Papers

We explore identifying partial differential equations (PDEs) from noisy observations of single time-space trajectories. Recent developments show the benefits of identifying PDEs in their weak forms. We investigate the use of differential Strong-form dictionaries for PDE IDENTification (S-IDENT), which enables finding more general linear and nonlinear PDEs. Building on an extensive exploration of integral-type denoised differentiation approaches, we propose to use Savitzky–Golay (SG) differentiation with an adaptive window length chosen based on Stein's Unbiased Risk Estimate (SURE). This offers a guaranteed order of accuracy while producing estimators with minimal variance. The identification process is further refined and stabilized through trimming and reduction-in-residual model selection. Numerical evidence shows that S-IDENT can successfully identify nonlinear PDEs at higher levels of noise than existing strong-form methods, while also yielding results comparable to weak-form approaches. We further verify the effectiveness of S-IDENT through comparisons with various strategies to approximate differential features. We provide numerical evidence that general differential-form dictionaries are larger and more ill-conditioned than those used for weak-form identification, yet S-IDENT does not significantly suffer from this combinatorial increase in dictionary size.

Let $G$ be a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture. Assuming that $b_1^{(2)}(G)=0$, we associate to each surjective integral character $\phi$ the quantity $b_1^{(2)}(\ker \phi)$. We prove that the resulting function extends to a seminorm on $H^1(G;\mathbb{R})$ and is the thickness function of a polytope in $H_1(G;\mathbb{R})$. More generally, assuming that $G$ is of type $\mathrm{FP}_k(\mathbb{Q})$ and has $b_k^{(2)}(G)=0$, we prove that the function assigning $b_k^{(2)}(\ker \phi)$ to each surjective integral character $\phi$ is likewise the thickness function of a polytope. This confirms a conjecture of Friedl, Lück, and Tillmann. For torsion-free virtually (finitely generated free)-by-cyclic groups, we relate the resulting invariant to the $L^2$-Betti complexity of a character, thereby confirming a conjecture of Gardam and Kielak. As an application, we establish the existence of an algorithm for computing the Bieri–Neumann–Strebel invariant of free-by-cyclic groups and discuss connections with the isomorphism problem for such groups.

The fast multipole method (FMM) is an important component for the boundary element method (BEM), because with the FMM the efficiency and feasibility of the BEM can be enhanced to a large degree. Part of the FMM is grouping the elements of the boundary element mesh into different clusters. The size of these clusters in terms of number of elements and spatial expansion has a huge impact on the efficiency and stability of the method. However, while the theory behind the multipole expansion has been broadly researched, the clustering process itself and its effect on the FMM has been neglected in comparison. Most of the time, for example, it is implicitly assumed that the elements of the mesh have about the same size, which is often not the case in practical applications, e.g., when calculating the sound field around the human head. In this study we compare different types of clustering approaches with respect to stability and efficiency of the underlying FMM applied to meshes that have uniform as well as non-uniform element sizes. Also, some examples are provided for cases where a wrong clustering can lead to numerical problems and instabilities of the FMM-BEM.

We study the large-scale behavior of the coincidence set of perturbations of global solutions to the classical obstacle problem in $\mathbb{R}^n\setminus B_1$, with blow-down invariant in the $e_n$ direction. In dimensions $n\geq 3$, we prove that, locally around regular points sufficiently far out, the cross-sections of $\{u=0\}$ perpendicular to $e_n$ are $C^2$ perturbations of ellipsoids. The main ingredient is a new large-scale almost monotonicity formula for the Alt–Caffarelli–Friedman functional. In contrast with the classical small-scale perturbative theory, our argument exploits the stability of the obstacle problem together with the fact that local perturbations vanish under blow-down. The method provides a model mechanism for controlling errors at infinity in stable free boundary problems.

Insecticide-treated nets (ITN) are an effective and low-cost intervention for controlling vector-borne disease (VBD), however, their use depends on individual decisions based on perceived cost and risk of infection. This study investigates a nonlinear multi-host model for the transmission of VBD with endogenous strategic control. We assume that hosts' adoption of ITN emerges from the payoff-based decision-making, creating a nonlinear coupling with disease prevalence. We model vector preference as a function of ITN coverage to probe the complex interplay among individual choices, disease prevalence, and its control in a multi-host setting. The qualitative behavior of the system is characterized by the thresholds $R_0$ and $R_c$, which determine the existence and local stability of the disease-free and endemic equilibria. The system exhibits rich dynamical behavior; hence, we provide a bifurcation analysis identifying the conditions for saddle-node and Hopf bifurcations. Our results demonstrate that the interaction between the perceived cost of ITN and the infection risk can induce critical transitions, including regime shift from stable endemic states to sustained periodic oscillations. Furthermore, we identify a counterintuitive effect whereby complete ITN adoption by the primary host can increase the overall prevalence in the secondary host due to adaptive shifts of vector feeding behavior.

M.V. Zaicev and S.K. Segal, as well as S. Dăscălescu, B. Ion, C. Năstăsescu, and D. Raios Montes studied certain gradings on matrix rings and algebras - 'elementary' gradings. However, examples of gradings on a matrix ring that are not elementary are known. In the present article, we show that any subalgebra of a full matrix ring over an arbitrary graded division ring can be embedded in an elementary graded matrix ring.

We construct an incremental prime-counting approximation from corrected right-endpoint sums on polygonal-number partitions. For each fixed polygonal family, retaining completed contributions gives $\mathcal O(1)$ numerical operations per consecutive node and $\mathcal O(\sqrt X)$ operations for the full traversal through $X$, in the unit-cost model. The scheme uses a fixed number of working numerical words rather than repeated summation. Asymptotic formulas for the discretization error and terminal scale yield an unconditional relation with the classical normalized prime-counting error. We prove that the logarithmic mean of the normalized estimator error converges to $1/\sqrt2$ if and only if the Riemann hypothesis holds, and give an independent Mellin proof of the converse. Under the hypothesis, this constant uniquely removes the limiting signed mean bias. Computations at powers of ten through $10^{19}$ with coefficient $0.7071$ show accuracy comparable to the Riemann approximation at the sampled arguments

We prove that the special linear Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ over a finite field of characteristic $p$ is generated by two random elements with high probability as $|\mathfrak{sl}_n(\textbf{F}_q)|$ tends to infinity, provided that $(n,p) \neq (3,3), (4,2)$.

We propose Regularized Newton-SLRA (RN-SLRA), a regularized Newton-type method for local manifold–affine intersection problems motivated by structured low-rank approximation. Classical Newton-SLRA achieves fast local convergence under transversality, but its tangent-space intersection step may become ill-defined, singular, or severely ill-conditioned when transversality fails. RN-SLRA overcomes this difficulty by replacing the exact tangent-space intersection step with a regularized quadratic subproblem over the affine space. Under intrinsic transversality, RN-SLRA with regularization parameter $\mu_k=c r_k^\rho$ converges locally with Q-order at least $1+\rho$ for $0<\rho\le1$, including quadratically for $\rho=1$, and linearly for $\rho=0$. We also analyze an inexact variant based on $\sigma$-quasioptimal manifold projections. It retains the $1+\rho$ order for $0<\rho\leq1$, and converges linearly for fixed regularization when $q\sigma<1$, where $q$ is the local one-step alternating-projection factor. Experiments support the predicted quadratic behavior on an analytically clean but nontransversal SLRA instance, where the classical Newton tangent system is severely ill-conditioned, and show reduced projection costs for the inexact method on large-scale Hankel problems.

In this paper, we establish a completed pre-Lie bialgebra structure on the tensor product of a Leibniz-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Zinbiel algebra. We also obtain such a structure on the tensor product of a Zinbiel-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Leibniz algebra. Moreover, a Zinbiel-dendriform bialgebra is precisely one whose affinization by a special quadratic $\mathbb{Z}$-graded Leibniz algebra is a completed pre-Lie bialgebra. Finally, using solutions of the ZD-YBE (resp.~LD-YBE) with invariant skew-symmetric parts in a Zinbiel-dendriform (resp.~ Leibniz-dendriform) algebra, we construct completed solutions possessing invariant symmetric parts of the $S$-equation in the induced pre-Lie algebra.

We study the ring-theoretic structure and representation theory of the super Jordan plane $\mathcal{J}$ over fields of characteristic different from $2$. We prove that $\mathcal{J}$ is prime and classify its prime, primitive, and maximal ideals. We determine its classical ring of quotients and classify the finite-dimensional simple modules, while relating infinite-dimensional simple modules to those of the first Weyl algebra. Our approach is based on showing that a localization of $\mathcal{J}$ is a matrix algebra over a localization of the first Weyl algebra.

Unresolved measurements occur in many inference problems where two or more hidden processes may, at times, jointly generate a single measurement. For instance, such phenomena are encountered in multiobject tracking owing to the limited resolution capabilities of practical sensors; or in camera-aided autonomous driving due to shadowing or occlusions. Substantial performance degradation, such as track losses, are incurred when unresolved measurements are not accounted for. In this paper, we address multiobject tracking under a generalized unresolved measurement model, where any subset of objects may generate a single unresolved measurement according to a probabilistic model. Our innovation lies both in modeling and algorithm-design directions. First, we develop a probability distribution for object partitions based on a model of pairwise coupling of objects and subsequently a probability distribution for object-to-measurement association variables. This generic model incorporates sensor resolution capabilities, sensor detection, and sensor noise characteristics for object groups. Second, a generic Loopy Belief Propagation (LBP) method as well as a specialized Gaussian-LBP (GLBP) algorithm are proposed that perform object state inference under the aforementioned model. In contrast to direct marginalization methods, which involve a computational complexity of $O(m^n)$, for $m$ measurements and $n$ objects, the proposed GLBP algorithm achieves a computational complexity on the order of $O(m n 2^{n})$. Numerical results demonstrate the effectiveness of our proposed GLBP, with estimation performance that closely matches that of exact marginalization for only a fraction of the computational resources.

We study Ornstein–Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure. Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root. The associated semigroups are symmetric, analytic and positivity preserving; the Dirichlet semigroup is sub-Markovian, while the Neumann semigroup is Markovian and admits the Gaussian measure as its unique invariant measure up to scalar multiples. We prove compactness of the resolvent and derive linear eigenvalue asymptotics. For regular rooted trees, we adapt the Naimark–Solomyak decomposition to the Gaussian weighted setting, reducing the operators to one-dimensional half-line problems and obtaining refined spectral localisation and lower bounds.

We propose a quantum collocation framework for approximating solutions of one-dimensional linear and nonlinear boundary value problems. The method formulates the search for admissible solutions as a residual-based quantum search over a discretized ansatz space, where candidate solutions are evaluated through residual conditions imposed at collocation points. A residual-threshold oracle is constructed that acts jointly on spatial and parameter registers. This joint oracle structure leads to amplification dynamics that decompose into a coherent superposition of spatially conditioned amplitude-amplification processes rather than a single global amplification mechanism. We derive the corresponding amplification geometry and show that the success probability is governed by a weighted combination of spatially dependent amplification angles. Furthermore, we prove that the reversible residual oracle can be implemented with gate complexity polynomial in the logarithm of the number of collocation points, while retaining the quadratic search acceleration associated with amplitude amplification in the parameter space. We analyze how the spatially dependent oracle structure influences the amplification dynamics and corresponding success probabilities. Furthermore, we investigate how discretization, ansatz expressivity, oracle tolerance, and finite-precision effects influence both approximation quality and amplification behavior. Numerical experiments validate the theoretical predictions and illustrate the resulting search dynamics across different discretization and precision regimes.

List-coloring, introduced independently by Vizing and by Erdős, Rubin, and Taylor in the 1970s, generalizes ordinary vertex coloring by assigning to each vertex its own set of admissible colors. A graph is chromatic-choosable if its list chromatic number equals its chromatic number. The previous survey on list-coloring by D R Woodall (2001), emphasized defective choosability, the list-coloring conjectures, and different methods used for list-coloring. This survey reviews major developments on list-coloring and chromatic-choosability, with emphasis on graph classes for which equality is known, graph classes exhibiting a nontrivial gap, and the principal methods used to prove such results. The survey covers embedded graphs, perfect graphs, complete bipartite and multipartite graphs, claw-free graphs, line graphs, powers of graphs, graph products, and selected variants of list-coloring.

We prove local-global compatibility results at $p \neq \ell$ for the automorphic group determinants constructed by Scholze, generalising the result of Varma to torsion classes appearing in Betti cohomology. Our argument combines the construction of Scholze with the theory of representations of $p$-adic general linear groups with $\mathbf{Z}_{\ell}$-coefficients.

We generalize the notion of twisting endomorphisms, first defined by Castryck-Panny-Vercauteran, to the setting of $\mathcal{O}$-oriented supersingular elliptic curves. We give an algorithm to find supersingular elliptic curves over $\mathbb{F}_p$ with a twisting endomorphism of prime degree $\ell$, and we use it to compute a basis of their full endomorphism rings.

Let I=[a,b] and consider the degree n Lagrange interpolation at the nodes x, where x\in S:={x=(x_0,x_1,...,x_n):a=x_0<x_1<...<x_n=b}. Then the norm of the Lagrange interpolation operator is the maximum of the Lebesgue function L(x,t) on I. Bernstein conjectured that the norm of the Lagrange interpolation operator becomes minimal exactly for node systems which exhibit an equioscillation property in that the interval maxima m_k(x):=max_{[x_{k-1},x_k]} L(x,.)}, (k=1,...,n) are all equal. Erdős added to the conjecture the sandwich property that if y is an extremal (minimal norm) system, then for any other node system x there have to be indices i,j with m_i(y)<m_i(x) and m_j(y)> m_j(x). The conjectures were proved by Kilgore and de Boor–Pinkus in 1978. Since then, analogous results were obtained only for a few cases when interpolation is made to certain very special spaces of polynomials, or when we apply weighted interpolation with rather special weights. Worse than that, it turned out that published proofs of results on infinite intervals and weighted interpolation were seriously flawed. Here we prove the Bernstein and Erd\H os Conjectures for the case of exponentially weighted polynomials on the halfline. This is the first proof of these conjectures in a situation where, contrary to all existing successful proofs, we encounter singularity of certain derivative matrices.

Let $Y$ be a prequantization bundle over an integral symplectic manifold $(\Sigma,\omega)$. Let $L$ be a closed monotone Lagrangian submanifold that admits a Legendrian lift $\mathcal{L}$ in $Y$. Under the assumption that the minimal Maslov number $N_L$ of $L$ is greater than 2, we define the Rabinowitz Floer homology of $\mathcal{L}$. We then establish an isomorphism between the $\mathbb{Z}_d$-equivariant Rabinowitz Floer homology of $\mathcal{L}$ and the quantum homology of $L$, where $d$ is the degree of the covering map $\mathcal{L}\to L$. Under a more restrictive condition on $N_L$, we show that this map is a ring isomorphism. Using this isomorphism, we compute the quantum homology ring of Lagrangian spheres in quadrics and two-step flag manifolds. Furthermore, we investigate the implications of the quantum invertibility of $\omega$ for the vanishing of the quantum homology of $L$ and the obstructions to topologically simple fillings of $\mathcal{L}$. We also show that if $(\Sigma,\omega)$ admits a polarization and $L$ is disjoint from the Lagrangian trace, the quantum homology of $L$ vanishes.

This paper is concerned with the existence of non-radial positive classical solutions for the critical Hénon equation \[ -\Delta u=|x|^\alpha u^{\frac{N+2+2\alpha}{N-2}} \qquad \text{in }\mathbb R^N, \] where \(\alpha>0\) and \(N\ge3\), satisfying the Newtonian-type decay condition at infinity. Gladiali, Grossi and Neves (2013) proved existence for the discrete sequence $\alpha_k=2(k-1)$, $k\in\mathbb N$, and conjectured that non-radial solutions may exist only at these special values. We disprove this conjecture by establishing existence for a continuum of exponents near each \(\alpha_k\): for every even $k>\frac{N-2}{2}$, non-radial solutions persist for parameters \(\alpha\) close to, and different from, \(\alpha_k\). We recast the problem as a semilinear elliptic equation with Sobolev-supercritical exponent on the cylinder via the Emden–Fowler change of variables. Our argument is formulated directly on the cylindrical domain, thereby streamlining the characterization of the kernel of the linearized operator via Pöschl–Teller spectral theory, avoiding the ball-exhaustion technique employed in the original work, and allowing us to compute the bifurcation slope and verify the non-verticality condition.

We give an overview of the theory of quasi-$F$-singularities, focusing on their connection with singularities in birational geometry.

In this manuscript, we introduce positivity-preserving correction methods for low-rank approximations of the Vlasov equation. The key idea is to formulate structural properties, including positivity-preservation, as constraints and to seek a minimal correction term that is added to the low-rank solution, by solving a quadratic programming problem. As a result, the corrected solution satisfies the constraints and preserve these properties, while remaining close to the original low-rank solution. Two positivity-preserving schemes are proposed in this work, and one of them also preserves the total mass and momentum of the system. We apply the proposed methods to a Vlasov–Poisson and Vlasov–Poisson-BGK employing a spectral discretization in space and an explicit Runge–Kutta scheme in time. Numerical experiments demonstrate the effectiveness of the proposed methods.

We develop a framework to predict whether a family of Selmer groups has average size that is bounded or unbounded. Applying this framework to certain geometric families of abelian varieties over $\mathbb{Q}$, we give a conjectural characterization of which such families have $\ell$-Selmer groups of unbounded average size for a given prime $\ell$. In the case that the $\ell$-torsion Galois module is constant across the family, we show that our characterization is correct. The key tool of our technique is the Greenberg–Wiles' formula, which expresses the ratio of the sizes of a Selmer group and the corresponding dual Selmer group as a product of local factors. This formula gives a purely local lower bound for the size of a Selmer group that we conjecture is close to sharp most of the time.

Scalar inequalities in an order parameter often arise as the $2\times2$ shadow of a stronger Hankel determinant statement. We record a moment-representation criterion: positive exponential and Mellin order representations, together with gamma-normalized completely monotone averages, generate totally nonnegative Hankel kernels, with strictness controlled by the support of the representing measure. The criterion packages the classical total-positivity mechanism as a recognition calculus for special-function inequalities, turning the order parameter into a moment exponent after the correct normalization. The applications include three named determinant lifts. First, we prove the positive Jackson $q$-gamma Hankel conjecture of Karp–Vishnyakova–Zhang: for $0<q<1$, the kernel $(x,y)\mapsto\Gamma_q(x+y)$ is $\mathrm{STP}_\infty$. This is an atomic Mellin-moment instance of the general criterion; the reciprocal sign-regularity problem for $1/\Gamma_q$ is separate and is not addressed here. Second, we answer Yang's continuous half-gamma Mills-ratio log-convexity question and strengthen it to strict total positivity, hence to all higher Hankel Turan determinants. Third, we treat Tricomi rays and the one-dimensional Coulomb regularization as all-minor Hankel determinant hierarchies. For the Coulomb regularization, the $2\times2$ minor gives the scalar log-convexity question recorded by Baricz–Pogany, and the full theorem supplies the corresponding all-minor strengthening.

Let $A/\mathbb{Q}$ be a modular abelian variety of analytic rank $0$. If $G$ is a non-trivial finite abelian group such that all prime factors of $\lvert G \rvert$ are sufficiently large in terms of $A$, we show that there are infinitely many $G$-extensions $F/\mathbb{Q}$ such that $A(F)$ is finite. When $A$ is a rational elliptic curve of analytic rank zero with no exceptional primes, or the product of two such curves, the same conclusion holds without any assumptions on $|G|$. Our proof relies on new simultaneous non-vanishing results for twisted central $L$-values of even-weight holomorphic newforms. These results are obtained via novel constructions related to horizontal $p$-adic $L$-functions and are of independent interest.

We develop a unified framework for Fujita-type blow-up of solutions to the inhomogeneous semilinear heat equation $$\partial_tu-\Delta u=|u|^p+\mathbf{w}(x), \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0, \cdot)=u_0.$$ The classical integrability assumptions on the forcing term are replaced by quantitative regular variation properties of its spatial mass $$F(R)=\int\limits_{|x|\le R}\mathbf{w}(x)\,dx.$$ Using techniques from regular variation theory together with the Mitidieri–Pohozaev test-function method, we establish sharp Fujita-type nonexistence results and identify the critical exponent in terms of the variation index of $F$. We prove that global solutions do not exist in the subcritical range and obtain critical-case blow-up under suitable slowly varying corrections. The regular variation framework further shows the optimality of the underlying mass condition, extends naturally to anisotropic settings through operator regular variation, and yields sufficient blow-up criteria for sign-changing forcings via the Gaussian-Laplace transform. The approach also applies to space-time dependent forcings, Riesz-potential type forcings, and equations involving the fractional Laplacian, providing a unified description of blow-up thresholds beyond the classical Fujita theory.

We study oriented surfaces in the Heisenberg space $\mathrm{Nil}_3$ whose mean curvature $H$ at each point is $H=\langle N,\partial_z\rangle+\lambda$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field and $\lambda\in\mathbb{R}$. These surfaces are known as $\lambda$-translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow. The objective in this paper is to classify $\lambda$-translators invariant by the following one-parameter groups of isometries of $\mathrm{Nil}_3$: left-translations, rotations and helicoidal motions.

Ziegler proved that every simplicial $d$-dimensional $0/1$-polytope has at most $2d$ vertices, and asked whether equality forces the polytope to be centrally symmetric and hence, equivalently, a $0/1$-realization of the $d$-dimensional cross polytope. In this note, we give a negative answer, exhibiting an explicit set of $14$ vertices in $\{0,1\}^7$ whose convex hull is a simplicial $7$-polytope and is not centrally symmetric. Moreover, via exhaustive enumeration we show that up to the symmetries of the cube, there are precisely five such polytopes in dimension $7$ (of two combinatorial types) that are not centrally symmetric.

In this article, we investigate V-line transforms for symmetric $m$-tensor fields whose support lies inside a disk of radius $R$ and centered at the origin. We provide an explicit characterization of the kernel of the V-line transforms acting on a symmetric $m$-tensor field and derive a new inversion formula using a decomposition result. In addition, we present a comprehensive numerical verification and validation of the inversion algorithms for these V-line transforms for vector fields and symmetric $2$-tensor fields, which were recently developed in \cite{bhardwaj_2024,bhardwaj2025tensor}. The reconstruction results obtained for various phantoms demonstrate the effectiveness and robustness of the proposed numerical methods, including in the presence of noise.

A $\lambda$-translator in $\mathbb{S}^2\times\mathbb{R}$ is an oriented surface whose mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle+\lambda$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field tangent to the fibers of the submersion and $\lambda\in\mathbb{R}$. When $\lambda=0$ we fall into the class of translators. In this paper, we classify all $\lambda$-translators that are invariant by a one-parameter group of rotations and by vertical translations of $\mathbb{S}^2\times\mathbb{R}$.