Papers

Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.

We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton $(M,g,J,f)$ and a real-valued pluriharmonic function $u$, we investigate conditions under which $u$ must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that $u$ is constant whenever $$ \int_M|\nabla u|^p\mathrm{d}v<\infty $$ for some $0<p<\infty$. In the shrinking case, we prove the same conclusion for $0<p\leq 2$. Finally, we construct a complete Kähler example showing that the extension to the range $0<p<1$ relies essentially on the soliton structure and does not hold on general complete Kähler manifolds.

We show that nuclear norm-attaining operators (resp.\ polynomials) are always $w^*$-dense in the space of integral operators (resp.\ polynomials). Besides, the denseness is in norm if the predual space does not contain any isomorphic copy of $\ell_1$. We also show that there are reflexive spaces for which the set of projective norm-attaining elements does not coincide with the whole projective tensor product (which is indeed also reflexive here). Next, we show that if $Y$ is a II-polyhedral space, then every nuclear operator from an arbitrary space $X$ to $Y^*$ attains its nuclear norm. As a consequence, if $X^*$ or $Y^*$ has the approximation property, then the set of norm-attaining operators from $X^*$ to $Y^{**}$ is dense. Finally, we study proximinality results of a natural subspace of the projective tensor product and obtain an application to integral projective norm-attaining tensors which solves a proposed open question.

The soybean pod borer (Leguminivora glycinivorella) poses a severe threat to global soybean production.In this study, we developed a stage-structured host-parasitoid dynamic model that explicitly couples the holometabolous life cycle of the pest with the obligate egg-parasitism mechanism of Trichogramma wasps. Utilizing field monitoring data from Changchun, Jilin Province, key biological parameters were rigorously estimated via the Markov Chain Monte Carlo (MCMC) method.This calibration facilitated the establishment of a precise Economic Injury Level ($Q_{EIL}$) of 0.0389 individuals/$m^2$, based solely on the destructive larval stage. Through theoretical and numerical analyses of different intervention scenarios, we identified an optimal continuous release rate ($C^* = 2.645$) that efficiently suppresses the outbreak without causing wasteful parasitoid accumulation. Furthermore, simulations demonstrate that a 5-day impulsive release interval provides the optimal balance between strict pest suppression and field operational costs. This study bridges the gap between theoretical population dynamics and applied agricultural management, providing a directly applicable mathematical decision-making tool for the precise biological control of crop pests.

Let \(q\) be an odd prime power. For \(\mu\in\mathbb F_q^\times\) and an additive coset \(\bar\alpha\in\mathbb F_{q^2}/\mathbb F_q\), consider the family \[ \{x^2+\mu x-\alpha:\alpha\in\bar\alpha\}, \] where each polynomial is regarded over its coefficient field \(\mathbb F_q(\alpha)\). We prove that, for \[ q\notin{7,11,13,19,29,31,41,43}, \] every such family contains a primitive polynomial. For the zero coset, the result follows from Cohen's prescribed-trace theorem. For a nonzero coset, after a natural normalization the roots are parameterized by two smooth projective conics arising from the two \(q^2\)-Frobenius eigenspaces in \(\mathbb F_{q^4}\). Their affine \(\mathbb F_q\)-point counts, \(q-1\) and \(q+1\), correspond respectively to the reducible and irreducible members of the family. On the irreducible root conic, \(q^2\)-Frobenius induces a fixed-point-free involution on rational points. Passing to the quotient conic allows the relative norm of the root function to descend to a rational function. Tensor induction then yields order-sensitive character-sum bounds with constants \(6,8\) on the root conic and the sharper constants \(2,4\) after norm descent. Combining these estimates with a double-core prime sieve and an exact residue-cover verification completes the finite range. As consequences, Gow and McGuire's Conjecture~1 holds for every odd prime power \(q\ne13\), while their Conjectures~2 and~3 hold for every odd prime power \(q>43\); the threshold \(43\) is sharp.

In this paper, we study portfolio utility maximization problem in a setting where the risky asset is driven by a multidimensional Brownian motion and an independent homogeneous Poisson random measure, and where strategies may incorporate jump signals. Following the same approach as in the one-dimensional case for the exponential utility function [17], we first represent the portfolio dynamics as semimartingale processes. We then use martingale optimality principle to derive the corresponding backward stochastic differential equation (BSDE) with jumps to characterize both value function and an optimal strategy in terms of its solution. We subsequently prove the existence and uniqueness of the solution to the related BSDE with jumps. We also address the idiosyncratic case i.e where the investor receive heterogeneous jump signals. Finally, we provide numerical illustrations with Gaussian signals for the logarithmic case.

We quantize the trigonometric spin Ruijsenaars-Schneider model of $N$ particles each with $\ell$ spin states using the recently developed description of the classical model in terms of the $K$-theoretic Coulomb branch of the 4d $\mathcal{N}=2$ quiver gauge theory for the necklace quiver with $\ell$ nodes of rank $N$. The main algebraic tool is an algebra of $L$-operators derived from abelianized monopole operators of minuscule charge, which turns the necklace quiver into an integrable spin chain by producing a family of commuting Hamiltonians. We show that the lowest Hamiltonian coincides with the first mode of the quantum determinant of the horizontal quantum loop algebra living inside the $K$-theoretic Coulomb branch algebra, whose Bethe subalgebra generates a maximal family of commuting Hamiltonians. Finally, we derive the commutation relations and quantum equations of motion of the quantized physical spin variables.

We establish global exponential turnpike properties for quadratic optimal tracking problems governed by the one-dimensional viscous Burgers equation with localized internal control. For every initial datum, finite-horizon optimal solutions approach the unique optimal periodic regime when the periodic tracking target is sufficiently small; the zero-target case yields a global steady turnpike at the origin, with no smallness assumption on the initial datum. To our knowledge, these are the first global exponential turnpike results for the viscous Burgers equation. The proof combines a local exponential turnpike, obtained through strict convexity and periodic Riccati theory, with a parabolic dissipation argument that provides an absorbing time independent of the horizon.

We investigate the spectral problem arising in the asymptotic analysis of vibrations of open-book structures with a mass density perturbed near the binding. The limiting problem is go\-ver\-ned by a non-self-adjoint block operator matrix coupling the macroscopic and microscopic components of the model. We describe the spectrum of this operator and completely characterize its eigenspaces and root subspaces. We further prove that generalized eigenvectors form chains of length at most two and derive an explicit criterion for the existence and number of Jordan blocks. This model provides a nontrivial example of a family of self-adjoint operators acting in varying Hilbert spaces whose limiting spectral behavior is described by a non-self-adjoint operator with a genuine Jordan structure.

Extending a recent result from [Bull. Belg. Math. Soc. Simon Stevin 33 (2026), 138-144], we prove that the space of continuous functions $C(X)$ on any countable compact space $X$ admits an isometric copy in $C([0,1])$ consisting, except for the zero function, entirely of Besicovitch functions, i.e., functions that have no one-sided derivative (finite or infinite) at any point.

It is well known that Newton's method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.

It is shown that intersections occur in the dragon curve up to an unfolding angle of 95.828{\deg}. To this end, two specific edges in the 18th iteration are examined.

We prove in detail how to construct the parametrix of a parameter-dependent family of pseudodifferential operators, appearing in the analysis of time-fractional partial differential equations. In particular, we perform a precise study of the dependence from the parameter of the corresponding asymptotic expansion terms, as well as of the smoothing remainders. Moreover, we provide some results about the Laplace transform of vector-valued distributions, also appearing in the analysis of time-fractional partial differential equations.

Many properties of classical graphs are defined in terms of subsets of the vertex set. Examples include connected components, which are subsets $X \subseteq V(G)$ such that $X \times X^c$ and $E(G)$ are disjoint, or independent sets, for which $X \times X$ and $E(G)$ are disjoint. Direct generalisations of these definitions to quantum graphs are difficult to achieve, since the natural notion of subsets of a quantum set is much too rigid. As a consequence, approaches to generalising these classical properties to the quantum setting have been eclectic. In some cases, multiple inequivalent definitions of the same notion are in use. We introduce a natural and well-motivated alternative definition of subsets of a quantum set. Building on this, we propose unified definitions of quantum graph properties as straightforward generalisations of the classical definitions. We recover this way the established notions of colourings and connected components. For independent sets and cliques, our approach suggests variations that diverge from existing definitions, but address some of their counterintuitive properties. We nevertheless show how to recover an important existing definition of independent sets in our framework.

By examining the relationship between the support function of convex geometry and the Euler Characteristic Transform (ECT) of topological data analysis, we develop new tools and suggest variations on some common ECT pipelines. Specifically, we put forward new definitions of ECT-induced pseudodistances, which have the advantage of being invariant under common euclidean isometries and require no cutoff parameter to compare shapes with distinct Euler characteristic. These definitions rely on a generalization of the convex geometric concept of the Steiner point, which we define in general as a distinguished point given by the ECT. We also show how convex geometry provides a path to recover interesting geometric information of a flat shape from its ECT, namely, its perimeter, for which we give an explicit formula. By building on these concepts and leveraging persistent homology, we define Steiner barcodes as an isometry invariant feature of shapes, as well as homological variants of the support function and Steiner point. Finally, we put these constructions to the test in shape classification tasks, providing lightweight features for aligned and misaligned datasets.

Reconfigurable intelligent surfaces (RISs) are a promising technology for improving the spectral and energy efficiency of future wireless networks, which make use of metasurfaces. However, optimizing RIS configurations typically leads to large-scale, non-convex problems whose complexity grows significantly with the number of scattering elements and the adoption of advanced metasurface architectures. In this paper, we develop a stochastic optimization framework for RIS-aided wireless networks based on continuous versions of the ($a$) cross-entropy (CE) and ($b$) Metropolis-Hastings (MH) methods. Unlike existing stochastic approaches that mainly focus on discrete optimization, the proposed framework directly handles continuous variables and can be readily applied to discrete settings through relaxation and projection. We provide a theoretical characterization of the proposed algorithms, including convergence guarantees and efficiency analysis. The framework is applied to ($i$) achievable-rate maximization with nearly-passive RISs and ($ii$) energy-efficiency maximization with active RISs. Numerical results show that the proposed methods achieve performance comparable to, or better than, state-of-the-art deterministic algorithms, while reducing execution times up to 10 times in representative scenarios.

In 2012, Kaufman and Lubotzky constructed the first family of symmetric LDPC good codes. Their construction used Cayley codes, as originally defined by Kaufman and Wigderson (2016). In this paper we present two generalisations to the Cayley code construction, which we call graph codes and digraph codes. We investigate both the algebraic, and combinatorial properties of these constructions and show that they possess the same desirable attributes as Cayley codes, but with added freedom. We analyse the relationship between the expansion properties of the ingredient (di)graphs and the parameters of the constructed codes; our analysis offers an improvement to the results of Kaufman and Lubotzky. As an application, we construct an infinite family of good digraph codes, and we propose a series of open problems.

Although neural-based machine learning models have received a lot of attention recently, tree-based models such as gradient boosting are competitive for tabular data and therefore remain widely used in various applications of AI. As when using other machine learning predictive models, they can however yield discriminative predictions across demographic groups, due to so-called algorithmic biases. These undesirable phenomena have motivated the emergence of new regulatory frameworks and various AI fairness strategies. While several pre-and post-processing methodologies exist to mitigate such bias on gradient boosting models, only a few in-processing methods have been proposed. To bridge this gap, we introduce OT-FairBoost, a novel in-processing framework that incorporates a Wasserstein-2 distance penalty directly into the objective function of gradient-boosted trees. This OT-based mitigation strategy has been shown to efficiently optimize group fairness criteria such as Demographic Parity and Equalized Odds on neural-based predictions. To adapt this approach for gradient boosting, we extend the sample-wise gradient estimation of the Wasserstein-2 distance between group predictions to discrete distributions and hessian diagonals. We then integrate our approach into the LightGBM training procedure and evaluate it across binary classification, regression, and multi-group sensitive attribute settings. Experimental results in each of these settings demonstrate that OT-FairBoost achieves best accuracy-fairness trade-offs against alternatives.

We give a homological proof of the famous Gorenstein Symmetry Conjecture, which asserts that one-sided finiteness of the self-injective dimension of an Artin algebra is sufficient for Gorensteinness.

Novel findings on nanostructures in semi-metal and semi-graphene materials are discussed regarding rectangular quantum dots with zigzag edges, MIT bag models, and Dirichlet boundary conditions. The article theoretically investigates some models of quantum dots for these hybrid materials and the point spectra of the semi-Dirac semi-Laplacians via one-dimensional Laplace and Dirac operators. We also prove the self-adjointness of the semi-Laplacian defined on a rectangle, subject to zigzag and hard-wall boundary conditions, and study its spectral properties.

We describe infinitesimal deformations of post-Lie algebras and post-Hopf algebras and prove that the universal enveloping algebra and primitive elements adjunction is compatible with the infinitesimal structure. In characteristic zero, this restricts to an equivalence for connected cocommutative infinitesimal post-Hopf algebras, extending the Cartier–Milnor–Moore theorem. We parametrize all infinitesimal post-Lie structures on $\mathfrak{sl}(2)$ and exhibit a second-order obstruction. From a geometric point of view, infinitesimal post-Lie algebras emerge from flat connections with covariantly constant torsion and an additional compatible symmetric $(1,2)$-tensor. This leads to explicit examples of infinitesimal post-Lie structures based on Heisenberg Lie algebras. On the Hopf side, we classify infinitesimal structures above the trivial action, prove rigidity for group algebras, and describe the structures on Sweedler's Hopf algebra. The subadjacent construction induces a Hopf algebra deformation with fixed coproduct. Finally, we prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, using a filtered distributive law between the operads of Lie algebras and bi-magmas.

Diffusion processes with reflection, possibly oblique, on the boundary of a given sufficiently regular closed convex domain in Euclidean space, are constructed in a novel way via instantaneous transformations (“foldings") of suitable unconstrained diffusions.

We compute the homotopy type of the classifying space for commutativity introduced by Adem, F. Cohen and Torres-Giese for the group $\SL_2(\Z[1/p])$, both as a discrete topological group and with the subspace topology from $\SL_2(\R)$. Doing so requires compiling detailed information on the maximal abelian subgroups of $\SL_2(\Z[1/p])$, which turn out to be precisely the centralizers of the non-central elements of the group, for which we employ methods from geometric group theory.

We prove that mapping degree sets are stable under multiplication by finite subsets of $\mathbb Z$ containing $0$ and by sets obtained from additive submonoids of $\mathbb Z$ through finitely many sums and products. In particular, every set of the latter type occurs as a mapping degree set. As a consequence, we obtain a broad family of infinite mapping degree sets, including finite unions of arithmetic progressions starting at $0$. These results extend previous work on the realization problem and are related to a question posed by Neofytidis, Wang, and Wang.

We investigate Full Support (FS), a Partizan Subtraction game in which the players can remove any number of pebbles from the heap up to certain bounds that are typically different for Left and Right. The player with the richer move set always wins for all but finitely many heap sizes. We confirm this advantage by finding the general canonical form and the atomic weights of this game. To restore fairness (and peace), we introduce Truncated Support (TS), which essentially trims the larger subtraction set from below. If the truncation is shallow, the unfairness persists above a certain heap size, and one player continues ruling. If the truncation is deep, another player starts ruling. Interestingly, there is one more balanced truncation level in the middle, for which a non-trivial periodicity emerges, and where it has infinitely many $\mathcal P$ and $\mathcal N$-positions. We also explore the atomic weights for the lightly trimmed scenarios.

Spatially distributed service systems rely on state-dependent routing to allocate users, tasks, or requests to less-loaded service nodes. In practice, a routing decision does not take effect immediately: the assigned job reaches the selected node only after a lag caused by travel time, communication latency, or actuation. We call this lag the action delay. Whereas delayed-information models treat such a lag as stale information and analyze the model via a delay differential equation, the decision uses the current state and only its execution is deferred, so the job experiencing an action delay must be tracked as part of the state. Representing the action delay by an Erlang phase structure, we obtain a finite-dimensional Markov jump process and build ordinary differential equations that explicitly track the jobs in the delay phase. Exploiting the two-server symmetry, we reduce the dynamics to a difference mode for the server imbalance and derive its characteristic equation for an arbitrary number of phases. This equation coincides with that of the delayed-information model, showing that the two different delays share the same linearized imbalance dynamics. Numerical experiments confirm the fluid approximation and illustrate how the number of phases, the routing sensitivity, and the mean delay govern the transient response of the server imbalance.

We derive finite-sample generalization bounds for Transformers trained with dynamic programming recursions. Building on the doubly lifted, measure-valued formulation of Transformer dynamics, we view data sets as probability laws on pairs of empirical input-output measures, allowing us to interpret the training problem as a finite-horizon Markovian control problem. We then analyze a quantized model, derived by quantizing the state, action, and measure-state spaces, and derive explicit finite-sample generalization bounds using concentration inequalities for empirical laws on finite metric spaces together with a Lipschitz stability estimate for the value function. These bounds are transferred to the base model at the cost of an explicit approximation error. Finally, we show that the same machinery yields a distributionally robust control formulation of the training problem, connecting Transformer generalization to Wasserstein distributionally robust optimization.

Recently, I. Castro, I. Castro-Infantes, and J. Castro-Infantes introduced a two-parameter family of helicoidal minimal surfaces in $\mathbb S^3$, denoted by $\operatorname{Hel}_c^h$, with the pitch $h\geq0$ and $c\in[0,1/2)$. At $(h,c)=(0,0)$, the surface $\operatorname{Hel}_0^0$ is the totally geodesic sphere, while the limiting surface as $c\to1/2^-$ is the Clifford torus. The subfamily $c=0$, $h>0$, consists of the Lawson spherical helicoids, whereas the subfamily $h=0$, $0<c<1/2$, consists of the spherical catenoids, whose compact members are the Otsuki tori. Castro et al. remarked that it is not an easy problem to determine when $\operatorname{Hel}_c^h$ is a compact surface. In this paper, we resolve this compactness problem, namely we prove that the compact members of the family are characterized by \[ \operatorname{Hel}_c^h \text{ is compact} \quad\Longleftrightarrow\quad \begin{cases} h\in\mathbb Q, & c=0,\\[1mm] h\in\mathbb Q\ \text{and}\ q(h,c)\in\mathbb Q, & 0<c<1/2, \end{cases} \] where $q(h,c)$ is given by an explicit integral. For $0<c<1/2$, every compact quotient surface induced by the parametrization is a torus. For $c=0$ and $h=j/\nu>0$ written in lowest terms, the quotient of the parameter plane by the full automorphism group is a torus when $j$ and $\nu$ are both odd and a Klein bottle otherwise. The Willmore energies of the corresponding compact immersed surfaces are computed explicitly. Along each Lawson associated family of a spherical catenoid, only finitely many parameter values yield compact helicoidal surfaces with Willmore energy below any prescribed bound.

In this paper, we introduce a new aspect of the F-system for the F-method arising in the case where the nilpotent radical is not necessarily abelian. For this, we define the symmetrization operator on the space of polynomial functions on the dual $\mathfrak{g}^\vee$ of a finite-dimensional Lie algebra $\mathfrak{g}$. In the context of the F-method, the conjugation by this operator of the F-system yields a new system, which we call the ordered F-system. These two techniques allow one to consider the symmetrized form (with symmetrization) and ordered form (without symmetrization) of differential symmetry breaking operators (DSBOs) uniformly. As an application of this theory, we classify and construct all the DSBOs between principal series representations for $(GL(3,\mathbb{R}), GL(2,\mathbb{R}))$ in both symmetrized and ordered forms. Here, we consider all embeddings $GL(2,\mathbb{R}) \hookrightarrow GL(3,\mathbb{R})$ corresponding to the positive roots of $\mathfrak{gl}(3,\mathbb{R})$. Furthermore, we utilize the DSBOs for the above pair to investigate differential intertwining operators (DIOs) for $GL(3,\mathbb{R})$ and DSBOs for $(SL(3,\mathbb{R}), SL(2,\mathbb{R}))$. The branching laws of the corresponding Verma modules are also studied to support our results of DSBOs.

We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2s$-stable Lévy operators, where $s\in (0,1)$. We first prove that every positive distributional solution of $-Lu=u^p$ in a punctured domain $D\setminus\{0\}$ satisfies $-Lu=u^p+k\delta_0$ in $D$ for some $k\ge0$, and that necessarily $k=0$ whenever $p\ge d/(d-2s)$. We then study the corresponding Dirichlet problem in which the Dirac mass is replaced by a bounded positive measure, and establish the existence of a critical parameter $k_\mu$: below this threshold minimal positive solutions exist, whereas above it the problem admits no solution. In the symmetric case, we further prove multiplicity below the threshold, as well as existence and uniqueness at the threshold itself.