Papers

We give a partial generalization of the Kurdyka theory of arc-analytic functions and arc-symmetric sets on Nash manifolds to the setting of arbitrary locally closed semialgebraic sets.

In this paper, we further investigate the role of the graded Grothendieck group $K_0^{\gr}$ and its positive cone (the talented monoid) as an effective tool for distinguishing structural types of algebras associated to higher-rank $k$-graphs. We study the prime spectrum (the space of all prime $\Gamma$-order ideals equipped with a Zariski-like topology) of a general commutative $\Gamma$-monoid and establish that this space is spectral in the sense of Hochster, provided the monoid has the refinement property and every $\Gamma$-order ideal is finitely generated. As a result we are able to show that the prime spectrum of the talented monoid of a row-finite $k$-graph without sources and with a finite set of vertices is spectral. For any row-finite $k$-graph $\Lambda$ without sources, one of our main results says that the space of all graded prime ideals of the Kumjian–Pask algebra $\KP(\Lambda)$ is homeomorphic to both the space of all prime $\mathbb{Z}^k$-order ideals and the space of all prime $\mathbb{Z}^k$-filters of the talented monoid $T_\Lambda$. Another main result of this paper provides a complete topological description of regular $\Gamma$-order ideals of a refinement $\Gamma$-monoid: a $\Gamma$-order ideal $J$ is regular if and only if the corresponding closed (resp., open) set $V(J)$ (resp., $D(J)$) is regular closed (resp., regular open) in the prime spectrum. As an application of these results, we establish a lattice isomorphism between the lattice of all regular $\mathbb{Z}^k$-order ideals of the talented monoid and the lattice of all regular graded ideals of the Kumjian–Pask algebra via a spectrum-theoretic approach.

For $n\geq3$, the group $\mathrm{SL}_n(\mathbb Z)$ has property $(T)$, so every finite generating set has a positive Kazhdan constant. However, it was recently shown that the infimum of these constants over all finite generating sets is $0$. This naturally raises the question of whether the infimum remains zero when the cardinality of the generating sets is bounded in advance. We prove that the infimum of the Kazhdan constants of $\mathrm{SL}_n(\mathbb Z)$ over all two-element generating sets is $0$ for every $n\geq3$.

We consider the two-species Lotka-Volterra competition-diffusion system where one species has a small diffusion rate relative to the other species and has a small competition coefficient. By the geometric singular perturbation theory, we prove the existence of the wavefront connecting the coexistence state to the trivial state, with traveling speed greater than the minimal speed. For such wavefronts, we show that the profiles of both species are monotone. Finally, we use the geometric singular perturbation theory to estimate the associated Evans function, and give the related stability results.

Parameter estimation in finite mixture models can exhibit highly heterogeneous convergence behavior: locally isolated components may be estimated substantially faster than groups of competing components. Existing analyses based on Wasserstein distances typically characterize only the worst-case rate and therefore do not fully capture this local heterogeneity. In this paper, we introduce a Voronoi-based partial optimal transport (VPOT) framework for obtaining refined local and global convergence guarantees for the maximum likelihood estimator of the mixing measure. The key geometric idea is to localize the comparison of two mixing measures to extended Voronoi neighborhoods and use partial optimal transport to accommodate the unequal masses of their local restrictions. Within each neighborhood, the first-order POT discrepancy is raised to a power determined by the number of locally competing atoms, allowing the resulting loss to adapt to the local degree of singularity. Under suitable regularity and strong identifiability conditions, we establish uniform local and global upper bounds for a maximum likelihood estimator under the VPOT loss. These bounds reveal a configuration-dependent form of parameter estimation: less singular local configurations admit faster convergence, whereas the most singular configuration recovers the classical worst-case behavior characterized by Wasserstein-based analyses. We further establish a minimax lower bound showing that the convergence rate for estimating the mixing measure under the VPOT loss is optimal. Our results hold in arbitrary fixed dimension without requiring mixing proportions to be uniformly bounded away from zero or prior knowledge of the true number of mixture components. Overall, VPOT provides a configuration-adaptive framework for capturing heterogeneous parameter-estimation behavior in finite mixture models.

Motivated by the problem of asymptotic stability for ground states of the Klein–Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic–quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic–quintic Klein–Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let $Q_\omega$ be the positive radial ground state of the three-dimensional cubic–quintic elliptic equation \[ -\Delta Q_\omega+\omega Q_\omega-Q_\omega^3+Q_\omega^5=0. \] We study two related, but logically distinct, spectral problems as $\omega$ approaches the endpoint $3/16$ of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian \[ L_+=-\Delta+\omega-3Q_\omega^2+5Q_\omega^4. \] This operator is also the exact linearized spatial operator around the static state $Q_\omega$ for the real scalar cubic–quintic Klein–Gordon equation with mass parameter $\omega$. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic–quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to $(3/16-\omega)^{-1}$. %The rotating complex Klein–Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic–quintic models.

This paper reports the proof of a strong converse for the unconstrained quantum capacity of the pure-loss bosonic channel. At every fixed rate above capacity, the entanglement-generation fidelity of every code is bounded by a constant times the reciprocal of the number of channel uses. The bound holds without an energy constraint and for arbitrary encoded states, including states correlated across all input modes, and arbitrary joint decoders. The proof combines quantum Chebyshev and hockey-stick testing inequalities with a uniform relative-entropy-variance bound for the balanced pure-loss channel, corresponding to transmissivity $\eta=1/2$. The variance bound follows by expressing the balanced beam splitter in bright and dark modes: the dark modes are exactly in vacuum, and any state orthogonal to that vacuum contains at least one dark photon. For general transmissivity, dilating the degrading attenuator reduces the problem to this balanced-channel setting and bounds the decoder test by precisely the factor that produces the known quantum-capacity threshold. The resulting argument establishes the strong converse at the unconstrained quantum capacity for every pure-loss bosonic channel.

Approximating the input-output behavior of a multivariable black-box function from limited data is challenging when blind to the importance of its inputs and their interactions. We introduce total sensitivity kernels (TSKs), a method based on families of weighted ANOVA kernels that learn and adapt to this multivariable structure. TSKs parameterize the weights on each multivariable component of the target function by factors for each input. We propose learning these factors directly from function evaluations by selecting the reproducing kernel Hilbert space (RKHS) in which the target function has minimum norm. Under suitable conditions, we show that this norm-minimization problem admits a unique solution, and we establish consistency of a finite-data formulation based on minimum-norm interpolation. The learned TSK factors characterize the participation of individual inputs across interactions and main effects, providing a kernel-dependent notion of input sensitivity related to total Sobol indices. Numerical experiments demonstrate that adapting the kernel to learned multivariable structure can substantially improve approximation accuracy over a standard product kernel.

We expand upon a previous study conducted by the author on the behavior of punctured log Gromov-Witten theory under log étale modifications $\widetilde{X} \rightarrow X$, giving expressions for log Gromov-Witten classes on $\widetilde{X}$ in terms of log Gromov-Witten classes on $X$, facilitating a complete reduction of the punctured log Gromov-Witten theory of any modification $\widetilde{X}$ of an snc log scheme $X$ to the punctured log Gromov-Witten theory of $X$. We apply this result in two settings. First, we prove a log-orbifold correspondence equating the logarithmic invariants of the canonical wall structure of Gross and Siebert with a class of orbifold invariants considered in the relative quantum cohomology ring of Tseng and You, and more generally construct a family of algebra homomorphisms from the intrinsic mirror algebra $R_{(X,D)}$ of Gross and Siebert to appropriate power series rings. Second, we show the punctured log Gromov-Witten classes of split toric bundles are effectively reconstructed in terms of the punctured log Gromov-Witten classes of the base. Additional input for the second application is the introduction and study of double ramification cycles with target log variety, generalizing the double ramification cycles with target variety investigated by Janda-Pandharipande-Pixton-Zvonkine.

Factor complexity counts the distinct factors of each length in an infinite word, while palindromic complexity counts those invariant under reversal. For recurrent aperiodic words with reversal-closed language, the total number of palindromic factors in two consecutive lengths is at most the one-step growth of factor complexity plus two. We prove that equality is forced whenever the factor complexity at a given length does not exceed three halves of that length plus one. We construct examples showing that the bound is optimal. We also obtain an adaptive local criterion and consequences for reversal-closed quasi-Sturmian words and palindromic defect.

We obtain a Furstenberg-Sárközy-type result for sets $A\subset [N]$ whose difference set $A-A$ does not contain the sum of $s$-many $k$-th powers of positive integers, with $k>1$ odd and $s>0$ even. Namely, we prove that such sets must satisfy a power-saving bound $|A| \, \ll \, N^{1-\frac1k\min\{s \, \sigma_k, \, 1/2\}+\epsilon}$ for any fixed $\epsilon>0$, where $\sigma_k >0 $ is any admissible saving in a classical one-variable Weyl estimate. In particular, we can take $\sigma_k=\max\left\{{2^{1-k}}, \, \frac{1}{k(k-1)}\right\}$ using the classical theory and the best currently available bounds for classical Weyl sums. A greedy construction produces a set $A\subset[N]$ with $|A|\gg N^{1-s/k}$ for which $A-A$ contains no sum of $s$-many positive $k$-th powers, so our power-saving bound is of the correct shape.

We study polynomial maps in three variables with nilpotent Jacobian over a field of characteristic zero. The proposed classification reduces maps with linearly independent components to a family determined by a univariate polynomial evaluated at a quadratic coordinate. The geometric part of the argument produces two algebraically dependent constant linear combinations of the components. A derivation argument then yields the normal form over the original field. We obtain explicit polynomial inverses and tame factorizations. We then study higher-dimensional maps in which all but the last three components depend on three variables. For this class, we give separate normal forms for the two three-variable blocks and deduce stable tameness from the residue-field criterion of Berson, van den Essen, and Wright.

A folding rational map is a postcritically finite rational map $f$ admitting an essential Jordan curve $\beta$, disjoint from the postcritical set $P_f$, whose preimage is disconnected and consists of Jordan curves homotopic to $\beta$ relative to $P_f$. Such $\beta$ is called a characteristic curve of $f$. We construct a real postcritically finite rational map $f$ of degree $6$ with $\#P_f=4$, which admits two non-homotopic characteristic curves $\beta$ and $\beta'$. It is hyperbolic with orbifold signature $(2,2,3,\infty)$. This non-uniqueness is not a Lattès phenomenon, nor does it arise from a mating decomposition. The map is constructed piecewise, and a numerical model is provided to illustrate its dynamics. Thurston obstructions are excluded by arc-lifting and intersection-number arguments.

We study the completion time of a task subject to independent reset (catastrophe) at each step. The completion-time PGF depends on the base-process PGF through an affine relation, and we exploit this structure systematically. Our main result shows that, among age-based catastrophe mechanisms, geometric-tail catastrophe is exactly the class that yields uniform affine PGF structure; in continuous time, the characterization sharpens to Poisson resetting. We establish a sharp two-sided Kolmogorov bound of order $p+|\alpha|$ for the exponential approximation $d_K(T/E[T], \mathrm{Exp}(1))$, thereby closing a logarithmic gap. Applications to the coupon collector with reset coupons reveal a discontinuous Gumbel-to-Exponential transition under resetting, while a multi-phase model exhibits a Gaussian-to-exponential transition with exponential convergence rate.

The computation of transport coefficients in interacting quantum many-body systems is rarely analytically accessible. Here, we develop a new Krylov-space mechanism that makes the leading density dependent correction to Green-Kubo diffusivity exactly calculable in the strong-dissipation regime of a one-dimensional noisy spin-$\frac{1}{2}$ XXZ chain. This is done by showing that the density-polarization-dressed bond coherence generates a Krylov subspace where repeated action of the dissipator closes exactly on an explicitly identifiable operator family. Within this subspace, the dissipative dynamics is then mapped onto a self-similar semi-infinite chain with a boundary defect. Surprisingly, all Lanczos coefficients of the Krylov chain and its boundary Green's function can be determined exactly. The latter then determines the exact leading density-dependent correction to the diffusivity. Finally, we calculate analytically the full boundary-to-bulk Green's function and find that it decays exponentially along the emergent Krylov chain.

We study the augmented Zarankiewicz problem, in which disjoint pairs of cells are added to a binary matrix with no all-one $2\times2$ submatrix. The pairs must satisfy compatibility conditions, and the objective counts each original occupied cell and each added pair once. We show that starting with a maximum $C_4$-free matrix can lower the final optimum, answering a question of Qi, Cui, and Xu. Let ${z_A}(m,n)$ be the optimum over all $C_4$-free initial matrices, and ${z_L}(m,n)$ the optimum when the initial matrix must have the maximum number of occupied cells. As $n\to\infty$ with $n\le m=o(n^2)$, we prove \[ {z_A}(m,n)-{z_L}(m,n)\ge\left(\frac1{30}-o(1)\right)mn \] and determine the sharp second-order term: \[ {z_A}(m,n)=\frac{mn}{3}+\left(\frac1{\sqrt6}+o(1)\right)n\sqrt m. \] An explicit construction gives a separation at $m=n=1893$. We also find a sharp density threshold: when $n\to\infty$ and $m/n^2\to c>0$, the limited density ${z_L}(m,n)/(mn)$ tends to $1/3$ if and only if $c\ge1/12$. The proofs combine density and stability estimates, combinatorial constructions, and an exact polynomial certificate.

We introduce new techniques for determining combinatorial properties of sets of multiple recurrence by considering weighted averages with quickly growing weights. Our main result is a far-reaching generalization of Szemerédi's Theorem which additionally confirms a conjecture of Bergelson-Moreira-Richter and contains as special cases both the Polynomial Szemerédi Theorem due to Bergelson-Leibman-Lesigne and the fact that if $f$ belongs to a broad class of smooth functions and satisfies $x^{d-1}\prec f(x)\prec x^d$ for some $d\in \mathbb{N}$ then for any $\ell\in \mathbb{N}$, any invertible measure preserving system $(X,\mathscr{B},\mu,T)$, and any $A\in \mathscr{B}$ with $\mu(A)>0$, the set $\{n\in \mathbb{N}: \mu(A\cap T^{-[f(n)]}A\cap T^{-2[f(n)]}A\cap \cdots\cap T^{-\ell[f(n)]}A )>0\}$ is thick, meaning that it contains arbitrarily long intervals of natural numbers. Additionally, we formulate and prove a generalization to weighted averages of Boshernitzan's criterion for uniform distribution which we use in the proof of our main result.

We prove that if we take a rooted product of a circulant graph with universal perfect state transfer with a path of fixed length whose end edge is weighted with a transcendental number, then there is pretty good state transfer between any pair of endpoints of these paths. As a consequence, in a path with an even number of vertices with transcendental weights on the two edges incident to the endpoints, there is pretty good state transfer.

Inverse problems are predominantly solved within the optimization-driven paradigm, which formulates the problem as objective-function minimization and approaches the solution by iterative search. Though universal, it suffers from limitations in efficiency, interpretability, and multi-parameter decoupling. This paper proposes a new paradigm—Structure-Driven Inversion (SDI). Instead of iterative search, SDI identifies and exploits the intrinsic structure of the problem to construct a solution method. It has two types of structure—mathematical structure and physical structure—and two corresponding driving inversion types: Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). The current representative method of MSDI is Mathematical Structure-Driven Pseudo-Inverse Inversion (MSDPII), which constructs a pseudo-inverse in the spectral domain via unitary diagonalization. The current representative methods of PSDI are Physical Structure-Driven Waveform Inversion Imaging (PSDWII), which performs inversion through virtual-source projection, and Physical Structure-Driven Back-Propagation (PSDBP), which implements three structural projections for deep neural networks via physical-system analogy. SDI is not a rejection but a complement and extension of Optimization-Driven Inversion (ODI). To the best of the author's knowledge, no existing study has systematically presented structure-driven inversion as an independent paradigm; this paper aims to fill that gap.

We study diagonal billiard trajectories inside the Young diagram of an integer partition $\lambda$. A trajectory has slope $\pm 1$, passes straight through sides shared by adjacent cells, and reflects from the exterior boundary until it closes. Let $\sigma(\lambda)$ be the number of closed orbits. This extends the mirror-curve model of Chokwe sona sand drawings studied by Gerdes from rectangular grids to arbitrary Young diagrams. Let $G_\lambda$ be the cell-adjacency graph of $\lambda$, with its natural planar embedding. We identify the billiard orbits with the components of the medial link of $G_\lambda$, and deduce that $\sigma(\lambda) = 1 + \dim \mathcal{B}(G_\lambda) = \mathrm{nullity}\, L(G_\lambda)$ over $\mathbb{F}_2$, where $\mathcal{B}$ is the binary bicycle space and $L$ the mod-2 Laplacian. Writing $\lambda^\square$ for the diagram obtained by deleting the first row and column of $\lambda$, we further prove $\sigma(\lambda) = 1 + \mathrm{nullity}_{\mathbb{F}_2} A(G_{\lambda^\square})$. For rectangles this recovers Gerdes' formula $\sigma(n^m) = \gcd(m,n)$ via identities for Fibonacci polynomials over $\mathbb{F}_2$, and in general it gives the characterization: $\sigma(\lambda) = 1$ if and only if $\lambda^\square$ has an odd number of domino tilings. Using the checkerboard bipartition of $\lambda^\square$, we decompose $\sigma(\lambda) - 1$ into a color-imbalance term, related to the BG-rank of Berkovich-Garvan, and an even rank-deficiency term. This yields parity restrictions and lower bounds for the orbit number, and shows that for any fixed $d$, asymptotically all partitions have more than $d$ orbits. We also prove that $\sigma(\lambda)$ is at most the Durfee length of $\lambda$, determine $\sigma(n, n-1, \ldots, 1) = \lceil n/2 \rceil$ for staircase partitions, and show that the adjacency-nullity formula is independent of the ground field.

Reduced-order modeling of transport-dominated systems remains challenging because moving fronts and shocks are poorly represented by low-dimensional linear subspaces. We develop a structure-informed data-driven reduced-order model(ROM) for scalar hyperbolic conservation laws based on the kinetic defect formulation. This formulation separates the nonlinear dynamics into known characteristic transport and a kinetic entropy defect localized on the shock manifold. We exploit this structure by first removing the known transport from the solution snapshots. We then extract and register the remaining defect-driven dynamics in a shock-attached coordinate system. Separate ROMs are used to evolve the shock geometry and the registered defect-driven source. During prediction, the predicted shock geometry is used to inverse-register the learned defect-driven source, which advances the kinetic state and recovers the physical solution. Numerical examples in one and two spatial dimensions demonstrate accurate reconstruction and prediction of nonlinear transport with shocks, including evolution beyond the training interval, while accurately capturing the mass and entropy-dissipation behavior of the reference solution.

In this manuscript, we study the master equation for mean field game of controls, assuming only that the data are Lipschitz continuous in the measure variable. Accordingly, we propose a weaker notion of solution called weak solutions for the master equation. We establish the global well-posedness of the master equation for such solution under the Lasry-Lions monotonicity and displacement $\lambda$-monotonicity conditions, respectively. The arguments rely on the analysis of the associated Pontryagin forward-backward stochastic differential equation system, especially the stability in the measure variable. Finally, we review several existing notions of non-smooth solution from the literature, namely good solutions, weak viscosity solutions, Lipschitz solutions, monotone solutions, and examine their relationship with our weak solution. We show that under appropriate assumptions all of them are equivalent by their definitions, and thus our manuscript provides a unification of weak solution notions for the master equations derived from mean field game of controls.

In this paper, we study singular positive solutions to the semilinear elliptic equation $$ - \Delta u = f(u) ~~~~~~ \textmd{in} ~ \Omega \setminus \Gamma, $$ where $\Omega \subset \R^n$ is a bounded or unbounded domain, and $\Gamma \subset \Omega$ is a singular closed set with zero Newtonian capacity. When $\Omega = \R^n$ and $\Gamma \subset \{ x_1 = 0 \}$, we establish the symmetry of singular solutions with respect to the hyperplane $\{ x_1 = 0 \}$ and their monotonicity in the $x_1$-direction. For the case where $\Omega \subset \R^n$ and $\Gamma$ is a smooth closed manifold of dimension $\leq n - 2$, we show the asymptotic symmetry of singular solutions with respect to the normal direction of $\Gamma$. These results significantly improve those of Chen-Lin (Duke Math. J. 1995; Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2001), Li (Invent. Math. 1996) and Sciunzi (J. Math. Pures Appl. 2017). Unlike their proofs, we employ an improved version of the moving sphere method, which also simplifies the analysis and extends its applicability.

We prove the conjecture of Ding and Mossel~\cite{DM}: the lazy random walk on the discrete cube, censored to any increasing set of fixed positive density, mixes in time $O(n\log n)$. More precisely, for every nonempty increasing set $A\subseteq\{0,1\}^n$, with $a=\mu(A)$ and $\mu$ uniform on the cube, we prove $$ t_{\mathrm{mix}}(P)\le C_a n\log(en), $$ where $C_a$ depends only on $a$. The proof uses hypercontractivity on the ambient cube to strengthen the Poincaré inequality on coordinate sections. An optimal stopping time occupation inequality for increasing sets converts the resulting local bound into a uniform bound on hitting times of large sets. An appendix by Y.~Peres improves the density dependence to $C_a=13a^{-1}\log(e/a)$, within a logarithmic factor of the conjectured $a^{-1}$ dependence.

We study the semiclassical limit from the magnetic Hartree–Fock equation to the magnetic Vlasov–Poisson equation in three spatial dimensions in the presence of a nonuniform, purely space-dependent magnetic field. We first establish propagation of regularity for solutions of the magnetic Vlasov–Poisson equation under exponentially weighted assumptions. Building on this result and a gauge-covariant magnetic Weyl quantization framework, we then derive explicit convergence bounds for the semiclassical limit in semiclassical Schatten norms. In particular, the estimates are formulated in terms of the magnetic field rather than a particular choice of vector potential and are therefore gauge covariant. The resulting semiclassical convergence holds for a suitable class of initial data on any fixed finite time interval.

We investigate finite-time field-induced ionization from a one-dimensional attractive delta-function well in a uniform dc electric field of arbitrary strength. Our aim is to determine the physical bound-state survival amplitude \(a_b(t)\) at any observation time without constructing the complete time-dependent propagator or wavefunction. In the gauge-equivalent Kramers–Henneberger representation, the field appears as motion of the contact point, and the resulting dynamics reduces to a closed Volterra equation. Exact endpoint-phase factorization organizes the full chronological rescattering history into a relative-time convolution hierarchy and an exact resolvent. By introducing the accumulated bound-state amplitude and reorganizing its two-time domain in relative and complementary center times, the final contact contribution becomes an explicit boundary integral. All spatial integrations are carried out analytically. The field-driven contact-free term is obtained in closed form using the Faddeeva function, while the direct and repeated-rescattering terms are expressed as explicitly evaluable time integrals. The result applies at arbitrary dc-field strength and finite time, with no weak-field expansion, rescattering truncation, or asymptotic-time approximation. It is verified through the exact field-free limit and an independent numerical solution of the original physical Volterra equation. The formulation gives an exact nonperturbative solution of a fundamental ionization model and reveals otherwise hidden analytical structure in driven quantum dynamics.

Local inference for spatial point distributions is dominated by Monte Carlo calibration. We develop an alternative based on the Tweedie–Miyasawa identities of empirical Bayes, which relate locally weighted moments of a point distribution under a Gaussian kernel to derivatives of its log-intensity in scale space. We first establish a rigidity theorem showing that the structure of these identities forces the Gaussian kernel. Under complete spatial randomness, we derive a closed-form null distribution for a bounded inter-scale contrast, yielding a calibrated simulation-free pointwise test. In experiments, the measured type I error is 0.070 at a nominal level of 0.05, with a calibration cost 199 times smaller than Monte Carlo for the same local statistic. We then derive a contamination law for structures of dimension m and width w embedded in a uniform background, together with an explicit detectability threshold. For a filament in three dimensions, the threshold is 16 pi. The resulting scale-resolved local dimension estimator has no free parameters. Applications to California seismicity, a trefoil knot, and 10,071 SDSS galaxies show that the method separates local structures across scales without a spatial mesh and reproduces published cosmic web fractions. All experiments are reproducible from a single public script.

Linear constraint systems (LCS) provide a compact algebraic language for nonlocal games and state-independent contextuality. In the binary case the Mermin–Peres square shows that inconsistent LCS can be solved by promoting variables to local Pauli operators, whereas a series of no-go results excludes analogous Pauli-, Clifford- and monomial unitary-based constructions at odd prime modulus. We first isolate the reason for this difference: for tensor-product quantum solutions, the obstruction to a classical solution of a LCS over $\mathbb{Z}_n$ decomposes into obstruction classes carried by the individual factors. As a consequence, a genuinely global obstruction cannot arise solely by tensoring locally unobstructed systems. The argument is specific to odd modulus; at even modulus the reordering phase need not vanish, as in Pauli-based examples. Given this distinction, we instead associate LCS to finite arrangements of rank-one projectors and resolutions of the identity in finite-dimensional vector spaces. Quantum solvability for such LCS is automatic and the search for a quantum-classical gap rests entirely on proving classical unsatisfiability of the underlying modular incidence problem of the arrangement. By relating such LCS with group-valued frame functions, we identify a family of examples in the work of [Harding, Jager, and Smith, Int. J. Theor. Phys. 44, 539 (2005)]. To establish the (non)existence of classically unsatisfiable LCS over $\mathbb{Z}_n$ with quantum solutions in all dimensions $d$ and for all $n\in\mathbb{N}$, we further construct explicit examples for the cases $d=n$ prime. Our main result thus positively resolves the existence problem of such LCS beyond the binary case: such systems exist if and only if $d\geq 3$ and $\gcd(n,d)>1$. Moreover, we establish the existence of LCS with a quantum-classical gap over finite fields if and only if $d\geq 3$ and $p\mid d$.

We develop a two-dimensional inviscid model for extensible membranes with continuous vortex shedding from both edges, coupled to a weak spectral Galerkin discretization of the membrane equation that permits direct computation at zero bending rigidity. Viewing the pressure jump as a flux for bound vorticity explains how membranes respond to concentrated fluid loading. For fixed-fixed membranes, compliance draws the body toward the leading-edge vortex, delaying its detachment and enhancing lift primarily during start-up; after periodic shedding develops, compliance mainly amplifies the force oscillations rather than mean lift. Vortex detachment can unload heavy membranes into compression, producing bending-regularized wrinkles. For $R_2>0$, the membrane equation remains well-posed independently of the sign of the tension, provided the membrane remains an immersion. Releasing the trailing edge creates an intrinsic degeneracy: the tension vanishes at the free endpoint, making the $R_2=0$ problem formally underdetermined, while the weak formulation selects its bounded-energy branch. This vanishing-tension endpoint traps pressure disturbances and drives rapid tip-snapping; the dominant frequency and effective wave number approach their zero-rigidity limits with an inverse-logarithmic correction arising from a singular Bessel branch. Finally, decreasing the stretching modulus destabilizes periodic flutter, and the repeated fluid-driven departure and elastic return suggest a homoclinic tangle underlying the transition from periodic to chaotic flutter.

The recent work \cite{HeWuXiaZhang2025} introduced a finite-data inverse nodal framework for regular one-dimensional Sturm-Liouville operators. Nevertheless, formulating a multidimensional inverse nodal theory for Schrödinger operators remains a challenging open problem. This paper addresses this gap by establishing a finite radial inverse nodal theory for the radial spectral branch of Schrödinger operators in balls. In contrast with the interval case, the nodal data are the radii of spherical nodal hypersurfaces of radial eigenfunctions. The exact radial reduction preserves this PDE nodal geometry and leads naturally to a singular weighted Sturm–Liouville problem with weight $w(r)=r^{n-1}$, rather than to a regular interval model. We minimize the PDE distance $\|Q-Q_0\|_{L^p(B_R)}$ , equivalently the weighted distance $\|q-q_0\|_{L^p_w(0,R)}$, among all radial potentials matching finitely many prescribed spherical nodal radii. For the associated nodal constraint sets, we prove spectral regularity, nodal differentiability, weak closedness, exact realization, and a finite-codimensional $C^1$ manifold structure. Crucially, without the variational selection imposed by \(q_0\), the prescribed finite nodal radii fail to determine the potential uniquely: the admissible potentials are invariant under constant shifts, and the finite nodal constraints yield locally infinite-dimensional level sets at regular points.