Papers

We extend the cone-ray exact-NMF pipeline of Ramteke (arXiv:2606.22451) from the uniform-support regime r_+ = r to the gap regime r_+ > r, and classify recoverable nonnegative factorisations by the rank of the W-factor into a three-regime taxonomy. Regime A (rank(W) = r_+, full column rank): a two-sided SVD-gauge cone-ray pipeline W = U_{r+}(G) Q, H = P V_{r+}(K)^T with G, K on Stiefel manifolds and square consistency Q P = diag(S_r, 0). On 10x10 dense random gap matrices it gives 100/100 recovery at r_+ = 5 and 6. We explain this via two geometric facts: slack enclosure (the data cone has codimension r_+ - r in the outer cone) and NRF-variety thickness (valid gauges form a positive-measure set, so the blind SVD lands on one with probability one). Regime B (rank(W) = r, W a column subset of M): a rank-deficient-W branch enumerating r_+-subsets of M's columns with per-column LP tests. On the block-diagonal family diag(C, J_k), where additivity of nonnegative rank collapses the valid gauges to a single point and the blind SVD pipeline fails, the column-subset branch restores recovery in milliseconds. Regime C (r < rank(W) < r_+, W not a column subset): exposed by the regular octagon's slack matrix. An exact size-6 NRF exists and is reachable by the symmetric formulation at an oracle gauge derived from a known factorisation (residual 1.5e-10), but the blind problem is open: 50 Haar random restarts and Riemannian gradient descent on the Stiefel/Grassmann gauge all stall, because the alt-LP residual is piecewise constant on cells of gauge-space, so local descent cannot cross cell walls. A combined toolkit (Regime B then A) covers regimes A and B with no regression on dense draws; Regime C remains open, with the regular octagon as the cleanest unsolved test case.

We study the Waring decomposition of symmetric tensors with nodes on a rational variety. We provide an explicit characterisation of the existence of such a decomposition under some technical assumption, and introduce an efficient algorithm to decompose this novel class of structured symmetric tensors. The framework directly generalizes Hankel tensors (Qi 2015) to the multivariate setting. We analyse in details the case of toric varieties and rational curves. Proving the existence of a quadrature formula of even strength 2N with at most N + 1 nodes, that avoids a prescribed finite set of points, we establish new sharp upper bounds on the minimal number of nodes for quadrature formulae on rational curves. Numerical experimentation demonstrates the gain of this approach, compared to classical direct approaches.

Addressing infeasibility in non-cooperative games has become an important topic, as many problems across different applications face this issue. In this paper, we propose a new solution concept for generalized games with possibly infeasible individual constraints. A solution is defined as the limit of a sequence of generalized Nash equilibria induced by games with penalty terms relaxing the individual constraints. Existence is established for a broad range of games and we provide conditions allowing to characterize a $\psi$-penalized solution as a strategy profile maximizing every player's utility over all her penalty minimizing strategies. A variation of Divide-the-Dollar serves as an illustrative example. We further establish the compatibility with the GNE and the solution to the Nash bargaining.

In this paper, we investigate a set of $k$-color partitions and examine how its subsets relates to the theory of the minimal excludant function introduced by Andrews and Newman. Furthermore, we provide generalization of the partition functions introduced by Andrews and Bachraoui and also define several new partition functions within this framework.

In this paper, we study geometrically continuous quadratic splines over triangulations. While a rich variety of $C^1$ quadratic splines is available over planar domains, and such splines can also be constructed on the torus, the problem becomes significantly more challenging on more general surfaces. We first construct a $G^1$ spline over a regular spherical $n$-gon, subdivided into $3n$ triangles. Based on this construction, we obtain a quadratic $G^1$ spline approximation of the sphere induced by an arbitrary uniform polyhedron, where each $n$-gonal face is subdivided into $3n$ triangles. The construction uses only quadratic triangular patches and yields explicit control points depending on the geometry of the underlying polyhedron and one free parameter. We also analyze the resulting approximation quality and curvature behavior, and illustrate the construction on Platonic and Archimedean solids.

Classical methods for solving linear ordinary differential equations, such as superposition, the method of undetermined coefficients, and the annihilator technique, are often presented as heuristic, procedural rules. In this article, we show that these methods admit a coherent algebraic interpretation when constant-coefficient linear differential operators are viewed as elements of the polynomial ring $\mathbb{C}[D]$, acting on spaces of smooth functions. Without invoking the formalism of $D$-modules or non-commutative operator algebras, we explain how homogeneous solution spaces arise as kernels of linear operators, how particular solutions form affine cosets, and how the search for solutions is an infinite-dimensional eigenvalue problem. Furthermore, we extend this algebraic framework to variable-coefficient equations, resolving the Euler equation through ring isomorphisms and framing d'Alembert's reduction of order as non-commutative operator factorization. We also explore the boundary of this linear theory, demonstrating how diffeomorphic linearization allows certain non-linear equations – such as those of Bernoulli and Riccati – to be mapped directly into the $\mathbb{C}[D]$-module framework. Finally, we contrast this framework with the multivariable ring $\mathbb{C}[D_x, D_y]$, using the loss of the principal ideal domain property to explain the intrinsic structural divergence of partial differential equations, and indicate further universal extensions to discrete difference equations and the Weyl algebra.

The preprojective algebra of a hereditary algebra $H$ can be defined as a certain orbit construction of the regular representation generated by the Auslander-Reiten translation. In this short survey, we will look at two important generalisations, namely, the contracted preprojective algebra and the total preprojective algebra. We will include several open problems and questions motivated by examples in the hope to stimulate future research on general orbit algebras of $H$-modules.

In this paper, we study the weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives. Firstly, we prove a family of the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities that complements the results in [C. Cazacu, J. Flynn and N. Lam, Calc. Var. Partial Differential Equations 62 (2023), no. 4, Paper No. 118, 26 pp.] and [A. T. Duong and V. H. Nguyen, On the sharp second order Caffarelli-Kohn-Nirenberg inequality. Ann. Fenn. Math., 50(1):275–286, 2025]. Secondly, we establish a stability version of the sharp weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields proved by Cazacu, Flynn and Lam. Finally, we prove a stability estimate for the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities established in this paper. Our approach is based on the spherical harmonic decomposition method, the one dimensional integral inequalities and their improvements.

Let $X$ be a finite left nilpotent skew brace and let $\pi$ be a set of primes. We show that every Hall $\pi$-subgroup of the multiplicative group $(X,\cdot)$ is a Hall $\pi$-subbrace of $X$. This extends \cite[Theorem 11]{CDDFT} by removing the solvability assumption. As an application, we obtain an upper bound for the left nilpotency class of $X$ in terms of the left nilpotency classes of its Sylow $p$-subbraces for every prime $p$ dividing the order of $X$. We also show that every $p$-subbrace of $X$ is contained in some Sylow $p$-subbrace.

We present a method of ensuring recursive feasibility and input-to-state stability of robust nonlinear Model Predictive Control (MPC) with multi-step predictors. Although feasibility guarantees are well-established for the case of single-step models applied recursively over a finite horizon, such guarantees are missing in naive MPC formulations that use distinct multi-step models to predict the system state at different future points in time. This issue arises because of potential inconsistencies in multi-step predictions generated at different times. Our approach performs an a priori sufficient feasibility check of the robust nonlinear MPC optimisation problem, and uses information from previous solutions to provide a fallback based on previously certified prediction sets. We illustrate the proposed predictor-substitution strategy with a simple numerical example.

In this article, we study the semi-linear elliptic problem: \begin{equation*} -\Delta_{\mathbb{B}^N} u \, - \, \lambda u = |u|^{p-1}u, \quad u \in H^{1}(\mathbb{B}^N), \end{equation*} where $N \geq 3$, $\lambda > \frac{N(N-2)}{4}$, and $1 < p \leq 2^*-1$. Here, $\mathbb{B}^N$ represents the Poincaré ball model of the hyperbolic space, and $H^{1}(\mathbb{B}^N)$ denotes the Sobolev space on $\mathbb{B}^N$. In the critical case $p = 2^*-1$, with $\lambda < \frac{(N-1)^2}{4}$ and $N \geq 4$, we establish the existence and multiplicity of nonradial sign-changing solutions. Moreover, our result answers an open question: “the existence of sign-changing solutions in the lower dimensions $4 \leq N \leq 6$". We also prove a partial non-existence result for a large class of symmetric solutions when $\lambda > \frac{(N-1)^2}{4}$, based on a quantitative orbit packing condition.

Opinions play a crucial role in shaping collective phenomena such as political polarization, cultural integration and demographic change. By continuously changing social environments in which opinions evolve, human migration serves as an important driver of collective opinion formation. While migration and opinion dynamics have both been extensively studied, the few existing models that couple the two are primarily deterministic and therefore cannot capture demographic fluctuations, finite-size effects or stochastic transitions between emergent collective states. To address this limitation, we introduce a unifying stochastic framework for opinion dynamics over migration networks that couples local opinion transitions, demographic processes and migration between communities. The dynamics are formulated through a spatio–temporal master equation, which provides a probabilistic description of the underlying population process. From this microscopic representation, we derive deterministic mean-field equations governing the co-evolution of community sizes and opinion compositions, thereby linking agent-level interactions to macroscopic population behavior. Using two representative case studies, we demonstrate how stochasticity and migration can qualitatively change the emergent dynamics and collective outcomes, including the emergence of consensus, polarization and the stabilization of oscillatory opinion dynamics. These examples highlight the rich interplay between social interactions, demographic change and migration in deterministic and stochastic settings, and they demonstrate that migration should be viewed as an integral component of collective opinion formation rather than only an external demographic process.

There are relatively few results on the convexity of solutions to complex equations. In this paper, We prove a strict real log-concavity theorem for the first eigenfunction of the complex $\sigma_{2}$-Hessian operator on smooth, bounded, real uniformly strictly convex domains in $\mathbb{C}^{n}$. As an application, we obtain a Brunn–Minkowski inequality for the first complex $\sigma_{2}$-Hessian eigenvalue. The proof combines a Bian–Guan constant-rank argument, a new inverse-convexity lemma for the compressed real Hessian, and Salani's viscosity admissible-test-function method.

Let $G$ be a finite group. We provide a description of the ordinary quiver of the complex monoid algebra of the wreath product $G \wr \mathrm{PT}_n$, where $\mathrm{PT}_n$ denotes the monoid of all partial functions on an $n$-element set. This description depends on the multiplicities of simple $G$-modules appearing in the decomposition of tensor products of simple $G$-modules. We also prove that the global dimension of this algebra is $n-1$. Both results are obtained by analyzing the associated Ehresmann EI-category related to the monoid. Finally, we describe the quiver of the algebra of the wreath product of $G$ with the submonoid of all order-preserving partial functions.

For a topological group, the quotient map modulo a subgroup is open and the quotient map modulo a compact subgroup is perfect. In this paper we prove and develop the corresponding compact-ideal theory for topological \(MV\)-algebras. We show that if \(I\) is an ideal of a topological \(MV\)-algebra \(A\), then the natural quotient homomorphism \(q:A\longrightarrow A/I\), where \(A/I\) is endowed with the quotient topology, is a continuous open quotient map and \(A/I\) is again a topological \(MV\)-algebra. If, in addition, \(I\) is compact, then \(q\) is perfect. As applications, we study three-space phenomena in topological \(MV\)-algebras. Under compact-kernel hypotheses we prove three-space theorems for compactness, local compactness, \(\sigma\)-compactness, Lindelöfness and paracompactness under the separation hypotheses stated below. We also prove a first-countability three-space theorem for locally convex topological \(MV\)-algebras.

Non-generalized Reed-Solomon (in short, non-GRS) type maximum distance separable (in short, MDS), near MDS (in short, NMDS), and linear complementary dual (in short, LCD) codes, as well as the hull of linear codes have interesting practical applications in cryptography and coding theory. In this paper, we focus on a class of non-GRS codes and its extended codes, i.e., modified generalized Reed-Solomon (MGRS) codes and extended MGRS (EMGRS) codes introduced by Wang et al. in 2026. Firstly, we prove that two classes of MGRS codes and EMGRS codes are either MDS or NMDS, derive the necessary and sufficient conditions for these codes to be NMDS, and then completely determine the weight distributions for one class of these NMDS MGRS or NMDS EMGRS codes. Secondly, we construct four classes of MGRS codes which are either Euclidean LCD codes or one-dimensional Euclidean hull codes. Thirdly, we constructively prove that there exist MGRS codes with flexible Hermitian hull dimensions and lengths. In addition, we illustrate the linearly inequivalence of NMDS MGRS codes and elliptic curve NMDS codes by Schur product. Finally, some corresponding examples are given.

Conventional quantum integrability is encoded in a commuting algebra of transfer matrices. By contrast, several models possess additional non-commuting conserved charges with important physical consequences, yet the nature of the corresponding symmetry has remained elusive. Focusing on the XXZ spin chain at roots of unity, we show that the non-Abelian analogue of the commuting transfer-matrix algebra is governed by quadratic relations following from a new class of unbalanced Yang–Baxter/RLL relations. This quadratic algebra is shown to encode precisely the Onsager algebra, for which we construct explicit matrix-product representations of both its generators and its duality defect line. The latter obeys $\mathbb{Z}_N$ Tambara–Yamagami fusion rules, thereby providing a lattice realization of the topological defect lines of the compactified boson conformal field theory. Our results identify non-Abelian transfer-matrix algebras as a microscopic origin for Onsager symmetry and dualities in lattice models.

In this article, using Halmos' two projections theorem, we completely characterize finite-rank commutators of orthogonal projections onto shift-invariant subspaces $\phi_1 H^2(\mathbb{D})$ and $\phi_2 H^2(\mathbb{D})$ of the Hardy space $H^2(\mathbb{D})$ corresponding to inner functions $\phi_1, \phi_2$ on the unit disc $\mathbb{D}$. Using our methods, we connect the finite-rank commutator with the Fredholmness of the projections $(P_{\phi_1}, P_{\phi_2})$ as introduced by Avron, Seiler and Simon. We conclude with several characterizations on the polydisc.

In 1996, Youngs proved a surprising theorem that quadrangulations of the projective plane could never have chromatic number exactly 3. This sparked a lot of interest, and the result has been further developed in many directions over the past decades. For example, the result is strengthened by considering the circular chromatic number, which is a real-valued lower bound on the chromatic number. The circular chromatic number of a quadrangulation cannot be in the interval (2,4). This parameter allows a generalization to larger even faces, for which a similar gap exists. In this work, we place these results into a framework based on the notion of winding number using extensions of colorings to continuous mappings. This yields unified and simplified proofs of gaps in the circular chromatic number for graphs with a distinguished set of directed even cycles. This generalizes the setting of graphs embedded on surfaces where every face is even. We further establish an analogous gap phenomenon when all faces are of a given odd length, previously known only in the case of triangulations. For example, we conclude that if G is a graph embedded on the projective plane such that all faces are 5-cycles, then either its circular chromatic number is 5/2 or at least 3, the former being the case only if G is Eulerian and every noncontractible facial walk is of odd length...

In this paper, based on the tensor form, we propose a class of economic cascadic tensor multigrid method(ECTMG) for solving high dimensional elliptic linear partial differential problems. Compared with traditional methods, the new method not only reduces the storage space but also lowers the computational complexity from $\mathcal{O}(n^{3})$ to $\mathcal{O}(n^{2})$. We analyze the convergence rate of the conjugate gradient method which is based on the tensor form($\mathrm{CG}\_\mathrm{BTF}$), and then provide the convergence analysis for the new method. Finally, the effectiveness of the new method is verified through numerical examples.

The nematic Helmholtz–Korteweg equation models the propagation of time-harmonic acoustic waves in nematic Korteweg fluids, such as nematic liquid crystals. The PDE augments the classical Helmholtz equation with two additional fourth-order terms, one of which is anisotropic in the direction of the nematic field. We refine the previous continuous analysis of Farrell et al. (2025) by using the Cordes condition and present a $C^0$-hybrid interior penalty discretization. The proposed discretization offers greater flexibility than $C^1$-conforming methods and is well-suited for applications in three dimensions and on curved domains. We prove stability of the method for any polynomial degree greater than or equal to two, independent of the spatial dimension, provided that the anisotropy is sufficiently small. Further, we show that the sequence of discrete solutions converges to the continuous solution under minimal regularity assumptions and derive convergence rates if the continuous solution has additional regularity. Finally, we illustrate the capabilities of the method through numerical examples.

In [8], a non-Archimedean analogue of the space of translation-invariant even valuations on convex sets was introduced. In [7], motivated by a further analogy with the classical theory, this space was equipped with two multiplicative structures, the product and the convolution. Both structures satisfy Poincare duality and the (non-mixed) hard Lefschetz theorem. In this paper, we formulate a conjecture concerning a more general mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations. We prove the non-mixed Hodge-Riemann relations in degree 1 for the product and, equivalently, in codegree 1 for the convolution.

We obtain reliable a posteriori residual-based error estimates for parabolic partial differential equations with small randomness in a Robin-type boundary condition. The uncertainty is addressed via a perturbation approach, transforming the problem with small random input data into a sequence of deterministic problems. Finite element approximations, combined with the backward Euler time discretization, are employed for the resulting problems. To ensure optimal spatial accuracy, the elliptic reconstruction framework is suitably adapted to this setting. This is achieved by introducing a parametric elliptic reconstruction operator that unifies the a posteriori analysis of deterministic parabolic problems with that of parabolic problems with small uncertainties. The obtained a posteriori error estimator is robust with respect to the parameter that describes the amount of uncertainty, in the sense that the constants appearing in the bounds are independent of the parameter, as well as of the mesh size and the time-step. In addition, numerical experiments are presented to validate the theoretical findings and illustrate the robustness of the proposed estimators.

In this paper, we provide a characterization of complex symmetric, self-adjoint, and normal generalized product-type operators on weighted Bergman space, with respect to conjugation $JW_{u_ {\alpha}, \vartheta_ {\alpha}}$.

We study the local existence and blow-up behavior of solutions, with possible growth at space infinity, for the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ ($p>1$), in a half-space with Dirichlet boundary conditions. Under optimal, polynomial growth assumptions on the initial data, characterized by the critical exponent $p/(p-1)$, we first prove local existence and uniqueness of a maximal classical solution and establish a blow-up alternative. This is complemented by a nonexistence result showing that $p/(p-1)$ is the sharp threshold for admissible growth at infinity. Next, for initial data with subcritical growth, we show global-in-time existence for $1<p\le2$ and that finite-time blow-up can occur only through gradient blow-up for $p>2$, and we derive sharp upper and lower estimates on the gradient, including a precise type~II boundary blow-up rate and a dichotomy in its behavior near the singular time. Finally, in the case $p>2$, for any type~II blow-up solution, we prove convergence after rescaling) to an explicit one-dimensional profile and providing a refined description of the asymptotic singularity formation.

In this paper, we consider a parabolic problem with constraint written as a differential inclusion, driven by a multiplicative colored noise and involving a p-Laplace operator (for $p \geq 2$), nonlinear random source terms and subject to Neumann boundary conditions on a bounded Lipschitz domain of $R^d$ with $d \geq 1$. This contribution aims at proving existence and uniqueness of a solution for such a multivalued problem. On one hand, the existence result is proved by the analysis of a semi-implicit time discretization scheme constructed on a smoother version of our problem, itself obtained by a regularization "à la Moreau-Yosida" of the subdifferential term. The key point of our approach consists in finding a clever relation between the time step denoted $\tau$ and the Moreau-Yosida regularization parameter denoted $\epsilon$ in view to pass simultaneously to the limit with respect to $\tau$ and $\epsilon$. On the other hand, the uniqueness of the solution is proved by standard arguments.

We consider proximity operator as a selector of the subgradient in the context of abstract convexity and characterize its properties in term of minimization problems. We also investigate the continuity and differentiability of proximity operator in the case of paraconvex and weakly convex functions. Supporting examples are given in each section.

Post-training hyperparameter selection is a critical step in the deployment of modern artificial intelligence systems, given the need to tune degrees of freedom of pre-trained models such as inference-time parameters, implementation-level settings, and thresholds driving decision rules. Despite its practical importance, hyperparameter selection is typically performed using best-effort empirical methods such as grid search or Bayesian optimization, which provide no formal statistical guarantees on reliability or safety. This monograph, intended for an audience of signal processing and machine learning researchers, presents a unified statistical framework for reliable post-training hyperparameter selection, centered on the learn-then-test (LTT) paradigm. LTT formulates the hyperparameter selection problem as multiple hypothesis testing over a candidate set of hyperparameters. The framework enables the choice of hyperparameters that provably satisfy application-specific reliability requirements—such as bounds on average risk, quantile risk, or information-theoretic constraints—with explicit, finite-sample control of error probabilities. The supporting statistical machinery, namely p-values, e-values, and concentration inequalities, is developed from first principles.

This work presents a transfinite version of the bar-recursion in the context of classical realizability models for set theory. Bar-recursion has been previously used to obtain realizability interpretations of countable choice and dependent choice, and was employed by Krivine to realize the continuum hypothesis in classical realizability. In this paper, we introduce a transfinite variant of bar-recursion and use it to construct realizability models validating uncountable fragments of the Axiom of Choice. Moreover, our construction reveals that this generalized bar-recursion is related to preservation of cardinals. To show this, we define an analogue of the forcing notion of $\kappa$-closure for classical realizability algebras that we call $\kappa$-fully-closed. We show that, in realizability algebras satisfying the $\kappa$-full-closure property, generalized bar-recursion realizes that any cardinal up to $\kappa$ admits a representative in the realizability model which remains a cardinal.

In two papers from 1995 P. Köhler and G. Nikolov showed that Gauss–type quadrature formulae associated with spaces of spline functions with equidistant knots are asymptotically optimal in certain Sobolev classes of functions. In particular, Gauss–type quadratures associated with the spaces of spline functions of degree $r-1$ with double equispaced knots are asymptotically optimal definite quadrature formulae of order $r$ when $r$ is even, and it is conjectured that the asymptotical optimality property persists also in the case of odd $r$. For $r=3,\,4$, these quadrature formulae have been constructed by G. Nikolov, who also proved estimates for their error constants. The aim of this note is to refine the estimates for the error constant in the case $r=3$, and to point out to some error estimates in both cases $r=3$ and $r=4$, which are easier to evaluate and could be sharper than those which involve the uniform norm of the $r$-th derivative of the integrand. \medskip \noindent Keywords and Phrases: Spline functions, monosplines, Peano representation of linear functionals, definite quadrature formulae, error estimation of quadratures, Bernoulli polynomials.\medskip \noindent Mathematics Subject Classification 2020: 41A55, 65D30, 65D32.