Papers

Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matrix analysis, it plays a central role in the study of eigenvalue inequalities, particularly those arising from classical determinantal inequalities such as those attributed to Hadamard and Fischer in the context of positive semidefinite matrices. A result of Fischer and Holbrook shows that equality in the Hardy–Littlewood–Pólya theorem for non-affine convex functions is closely linked to block structure in the associated doubly stochastic transformations. Motivated by this, we introduce $*$-majorization, a structured extension of majorization that respects prescribed block decompositions of vectors. This framework naturally corresponds to block diagonal doubly stochastic matrices and provides a refinement of the classical Hardy–Littlewood–Pólya and Rado theorem. We show that such transformations are precisely the linear operators that preserve $*$-majorization, and we extend fundamental constructions such as $T$-transforms and convex combinations to this setting. In an application, we study the eigenvalue relations associated with the principal submatrices of positive definite matrices. Classical majorization does not, in general, capture determinantal inequalities such as those of Koteljanskii, whereas $*$-majorization provides a natural framework for structured comparisons of eigenvalue vectors. This leads to new insights into the interplay between majorization theory, determinantal inequalities, and spectral properties of matrices.

xCPS is an xAct tensor algebra package for symbolic computations within the covariant phase space formalism of field theories. From a generic Lagrangian, xCPS automates the derivation of equations of motion and symplectic currents. It systematically determines whether an infinitesimal transformation in the space of fields is a Noether symmetry and computes the associated Noether charge. Additionally, xCPS can in many cases determine whether a tensorial expression is a divergence and, if so, find its divergence potential. By implementing vertical exterior calculus through a graded, supercommutative wedge product and vertical operators, the package enables efficient computations in gauge theories and higher-derivative models of gravity, including the derivation of thermodynamic quantities like Wald's entropy. xCPS is open-source under the GPL license and available at https://github.com/juanmargalef.

Boolean networks are widely used to model gene regulatory systems. Their structural and dynamical properties are commonly interpreted by comparison with ensembles of random Boolean networks generated by sampling Boolean functions for individual nodes. Canalizing and nested canalizing functions, in which one or more regulatory inputs dominate the output, capture an important feature of gene regulation. These functions are typically generated by sampling their defining parameters uniformly at random. Because multiple parameterizations can represent the same Boolean function, however, this procedure induces a biased distribution over functions and consequently over null models. We develop efficient algorithms for uniformly sampling Boolean functions with prescribed canalizing depth, thereby correcting this systematic bias. Using these unbiased null models, we show that the sampling measure substantially alters function- and network-level properties. Whereas parameter-uniform sampling yields nested canalizing functions with expected average sensitivity one, these sensitivities increase with degree under function-uniform sampling and approach 1.183. These differences alter expectations for robustness, attractor structure, and stability. Reanalysis of 122 published Boolean gene regulatory network models reveals substantially stronger enrichment of low-sensitivity canalizing architectures than previously recognized. Widely used parameter-based null models therefore systematically underestimate baseline sensitivity and overestimate the stabilizing role of canalization.

This is an incomplete attempt to show that the upper bound of $\lesssim n^\frac{4}{3}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$ points and $n$ lines that attain the sharp bound of the Szemerédi-Trotter point-line incidence bound.

When does a crisis-induced surge in radical-party support fade away, and when does it become a durable realignment? We address this in a mathematical sociology threshold model on a conserved population. The baseline admits a global classification through a Perron–Frobenius threshold. Adding a crisis-induced disengagement compartment, we separate state shocks (which alter the current state) from structural shocks (which alter parameters). State shocks affect transients but cannot move the long-run attractor; durable realignment requires structural threshold-crossing. We derive a critical shock amplitude and a finite mobilisation-window bound, and show that cumulative structural shifts can produce staircase realignment. A stylised illustration uses German federal elections, 2013–2025.

We prove sharp Heron-type majorization inequalities for two quadratic matrix expressions associated with the spectral and Kubo-Ando geometric means. For the spectral geometric mean cross term, we show that \[ \lambda\bigl(a^2A+b^2B+c(A\natural B)\bigr) \prec_w \lambda\bigl(W_{a,b}(A,B)\bigr), \qquad 0\le c\le 2ab, \] where $W_{a,b}(A,B)$ is the weighted Bures-Wasserstein expression. The coefficient $2ab$ is sharp, and at this endpoint the weak majorization becomes majorization. For the Kubo-Ando geometric mean, we prove the direct comparison \[ \lambda\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w \lambda\bigl(W_{a,b}(A,B)\bigr). \] This settles, in the two-variable setting, Bhatia's question of whether the Heron-type norm inequality of Bhatia-Lim-Yamazaki admits a weak-majorization refinement. More precisely, we prove \[ \lambda\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w \lambda\bigl((aA^{1/2}+bB^{1/2})^2\bigr), \] and consequently obtain the corresponding inequality for all unitarily invariant norms.

The Hamiltonian Path Problem is formulated as a continuous minimization problem on conductances assigned to an Ohmic network associated with the given graph. The objective function is a sum of two penalty terms that collectively enforce a set of conditions sufficient for a subgraph of the original graph to be a Hamiltonian path. The objective function is nonconvex. The main result (Theorem 1) shows that, provided the graph has a Hamiltonian path from $h^{start}$ to $h^{end}$, a conductance configuration is a global minimizer of the objective if and only if it corresponds to a Hamiltonian path.

For an integer $t \geq 1$, a homomorphism of a digraph G to a digraph $H$ is $t$-frugal if no more than $t$ in-neighbours of any vertex of $G$ have the same image. There is a dichotomy theorem based on structural properties when $t=1$ and $H$ has a loop at each vertex, but there is unlikely to be such a theorem for general digraphs $H$. For $t \geq 2$ we describe a structural dichotomy theorem for $t$-frugal homomorphisms of general digraphs.

Young tableaux are fundamental objects in algebraic combinatorics and representation theory, with operations such as promotion and jeu de taquin playing a central role in their structure and applications. While these operations are well understood for finite tableaux, their behavior on infinite tableaux has so far been studied mainly within probabilistic frameworks. In this paper, we investigate jeu de taquin on infinite standard Young tableaux from a purely combinatorial and dynamical point of view. We analyze the action of jeu de taquin on infinite shapes, describe the structure of inverse images, and classify tableaux exhibiting periodic, pre-periodic, and recurrent behavior. We also introduce a natural metric on the space of infinite tableaux and show that jeu de taquin defines a chaotic dynamical system in the sense of Devaney. These results extend classical tableau theory to infinite settings and identify connections between combinatorial dynamics and infinite representation-theoretic structures.

In this paper, we study crossing limit cycles of planar discontinuous piecewise differential systems separated by a nonregular switching line, where one subsystem is a linear differential center and the other belongs to one of six families of Hamiltonian nilpotent saddle systems. This work extends previous results for discontinuous piecewise differential systems separated by a straight line to the more general setting of nonregular switching boundaries. The problem is closely related to the second part of Hilbert's 16th problem for discontinuous piecewise differential systems. We prove that the systems under consideration admit at most seven crossing limit cycles. Moreover, for each family of Hamiltonian nilpotent saddle systems, we provide explicit examples exhibiting four crossing limit cycles.

This paper develops a unified explicit solution theory for optimal execution through sequential limit-order placement in a limit order book. Rather than controlling only the trading speed of a metaorder, we determine how individual limit orders should be quoted over time. The model incorporates signal-dependent drift, price impact, inventory risk, and execution risk, with fills modeled by point processes whose intensities depend on the submitted quotes. We formulate four execution criteria: expected terminal wealth, expected terminal wealth with running inventory penalty, CARA utility of terminal wealth, and CARA utility with running inventory penalty. For general price-impact and inventory-penalty functions, we derive the corresponding HJB equations and show that all four problems reduce to a triangular finite-dimensional structure which can be solved explicitly, leading to fully explicit value functions and optimal quotes across all cases. We also prove well-posedness, admissibility, and verification results. The explicit formulas reveal connections between quoting strategies under different criteria, support long-horizon asymptotic analysis, and show numerically that signal-dependent drift can substantially affect optimal execution.

In January 2023, the obligation for petrol stations to display the average fuel price calculated on a regional basis was introduced by the Italian Government. A mean field game model is here proposed to describe the evolution of the fuel price. In particular, each petrol station decides its own price, for the next day, taking account of an estimate of the average price, to be communicated by the Local Authority. The existence and the evolution of a mean field equilibrium are analytically investigated. Some simulations are compared with real data from the Autonomous Province of Trento, Italy.

A limit of a (small) diagram $d : J \to E$ in a complete category $E$ can be thought of as specifying a set of equations involving the objects of $E$. To motivate this intuitively, one can think of each object $d(j)$ as a "variable" and each morphism in $J$ as a "constraint" connecting these variables. If $E$ has an initial object, a natural question arises: does our set of equations have any solution at all? Equivalently, we can ask: is the limit of $d$ initial? In this paper we consider the computational problem that, given finite diagram $d$ in a finitely complete category $E$, asks whether its limit is empty. We construct a fast algorithm (in the sense of parameterized complexity theory) that solves this problem when $E$ is of the form $\mathbf{FinSet}^{J}$ for a finite category $J$ and $d$ is a structured co-decomposition, i.e. a diagram arising from the opposite of the Grothendieck construction of a simple graph.

The genus-g Fuss–Catalan (FC) number counts the number of ways to obtain a genus-g surface by identifying the edges of a pn-gon via p-valent hyperedges. For p=2 these are the genus–g Catalan numbers which are generated as the trace correlations in the Gaussian matrix model (GUE). Here we construct a simple two-matrix model which generates the higher-genus Fuss-Catalan numbers for any p as the coefficients of its 1/N-expansion. We obtain exact sum rules and an explicit formula for the higher-genus Fuss–Catalan numbers which generalises the Harer–Zagier formula to p>2. We discuss the relation of the higher-genus FC numbers to the intersection numbers and the Euler characteristic of the moduli space of spin-p curves.

We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under Hölder regularity assumptions, we prove Lipschitz $L^{2}$ (respectively $C^{0,\alpha}$) stability estimates for optimal transport maps in terms of the $2$-Monge–Kantorovich distance (respectively $L^{p}$ distances) between pairs of source and target densities. When the continuity assumption is removed, we obtain a Lipschitz $L^{2}$ stability estimate for the Brenier potentials in terms of the $L^{2}$ distance between the source and target densities. The proofs rely on a precise characterization of the linear response of the Brenier potential along smooth interpolations of the data, obtained by linearizing the Monge–Ampère equation in divergence form. As a further application of this approach, we derive an explicit formula for the second variation of the quadratic Monge–Kantorovich distance.

For some kernel matrices, low-rank approximations can be quickly obtained via analytic techniques. One important class of analytic methods that has received attention in recent years is based on the use of proxy points. Accuracy analysis for various proxy point methods has often been heuristic in nature, other than for certain special kernels. For more general cases, the methods lack an explicit number or location of proxy points required to yield a particular approximation accuracy. In this work, we carry out new analysis of a proxy point method that is applicable to general complex-analytic kernels. An intuitive way of choosing proxy points is used to show explicit error bounds. Such bounds decay exponentially with regard to the number of proxy points. This also leads to convenient estimates of numerical ranks of relevant kernel matrices. To showcase the utility of this new analysis, we apply it to design a new sublinear-time hierarchically semiseparable approximation method for certain Toeplitz matrices, including ones that frequently arise from real-world applications. This allows, for example, inversion of such matrices with lower computational complexity compared with existing direct methods. Some extensions of these ideas are also discussed.

Entropy functions played a key role in the development of mathematical theory for hyperbolic conservation laws. The notion of entropy, which is intimately connected with symmetry, is an extension imposed on nonlinear systems of conservation laws. In this context, Friedrichs raised the question whether the assumed symmetries can also be derived. We introduce a variational formulation that addresses Friedrichs' question: an entropy function is derived as an extremal object, from which we deduce, rather than impose, the maximum entropy production principle of Dafermos.

We study of the formation of pattern-forming fronts in the presence of a rigidly-propagating parameter ramp which is slowly-varying in space. In the context of the prototypical supercritical complex Ginzburg-Landau equation, we show that not only the leading order front interface, but also the selected spatial wave number is governed by the transition of the ramp between absolute and convective instability. The slow ramp then induces a further delay of the front interface and perturbation of the selected wave number, controlled by the slow passage near a complex fold of strong- and weak-stable eigenspaces. To analyze the behavior near this fold, we perform a multiple scales analysis to predict the higher-order front interface delay in terms of zeros and poles of a complex Airy quotient inner solution. We confirm these predictions with numerical continuation of heteroclinics in the associated traveling wave equation. We also numerically characterize their spectral stability, finding accumulation of eigenvalues consistent with previous results on slow absolute spectrum. We then show the leading-order absolute/convective instability heuristic accurately describes selected wave numbers in an analogous slowly-ramped Swift-Hohenberg equation.

We introduce odd Koschorke classes in odd (K)-theory by using degeneracy loci of self-adjoint Fredholm operators. These classes are characteristic classes analogous to the even Koschorke classes in even (K)-theory. We study two aspects of these classes: their role as obstruction classes and their realization as characteristic classes with real coefficients for odd twisted (K)-theory. On the even side, we introduce generalized Koschorke classes indexed by arbitrary partitions via Cibotaru's notion of a quasi-manifold. These classes form a (\mathbb{C})-basis of (H^*(\mathrm{Fred}_0;\mathbb{C})) and recover the usual Koschorke classes for rectangular partitions. Finally, by analogy with the correspondence between even Koschorke classes and singular vectors in a representation of the Virasoro algebra, we state a result about a representation of a super-Virasoro algebra: each singular vector in that representation has a unique finite expansion in terms of generalized Koschorke classes and generalized odd Koschorke classes.

This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate that in the Riemannian (positive-definite) setting, the associativity of $U$ is an extremely restrictive condition that forces the tensor to vanish identically, thereby recovering the class of bi-invariant metrics. In contrast, in the pseudo-Riemannian setting, we focus on the subclass where $(\mathfrak{g}, U)$ is associative and unimodular. As a primary result, we establish that every connected Lie group endowed with a left-invariant pseudo-Riemannian metric whose $U$-tensor is associative and unimodular is geodesically complete. Finally, we explore the families of 2-step nilpotent and almost-abelian Lie algebras. For the latter, we obtain some rigid structural classifications, showing that the paradigmatic models are the $3$-dimensional Heisenberg algebra with certain (anti)-Lorentzian metrics or a semi-direct extension involving a nondegenerate infinitesimal $\beta$-transformation on the canonical neutral space $W \oplus W^*$.

In this paper we study the large distance asymptotics of small steady solutions of the 3d Navier Stokes equation in exterior domains. It was proved by Korolev and the second author \cite{SverakKorolev} that the leading term is given by the Landau solution, and it was conjectured that the next order term should be $O(1/|x|^2)$ as $x\to\infty$. We confirm that this is indeed the case and we compute the next order asymptotics in terms of eigenvalues of a suitably constructed linearized operator around the Landau solution on the unit sphere. While the decay of some of the terms is precisely $O(1/|x|^2)$, the the decay of other terms is slightly accelerated.

In this work, we introduce Orlicz-Hardy type spaces and Orlicz-Calderón Hardy type spaces on the Heisenberg group $\mathbb{H}^{n}$ and study the relationship between them by means of the Heisenberg sub-Laplacian $\mathcal{L}$. More precisely, we show, under suitable assumptions, that every distribution in the Orlicz-Hardy space $H^{\Phi}(\mathbb{H}^{n})$ can be represented uniquely as the sub-Laplacian of a function in an appropriate Orlicz-Calderón Hardy space. In this way, for any $f \in H^{\Phi}(\mathbb{H}^{n})$, we obtain a uniqueness and solvability result for the equation $\mathcal{L}F=f$.

We study two new classes of inverse problems for a time-switched system in which a fractional wave equation (with Caputo derivative of order $\alpha \in (1,2)$) governs the dynamics on the interval $[0,a)$, and a fractional diffusion equation (with Caputo derivative of order $\beta \in (0,1)$ taken with respect to the switching point $t=a$) governs the dynamics on $(a,b]$. The two problems differ in which part of the transmitting condition at the interface $t=a$ is regarded as unknown. In both cases the overdetermination data consist of a single spatial measurement of the solution at a fixed time $\xi \in (a,b)$. Using the spectral expansion method with respect to the classical Sturm-Liouville eigensystem on $[0,1]$, we reduce each problem to a sequence of coupled scalar Cauchy problems involving the two-parameter Mittag-Leffler function. Explicit series representations for the solution $u(t,x)$ and the unknown interface functions $h(x)$ and $\bar{h}(x)$ are derived. Uniform convergence of the resulting infinite series and their relevant derivatives is established through four auxiliary lemmas, using the decay estimates for the Mittag-Leffler function, integration-by-parts arguments, the Cauchy–Schwarz inequality, and the Weierstrass $M$-test. A uniqueness and existence theorem is stated for Problem~1 under explicit Sobolev-type regularity conditions on the data, with an analogous result for Problem~2.

We consider the magnetic Schrödinger operator in the unit disk with constant magnetic field of strength $b>0$ and magnetic Neumann boundary condition. If $\lambda_1(b)$ denotes its lowest eigenvalue, then we prove that $\lambda_1(b) < \Theta_0 b$ for all $b>0$, where $\Theta_0$ is the de Gennes constant. The proof has two parts, both based on Rayleigh's principle. For large $b$, we use a trial state built from the de Gennes ground state. For the remaining bounded range of $b$, we divide the interval into finitely many overlapping subintervals and, on each of them, choose a trial state from a finite-dimensional space. This reduces the proof to finitely many inequalities between rational numbers.

We formulate a deterministic threshold-safety problem for a reduced compartmental voter-flow model. An exogenous load enters an alienation reservoir; between releases the reservoir recovers exponentially. Near the mainstream baseline the compartmental dynamics have a linear-stability threshold \(\Delta_c\): below this level the mobilised component contracts, while above it transient amplification is possible. The paper introduces an impulse-control layer for this threshold mechanism. The threshold is obtained from the local stability boundary of the reduced dynamical system, the exposure functional is tied to positive logarithmic amplification, and the scalar reservoir model is proved to be a conservative envelope of the nonlinear voter-flow dynamics. This bridge yields explicit safety benchmarks: the single-release exposure and its zero buffer, the complete-relaxation splitting problem with fixed per-release overhead, the finite-recovery constant-peak profile, and the fixed-horizon capacity frontier \(\Delta_c(1+\rho T)\). The scalar recurrence used after the reduction is the familiar leaky-reservoir skeleton also found in multiple-dose pharmacokinetics, fractionated radiotherapy, reservoir operation, and setup-cost scheduling. Its role here is to make the threshold regimes of the compartmental shock-absorption model analytically transparent.

Neural ideals were introduced by Curto, Itskov, et al as an algebraic tool to study neural codes. In this paper, we use the notion of polarization introduced by Güntürkün, Jeffries, and Sun to compute Betti numbers of inductively pierced neural codes. We prove that quadratically generated neural ideals are inductively pierced if and only if they have regularity 2. Further, we demonstrate how the Betti numbers yield information on the number and type of piercings of the original code. This work shows the utility of algebraic invariants of the neural ideal in detecting geometric features of the associated receptive fields.

The scalability and interpretability of message-passing (MP) decoding, such as (quaternary) Belief Propagation, remain open challenges in quantum error correction. Even for surface codes, arguably the first testbed for decoding methods, studies of improved MP decoders have mostly been restricted to small distances ($d \lesssim 19$). Moreover, the mismatch with established message-passing theory limits the decoder's interpretability, making it unclear whether MP decoding can sustain its effectiveness at large system sizes. This work takes a step toward a more principled and interpretable MP decoding framework, with the goal of making MP-based decoding more reliable and bridging theory and practice. We introduce a dilution method, which allows a quaternary Min-Sum (MS) decoder to exhibit an apparent depolarizing threshold of $16\%$ up to distance $20$, outperforming Minimum-Weight Perfect Matching in finite-length regimes. Notably, for $X$-noise, the standard MS decoder under dilution has worst-case complexity $O(N \log^2 d)$ and outperforms BP-OSD at $d=65$. The observed $\sim 9\%$ threshold may correspond to a true asymptotic threshold. Finally, we give a graph-dilution argument that interprets the success of the dilution method and offers insight into when MP algorithms can genuinely scale. Taken together, these results provide encouraging progress toward scalable and interpretable MP decoding in quantum error correction.

Ordinary differential equation models of biochemical reactions are often formulated as stoichiometric systems in which the dynamics arise from a collection of interacting processes. A central challenge is that the functional form of each process is rarely known a priori and may be difficult to infer from data. We propose biochemically informed neural ordinary differential equations (BINODEs), a neural-ODE framework that retains the stoichiometric structure of mechanistic models while representing individual processes by neural networks. In BINODEs, the outputs of neural network processes (NNPs) are mapped to state derivatives through a linear layer analogous to a stoichiometric matrix. This architecture allows biological side information, such as process-specific inputs, sign constraints, and monotonicity assumptions, to be built directly into the model. We characterize the approximation properties of NNPs for several standard biochemical rate laws and show that the proposed framework recovers both trajectories and process-level structure in Monod, Lotka–Volterra, pharmacokinetic, and ultradian endocrine models. These results suggest that BINODEs offer a useful compromise between mechanistic interpretability and data-driven flexibility for modeling partially known biochemical or biological dynamical systems.

We prove that the space of symplectic embeddings of $n\geq 1$ standard balls into the standard complex projective plane $\mathbb{C}\mathrm{P}^2$, normalized so that a line has symplectic area $1$, is homotopy equivalent to the configuration space of $n$ points in $\mathbb{C}\mathrm{P}^2$, provided that the sum of the ball capacities is strictly less than $1$. Our techniques further suggest that, for $n=9$, there are infinitely many homotopy types of spaces of symplectic ball embeddings, depending on the ball capacities. Moreover, for each $n\geq 5$, we exhibit capacities for which the embedding spaces are not simply connected, in contrast with the case $n \leq 4$. As an application, we show that, for $n\geq 9$ equal balls of capacity $c<1/n$, the symplectomorphism group of the blow-up has the homotopy type of the stabilizer of $n$ distinct points in $\mathbb{C}\mathrm{P}^2$.

In a landmark paper on arithmetical properties of Lambert series, Erdős proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erdős-Borwein constant. Crandall, in 2012, studied properties of the base-2 expansion of this constant, and left the following as an open problem: Does the string $11$ occur infinitely often in the base-2 expansion of $E$? This open problem was also subsequently noted by Shallit. We succeed in introducing a full proof that solves Crandall's problem in the affirmative. Our proof combines a congruence construction in the spirit of Erdős and an estimate due to Alford, Granville, and Pomerance for the counting function for primes in arithmetic progressions. Our argument was developed through extensive interactions with GPT-5.5 Pro.