use exact Mellin–Bessel integral representations and uniform vector asymptotics on sectors wider than $\pi$, followed by analytic continuation that preserves the Gamma framing. Cohomological chiral symmetry and mutations of exceptional collections put all eight sections at a common phase. This proves Gamma conjecture II for $X$ and determines a full exceptional collection and its Stokes and central connection matrices at the unit Novikov point and a specified admissible phase. The chiral symmetry also gives a flat symmetric bilinear pairing, an orthogonality identity for the Stokes matrix, and an integral unipotent square root of the monodromy matrix for a full turn. We obtain explicit formulas for the connection data at the opposite phase; on the Gamma lattice, the half-turn acts by tensoring with a square root of the anticanonical bundle. The same spinor integral transform constructs two canonical Gamma sections for $\PP_{Q^{2m+1}}(\OO\oplus\OO(1))$, $m\geq1$, at positive parameters with distinct spectrum. Their sectors have opening greater than $\pi$, and their leading constants are independent of the dimension.

The paper is devoted to a broad class of optimal control problems governed by constrained sweeping processes with controlled dynamics, controlled prox-regular moving sets, and variable time included in the optimization process. Employing an advanced version of the method of discrete approximations, we verify its well-posedness with appropriate strong convergence of feasible and optimal solutions. This approach leads us to considering a parametric family of discrete-time controlled sweeping processes, which are reduced in turn to nondynamic mathematical programs with special types of geometric and functional constraints. Invoking powerful tools of first-order and second-order variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete-time and then—by passing to the limit over the diminishing step of discretization—continuous-time controlled sweeping processes. A crucial role in our device is played by explicit coderivative evaluations of set-valued mapping associated with the sweeping dynamics. In this way, we establish novel necessary optimality conditions for local minimizers of continuous-time controlled sweeping processes that contain absolutely continuous and BV adjoint arcs, local and global maximization conditions, $L^2$ active set multipliers, vector measures generated by sweeping and control constraints, and free-time transversality condition expressed through the averaged Hamiltonian. For suitably selected multipliers, we also obtain nondegenerate forms of these conditions when the optimal trajectory starts on the boundary of the moving set. The efficiency of the obtained results is illustrated by some examples related to practical modeling.

For $0\leq\alpha<n$, let $A_{*}^{\alpha}$ denote the fractional spherical maximal operator. In this paper we characterise the range of $\beta$ for which the $L^p-L^q$ power-weight estimate \[\left\|A_{*}^{\alpha} f\right\|_{L^q(|x|^{q\beta})} \leq C \left\|f\right\|_{L^p(|x|^{p\beta})}\] holds. In the diagonal case $p=q$, our work gives an affirmative answer to a conjecture of Lee. We prove analogous results for the lacunary fractional spherical maximal operator.

We discuss General Relativity and some of its modifications as effective field theories of the breaking of the affine general linear group using the coset approach and the Maurer-Cartan form. While the ultimate goal is to break the symmetry down to the Lorentz subgroup, we argue that there are three interesting breaking cascades in which the affine general linear group is first broken to a larger subgroup that contains the Lorentz group. In all cases, the breaking of translations always generates Goldstone modes that can be interpreted as metric degrees of freedom. We argue that the effective field theories that emerge as a result of the different breaking patterns are all dynamically equivalent at the lowest order of the derivative expansion, e.g., using Einstein-Hilbert-like actions, but can differ beyond the leading order.

This paper considers channel state information (CSI) prediction using neural networks for time-varying multi-antenna channels from noisy historical observations. The main challenge is to exploit temporal correlation while distinguishing channel evolution from observation noise. We propose JEPA-kMMVAE, which combines a joint-embedding predictive architecture with a k-memory Markov variational autoencoder to predict future CSI through temporally structured latent representations. The framework supports direct prediction and residual corrections relative to the latest latent state and noisy CSI observation. Simulation results show that the residual configuration outperforms the evaluated baselines at moderate and high input CSI signal-to-noise ratios, while sequence baselines remain more accurate under stronger noise.

Quantitative Gaussian approximation estimates on path space are established for the nonequilibrium density fluctuations of the simple symmetric exclusion process (SSEP) on a $d$-dimensional discrete torus of mesh size $1/n$, started from a product measure with a smooth density profile. A novel $n$-dependent modified Ornstein-Uhlenbeck process is constructed that incorporates higher-order lattice information while remaining linear and Gaussian. It is shown that, for sufficiently high modification orders, this modification yields quantitative Gaussian approximation error bounds of order $n^{-\min\{d/2,2\}}$ for a class of smooth functionals of the entire fluctuation trajectory. In addition, it is shown that retaining an explicit microscopic correlation correction leaves a remainder of order $n^{-d/2}$, thereby distinguishing contributions to the bound due to correlations from those due to spatial approximation. For nonconstant initial profiles in dimensions $d\ge3$, a separate argument yields matching upper and lower bounds of order $n^{-2}$ for the direct weak error of suitable quadratic test functionals, provided that the modification orders are sufficiently high. The proof is based on a novel path-dependent Trotter formula. Its derivation builds on the introduction of a concept of horizontal absolute continuity for non-anticipative functionals and a corresponding martingale formula for Markov jump processes. This formula is subsequently combined with an integral backward Kolmogorov equation for Gaussian continuation functionals. A second ingredient of the proof is a refinement of the exchangeable-pair argument introduced by the authors in a previous work, thereby improving the central-limit rate $n^{-\min\{d/2,1\}}$ to $n^{-d/2}$ for the initial empirical fluctuations.

The escalating environmental footprint of the Information and Communication Technol- ogy (ICT) sectorwith global Data Center electric- ity usage potentially reaching 3% by 2030 and the imperative for real-time responsiveness in smart buildings demand a foundational shift towards Edge Computing. This article proposes a holis- tic architectural and algorithmic approach for cre- ating autonomous, resilient, and privacy-preserving learning systems within smart buildings, advancing the core concept of the building itself as the data center. Our contributions, demonstrating originality compared to conventional centralized systems, are structured around three main pillars: 1) Validation of resilient data acquisition chains using the af- fordable Arduino ecosystem (Uno R4 WiFi/BME280) and secure persistence solutions including centralized archiving (Node-RED/MySQL) and secure remote transfer (Python/rsync/SSH). 2). The proposal of a lightweight, KOptim orchestration framework, inter- operable with the FIWARE standard, for the dynamic management of Service Level Agreements (SLA). 3) The adaptation and experimental valida- tion of the Generalized Hebbian Algorithm (GHA) for Principal Component Analysis (PCA) on low- power microcontrollers,We introduce and compare an iterative learning strategy using data buffers, featuring sorting and explicit preservation of extreme samples to maintain diversity. Experimental results on the high-dimensional Tour Perret sensor data confirm that this novel method, despite introducing a necessary temporal complexity of O(W log W ) due to sorting, achieves convergence and final result quality that is visually equivalent to processing the entire dataset in full memory, reliably identifying two distinct operational states.

Motivated by semi-Markov models of anomalous diffusion, we study obliquely reflected diffusions in generalized parabolic domains under independent inverse-subordinator time changes. We focus on infinite-mean subordinators, whose inverse clocks can slow the superdiffusive growth induced by reflection. For domains with cross-sectional radius asymptotic to $a_\infty x^\beta$, where $-1<\beta<1$, we strengthen the almost-sure growth laws of [MMW23] to obtain precise second-moment asymptotics when the subordinator's Laplace exponent is regularly varying at zero, and matching bounds more generally. Inverse $\alpha$-stable clocks yield mean-square growth of order $t^{2\alpha/(1+\beta)}$, giving subdiffusive, diffusive, or superdiffusive regimes. For $\beta<-1$, explosion persists almost surely, with finite mean explosion time if and only if the subordinator has finite mean. Supporting results establish strong existence, pathwise uniqueness, and a time-change representation, also allowing dependent clocks.

We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected' strata of the PL-space, where normal Morse Data determine the change of topology. On the homology level there exist indices with multiplicities, combined in a Morse vector. We focus mainly on PL-manifolds. We give bounds, depending on the dimension of the strata. In low dimensions we describe singularity types. At the end we discuss relations with Discrete Morse Theory.

We consider Hecke modules of orthogonal modular forms attached to definite ternary lattices of half-discriminant $p^2$ with $p$ prime and $4p^2$ with $p$ an odd prime. Where possible, we prove explicit Jacquet-Langlands style correspondences with spaces of classical modular forms (in the spirit of Birch and Hein-Tornaria-Voight), and give conjectures for the remaining genera. This is done by applying sequences of Watson-style transformations that reduce the discriminant while preserving the adelic orthogonal group.

Let $D_{m,n}\subset\C^{m\times n}$ be the union of the zero loci of the maximal minors of a generic $m\times n$ matrix, and let $U_{m,n}$ be its complement. We show that $U_{m,n}\cong\GL_m(\C)\times\Gr(m,n)^\circ$, where $\Gr(m,n)^\circ$ is the open stratum of the Grassmannian on which all Plücker coordinates are nonzero. For $m\in\{2,n-2\}$ this gives closed formulas for the topology of $U_{m,n}$ and recovers results of Yim. Our main result, which is computer-assisted, is that $D_{3,5}$ is not a free divisor. This answers a question of Yim. We determine $\Der(-\log D_{3,5})$ explicitly. It has projective dimension one, and it fails to be free exactly on the matrices of rank at most two. There the failure comes from a non-free transversal arrangement: the ten planes spanned by pairs of five general vectors in $\C^3$. Computations in further cases support the conjecture that $D_{m,n}$ is free only for $m\le 2$ or $n=m+1$.

Let p be an odd prime. We study the Artin-Schreier analogue of Poonen's problem of constructing curves without extra automorphisms. An Artin-Schreier curve carries an unavoidable deck group Cp, so the smallest possible full geometric automorphism group inside the Artin-Schreier locus is Cp. We determine the exact numerical range in which this minimum is attained, for every p-rank. More precisely, for integers r >= 0 and N >= 1, there exists an Artin-Schreier curve X/Fp with g(X) = (p - 1)N/2, gamma(X) = r(p - 1), Aut_Fp(X) is isomorphic to Cp if and only if either r = 0, p does not divide N + 1, N >= 3 and (p,N) not equal to (3, 3), or r >= 1, N >= 2r and (r,N) not equal to (1, 2). All models are defined over the prime field by explicit equations together with elementary choices of auxiliary polynomials, and every excluded admissible case is shown to carry an extra geometric automorphism. The main structural input is the theorem of Valentini and Madan on the uniqueness of the rational subfield of index p outside three exceptional families. We exclude these families by intrinsic arguments: the genus and p-rank in most cases, Nakajima's bound on p-subgroups of automorphism groups of ordinary curves for the ordinary family, and the Lehr-Matignon normal form for isomorphisms of one-point covers for the monomial family.

We investigate max-phase isometries on Schatten quasi-classes in the range $0<p<1$. Motivated by the rigidity theory for smooth normed spaces [Results Math. 80 (2025), Article 126], we establish its operator-theoretic counterpart in the quasi-Banach setting. We prove that every surjective max-phase isometry on a finite-dimensional Schatten class is a standard real-linear Schatten isometry multiplied by an even sign-valued function, without any continuity assumption. This extends the scope of the classical Schatten isometry classification [Proc. Edinburgh Math. Soc. 37 (1994), 239–242] to nonlinear mappings preserving only unsigned two-point metric data. Our approach combines support orthogonality, a symmetrized rank-one spectral formula, and a max-profile uniqueness principle. For arbitrary Hilbert spaces, we obtain the corresponding classification on finite-rank operators, while continuity yields the full-space result. We also establish preservation of rank, singular values, and two-sided support orthogonality.

Brambles of constant congestion in directed graphs have interesting structural and algorithmic applications, and it is desirable to find the smallest function $f_c(k)$ such that if a digraph has directed treewidth at least $f_c(k)$, then it contains a bramble of size at least $k$ and congestion at most $c$. By the Directed Grid Theorem of Kawarabayashi and Kreutzer [STOC 2015], such a function $f_c(k)$ exists for every value $c \geq 2$ of the congestion. However, even using the recent improvements due to Hatzel et al. [FOCS 2024], the resulting function is a power tower of height $22$. Recently, Masařík et al. [SIAM J. Discrete Math. 2022] proved that, for congestion $8$, the function $f_8(k)$ can be made polynomial; more precisely, they showed that $f_8(k) = \mathcal{O}(k^{48} \cdot \log^{13} k)$. They asked whether it is possible to decrease the congestion, ideally to $2$, even at the cost of a higher polynomial function. In this work we answer their question affirmatively with the best possible value of the congestion, by proving that $f_2(k) = \mathcal{O}(k^{96}\cdot \log^{23} k)$. We also obtain smaller polynomials for higher congestion, namely $f_4(k) = \mathcal{O}(k^{72} \cdot \log^{21} k)$ and $f_6(k) = \mathcal{O}(k^{66} \cdot \log^{18} k)$. Our results are inspired by the proof for congestion $8$, which we improve by introducing walkages. Walkages allow us to avoid technicalities of the original proof by translating ideas due to Reed and Wood [Eur. J. Comb. 2012] that work for undirected graphs to the directed setting. Our proofs are constructive and naturally yield polynomial-time algorithms to find the brambles assuming a certificate for large directed treewidth is given. Moreover, we discuss algorithmic applications of our results for the Directed Disjoint Paths with Congestion problem, and to half-integral packings of subgraphs with large directed treewidth.

We consider the Navier-Stokes equations in a smooth bounded domain $D\subset \mathbb R^3$. It has been proved in arXiv:2609.08284 that these equations have a unique solution - globally defined in time - when a superlinear noise is introduced and the initial velocity belongs to the space $V$ of divergence-free $H^1_0(D;\mathbb R^3)$-vector fields. Here we show that the same result holds when there is also an additive noise and a deterministic forcing term. Moreover, when the deterministic forcing term is time-independent we prove the existence of at least one invariant measure for this stochastic equation. This is obtained by the Krylov-Bogoliubov method, based on tightness and Feller property in $V$. Finally, we show that the set of invariant measures contains one ergodic invariant measure.

For an oriented tame knot $K$, let $G_K$ be its group, with positive meridian $\mu_K$ and preferred longitude $\lambda_K$. We prove that the profinite completion $\widehat{G_K}$, marked by the conjugacy class of $\mu_K\lambda_K$, determines $K$ up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.

We derive second order perturbation expansions of volume-normalized sums of the first $n$ Dirichlet eigenvalues for near-circular and near-spherical domains. The disk (resp. ball) is only a critical shape when $n$ is the last index of an eigenvalue cluster, and in dimension two and three our expansions yield a finitely checkable criterion deciding whether the disk or ball is a weak local minimizer or a saddle point. In two dimensions, this allows us to show that there is an infinite number of cases in which the disk is a saddle point for finite eigenvalue sums.

We establish a structure theorem for smooth random walks on the circle and study the dimensional properties of their stationary measures, including exact dimensionality and a Ledrappier-Young-type dimension formula. These results provide the technical tools needed for the main theorems in Part II "Dimension theory of group actions by circle diffeomorphisms II: Minimal sets". They also yield an approximation theorem in the spirit of Katok, Avila-Crovisier-Wilkinson, and Morris-Shmerkin, together with a time-reversal symmetry for circle actions: every semigroup $T$ contained in $\mathrm{Diff}_+^{1+\alpha}(\mathbb{S}^1)$ without finite orbits has the same number of minimal sets as its inverse semigroup $T^{-1}$.

We consider the Cauchy problem on the real line for a one-dimensional transverse-field isentropic non-resistive magnetohydrodynamic system with viscosity $\mu(\rho)=\alpha\rho^\delta$, $0<\delta\leq1$, and vacuum at spatial infinity. The viscosity therefore vanishes precisely where the density vanishes. We establish two global large-data theorems, one for the sublinear range $0<\delta<1$ and one for the linear case $\delta=1$, in a nontrivial invariant class of magnetic fields. The class is described by the frozen-in flux $h=b/\rho$ and its mass derivative $k=\rho^{-1}h_x$. The key identity $[\partial_t+u\partial_x,\rho^{-1}\partial_x]=0$ transports both $h$ and $k$, removes all negative powers of the density from the Lorentz force, and propagates the magnetic regularity. A modified effective-velocity entropy contains an additional coercive magnetic term, while its only error is controlled by $k$. Magnetic pressure also strengthens the density upper bound. These estimates close the large-data continuation argument in the ranges $0<\delta<1$, $\gamma\geq\delta+\tfrac12$, and $\delta=1$, $\gamma>\tfrac32$. As a result, we extend the work of Cao, Li, and Zhu (SIAM J. Math. Anal., 2022) from the Navier-Stokes equations to the MHD equations.

We present high order accurate mixed explicit-implicit discontinuous Galerkin (DG) schemes for the solution of hyperbolic systems of conservation laws on Cartesian cut cell meshes. Such meshes may contain arbitrarily small cut cells, which leads to the well-known small cell problem: the CFL condition of explicit schemes would severely restrict the admissible time step. To overcome this difficulty, we use high order explicit fully-discrete one-step ADER-DG schemes only on Cartesian cells away from the embedded boundaries. The zone close to the embedded boundary, which consists of cut cells and their direct face neighbors, is discretized with an implicit space-time DG scheme that is coupled to the explicit scheme directly via the numerical flux. The resulting scheme, called ADER-ZIST (ADER-DG with Zonally Implicit Space-Time DG), is conservative by construction, and its time step is determined only by the size of the Cartesian background cells. Curved embedded boundaries are represented by high order isoparametric sub-triangles inside the cut cells, and the nonlinear system of the implicit cells is solved via an inexact matrix-free Newton-Krylov method. For problems with shock waves, we add an artificial viscosity term whose strength is controlled by an oscillation-eliminating smoothness sensor. ADER-ZIST is assessed on a comprehensive set of test problems for the compressible Euler equations, ranging from smooth flows around straight and curved embedded boundaries to acoustic scattering and shock-dominated flows. The scheme essentially reaches its nominal order of accuracy N+1 for smooth solutions and for polynomial degrees up to N=5, on meshes with minimal volume fractions of the small cut cells down to the order of 10^-7.

For diffusions whose coefficients are smooth, with at most linear growth and bounded derivatives of higher order, we establish explicit pointwise bounds on the spatial derivatives of the classical solution to the Cauchy problem with polynomially growing initial data. We also obtain explicit growth estimates for the derivatives of the SDE with respect to the initial data. The proofs combine Lyapunov functions and Markov semigroups on weighted spaces with Kunita's theory of the differentiable dependence of SDE solutions on the initial data.

In this paper, we study the relation between different notions of monotonicity for isotropic hyperelastic compressible and incompressible materials. We study how to define the monotonicity concept on the manifold $\rm{Sym}^{++}(n)\cap \rm{SL}(n)$ of positive definite symmetric tensors with unit determinant and the corresponding spectral manifold $\mathcal S=\left\{(\lambda_1,\dotsc,\lambda_n)\in \mathbb{R}^n_+\,\middle|\, \lambda_1\lambda_2\cdots\lambda_n=1\right\}$, and give a precise characterisation of monotonicity in the case of hyperelastic materials. In particular, we consider the monotonicity properties of the Biot stress tensor $T_{\rm Biot}(U)$, the engineering stress, or the load tensor.

Polynomial first-order projective differential moment invariants are investigated. The main result establishes an exact correspondence between such moment invariants and symmetric polynomial multipoint relative projective differential invariants of weight $-1$. This correspondence describes each homogeneous component with respect to moment degree and allows previously obtained results on relative projective invariants to be used for systematic construction. It is also proved that passing from the formal algebra of moments to their integral definitions introduces no additional algebraic relations among the constructed invariants. For low moment degrees, the absence of nontrivial invariants of degrees $1$ and $2$ is established, and a complete description of degree $3$ invariants is obtained. The ten cubic differential moments are naturally identified with the coefficients of a ternary cubic, connecting the construction with classical invariant theory. In particular, the Aronhold invariants $S$ and $T$ yield projective differential moment invariants of degrees $4$ and $6$. One of the known moment invariants of Wang et al. is shown to be proportional to $S$. The algebra of polynomial moment invariants is proved not to be finitely generated. The field of rational moment invariants coincides with the field of fractions of the polynomial invariant algebra. For the explicit constructions, compact multipoint representations define the corresponding moment polynomials without requiring their full expansions. These representations provide a basis for algorithmic construction and software computation of the invariants.

We present an explicit quantum circuit that constructs a block encoding of a generator-representable semiseparable (one-pair) matrix directly from its two generators. For an $N\times N$ semiseparable matrix with $N=2^{n}$, the circuit uses $2n+2$ ancilla qubits and has $O({\rm polylog} N)$ depth, and it realizes a $(2\sqrt{N},\,2n+2,\,2 \sqrt{2}\ \varepsilon)$-block encoding, where $\varepsilon$ is controlled by the accuracy of the state-preparation routines that load the generators. The building blocks of the circuit are cyclic shifts, swaps, and Hadamard gates, each admitting an efficient implementation through the binary decomposition of the relevant indices.

In this paper, we prove global pointwise estimates on the decreasing rearrangement \(u^{\ast}\) and \([H(\nabla u)]^{\ast}\) of SOLAs \(u\) of an elliptic Poisson equation driven by the anisotropic \(p\)-Laplace operator \(\Delta_p^H\) for given \(L^1\)-data and equipped with either homogeneous Dirichlet or Neumann boundary conditions. Here, we assume that the dimension \(n\ge 2\), \(1<p<n\) and \(H\) is norm on \(\R^n\). In the degenerate range \(2 \leq p <n\), this gives an anisotropic extension and a full-interval completion of the rearrangement theory of Cianchi-Maz'ya [JEMS, 2014]. In particular, we obtain a new rearrangement estimate for gradients by decomposing the data for singular range \(1<p<2\), where the nonlinear interpolation method aren't available. Moreover, the endpoint weak-type gradient estimate is obtained for all \(1<p<n\). In the range \(p_*(n)<p<n\) with \(p_*(n):=\frac{3n-2}{2n-1}\), we prove a stability estimate for differences of gradients. These results extends the known ones in the literature to the singular range \(p_*(n)<p<2\).