Papers

Let $w_1, \dots, w_m$ be positive real weights whose sum is $1$, and let $v_1, \dots, v_m$ be i.i.d. Bernoulli$(p)$ random variables. If we let $X=\sum_{i=1}^m w_i v_i$, then we conjecture that for all $0\leq p\leq 1/3$ we have \[\mathbb{P}\big[X\geq \mathbb{E}[X]\big]\geq p.\] In this short note, we observe a connection of this conjecture with a version of the Manickam-Miklós-Singhi conjecture, which allows one to prove it for sufficiently small values of $p$.

One proves that the the vorticity flow of 2D Navier Stokes equation can be identified with an absolutely continuous curve in Wasserstein space W_{p} where p\in [1,2}.

We study the scattering problem for a long range potential, which is time dependent. We prove the existence and completeness of the scattering wave operators, and find some properties of the weakly localized, non-scattering part of the solution. The method we use follows recent methods introduced and applied to short range systems.

We show that list $3$-coloring a~$C_4$-free graph of diameter-$2$ can be done in polynomial-time. Our algorithm is based on a structural characterization showing that many such graphs are not~$3$-colorable. In particular, we show that~$C_4$-free graphs of diameter-$2$ without universal vertices, where the maximum degree is at least~$17$, are not~$3$-colorable.

The aim of this article is twofold: (a) to revisit the blow-up theory of weakly superlinear heat equations; (b) to explore the notion of internal global/regional blow-up controllability for the linear heat equation.

In this paper, we study interacting bosons at zero temperature in a random and higher-dimensional continuum model introduced by Kac and Luttinger. For weak interactions we prove that there is condensation in the lowest eigenstate of the one-particle Hamiltonian (type-I BEC). For strong interactions, however, we show that condensation in a localized state cannot occur. We also prove generalized condensation, where a family of eigenstates of the one-particle Hamiltonian is macroscopically occupied as a whole. Combining these results yields a scenario where there is generalized condensation into a family of eigenstates of the one-particle Hamiltonian, but none of them is macroscopically occupied itself (type-III BEC). This proves a transition in the type of condensation. To the best of our knowledge, this is the first rigorous result in this direction for a random continuum model in higher dimensions.

The Bose-Hubbard system has been studied extensively both theoretically and experimentally, in particular in the context of ultracold atomic gases in optical lattices. Even in the two-mode case the many-particle dynamics display complex interference effects resulting in revival and breakdown phenomena as well as tunnelling. The most basic theoretical description is the mean-field approximation, which can be derived from a time-dependent variational principle assuming the many-particle wave function is an SU(2) coherent state. Here we build on this to construct a simple initial-value coherent state propagator, summing over mean-field trajectories and keeping track of their phases, given by the corresponding mean-field actions. This yields an approximation to the full time-dependent many-particle state, and is able to reproduce breakdown and revival dynamics. Applying a time-slicing procedure on top of this, we are able to accurately capture many-particle tunnelling effects. While in this paper we focus our analysis on the Bose-Hubbard dimer, the methods developed can be applied to more general SU(2) Hamiltonians, and can be extended to SU(M) systems.

We compute invariants of Carrollian spacetimes, deriving them from the geometry of the screen bundle. For generic Carrollian structures we specify how to generate the entire algebra of differential invariants, with emphasis on dimension 3, which has special physical relevance. Then, in the framework of jet-spaces, we compute the numerology behind these invariants: the Hilbert and Poincaré functions that govern their numbers according to order. Finally, we compute the Spencer cohomology behind the Carrollian geometry that, in particular, contains the spaces of intrinsic torsion and intrinsic curvature, which are fundamental invariants, important in the equivalence problem and symmetry analysis. Thus, we also discuss symmetry sizes of Carrollian spacetimes.

Signed measures are traditionally introduced as countably additive set functions that may take both positive and negative values. The classical Jordan decomposition theorem shows that every finite signed measure can be expressed uniquely as the difference of two mutually singular positive measures. While this theorem provides a structural description of signed measures, it does not characterize them by a universal property. We show that, for every measurable space, the abelian group of finite signed measures satisfies a universal property with respect to the commutative monoid of finite positive measures: every additive map from positive measures into an abelian group extends uniquely to a group homomorphism on signed measures. In this sense, signed measures are the canonical additive extension of positive measure theory. We compare this characterization with classical Grothendieck completion, clarifying both the analogy and the additional structure arising from countable additivity and Jordan decomposition. This places signed measures within the familiar framework of additive completion and linearization, providing a conceptual explanation for their role in analysis and probability.

A graph $G$ is $(t+1)K_{r+1}$-free if it contains no $t+1$ pairwise vertex-disjoint copies of $K_{r+1}$. Moon [Canad. J. Math. 20 (1968) 95-102] and Simonovits [Theory of Graphs (Proc. Colloq., Tihany, 1966)] independently determined that, for sufficiently large $n$, $K_{t}\vee T_{r}(n-t)$ is the unique $n$-vertex $(t+1)K_{r+1}$-free graph with the maximum number of edges. In 2023, Ni, Wang and Kang [Electron. J. Combin. 30 (2023) \#P1.20] showed that the graph $K_{t}\vee T_{r}(n-t)$ is also the unique adjacency spectral extremal graph over all $n$-vertex $(t+1)K_{r+1}$-free graphs for sufficiently large $n$. In this paper, for $r\geq 3$ and $t\geq 0$, we prove that $K_{t}\vee T_r(n-t)$ is the unique graph attaining the maximum signless Laplacian spectral radius among all $(t+1)K_{r+1}$-free graphs of sufficiently large order $n$.

In this paper, we study how partial observability and inexact latent-state inference affect reward learning from preferences. To that end, we study preference-based reward learning under partial observability, where the learner forms latent-state estimates using an inexact learned POMDP model, so model error can accumulate over time. For finite log-linear POMDPs, we characterize this error term by establishing the stability of the belief filter to parametric model error under certain mixing conditions, yielding bounds on the belief mismatch in expectation and in high probability. We further extend this stability mechanism beyond the log-linear setting to neural-softmax POMDP models with overparameterized neural networks. We then propagate these errors into trajectory-level feature perturbations and derive finite-sample guarantees for constrained Bradley-Terry reward estimation from preferences. Our results decouple statistical error from an irreducible model-mismatch bias, and clarify when preference-based reward learning remains feasible under partial observability with imperfect dynamics.

Boolean networks (BNs) are finite models of interacting systems whose long-term behaviour is organised by attractors. After a transient phase, every synchronously updated BN reaches either a fixed state or a cyclic attractor. Therefore, deciding whether a prescribed cyclic behaviour exists is a basic computational task in finite-state network dynamics. Here we study this task for networks with restricted local rules: each node is updated by a Boolean rule from a fixed closed rule family, and the period is fixed in advance. We prove a complete dichotomy for every fixed period at least two. For each closed rule family, cyclic-attractor detection is either polynomial-time decidable or $\mathrm{NP}$-complete. The intractable cases are precisely those that can implement majority-like self-dual behaviour or mixed monotone conjunction-disjunction mechanisms. In the tractable cases, affine, purely conjunctive, or purely disjunctive structure reduces the problem to linear algebra or graph reachability. These criteria identify which restricted local rule families preserve computational tractability in finite Boolean network models.

In this paper, we focus on the four-dimensional Thurston geometries whose isometry groups are four-dimensional, namely $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}^4$ and $\mathrm{Nil}^4$. We classify homogeneous hypersurfaces in the above three manifolds.

We extend the notion of computads for weak \(\omega\)-categories to allow marking certain generators as invertible, and describe inductively the free \(\omega\)-categories they generate. This gives a simple, finite description of the walking equivalences, the \(\omega\)-categories classifying invertible cells. We then construct a coreflection from generalised to ordinary computads, preserving the generated \(\omega\)-categories, and conclude that \(\omega\)-categories generated by generalised computads are cofibrant. Finally, we study the subcategory of generalised computads and generator-preserving morphisms, and show that it is a presheaf topos, similarly to the case of ordinary computads.

We investigate a regularized Newton method for unconstrained convex multi-objective optimization with twice continuously differentiable objectives whose Hessians are Lipschitz continuous. At each iteration, the method minimizes the quadratically regularized max-envelope of the local quadratic models. Using a Tanabe-type merit function, we prove that this merit decays at the global asymptotic rate $o(1/k^2)$ under the compactness assumption on the initial component-wise lower level set. This result also covers the single-objective case as a special case. Finally, we construct an explicit one-dimensional convex bi-objective family showing that no uniform merit estimate of order $\mathcal O(k^{-(2+\delta)})$ can hold for any fixed $\delta>0$. Thus the exponent $2$ is essentially sharp in the uniform polynomial sense, despite the $o(1/k^2)$ decay on each fixed trajectory.

Transport networks often lose functionality through gradual degradation in link operating conditions before topological disconnection occurs. Link-centred and binary percolation measures identify important facilities or connectivity failures, but they provide limited information on which spatial areas cause the largest loss of network-wide propagation capability. This paper develops a Dynamic Propagation Connectivity (DPC) metric based on quantum percolation for critical-area identification in transport networks. Time-varying link travel times are converted into continuous propagation strengths, which define a Hermitian propagation operator at each observation time. Candidate regions are then evaluated by a regional degradation experiment that measures the resulting loss of DPC. The method is applied to a benchmark Sioux Falls network and six Florida road networks during the post-Hurricane Irma disruption and recovery period, using 1,281 five-minute observation times. The benchmark confirms that the regional DPC score identifies a predefined structurally critical corridor. In the Florida networks, the identified critical areas differ from regions selected by link count, local degradation, edge betweenness, algebraic connectivity, and classical percolation. In Networks 1 to 4, DPC and classical percolation rankings have negative Spearman correlations, showing that continuous propagation degradation and binary fragmentation reveal different vulnerability patterns. Robustness tests under alternative travel time scaling, degradation strength, and grid size show stable results, with mean rank agreement between 0.84 and 0.96. The findings extend transport resilience analysis based on percolation from binary connectivity loss to continuous propagation degradation and provide a spatial diagnostic tool for regional monitoring, emergency planning, and recovery prioritisation.

Perel'man in 1987 and independently Pukhov in 1979 proved that the quotient between the $(n-i+1)$-th successive outer radius and the $i$-th successive inner radius of a convex body in $n$-dimensions is not larger than $i+1$. Apart from the solved cases by Jung 1901 $(i=1)$ and Steinhagen 1921 $(i=n)$, only Perel'man (1987, $n=3$, $i=2$) and González Merino (2017, $n\geq 4$, $i=2$ and $i=n-1$) provided small improvements that beat this bound. In this paper, we obtain sharper inequalities using relations between these inner and outer measures with the diameter and minimal width. We improve the current bounds in the following cases: $i=3$ when $4\leq n \leq 8$, $i=4$ when $n=5$, $6$, $i=5$ when $n=6$, $i=6$ when $n=7$, and for every $i\geq n-\Theta(\log n)$. Notably, our bounds provide the right order in $n$ when $i=n-m$, with $m$ constant and $n$ arbitrarily large. Additionally, we improve the case $n=5$, $i=3$ even further by refining an idea of Perel'man and using the optimal lower bound of the inradius in terms of the circumradius and the diameter in 3-space (see [7]).

We determine the continuous mod $p$ homology of the topological periodic homology $TP(MU)$ of the complex cobordism spectrum, as a graded algebra with Steenrod operations. The answer is given in terms of an explicit and purely algebraic construction $C_+$, analogous to Singer's construction $R_+$. Its $Ext$-algebra provides the $E_2$-term for a multiplicative Adams-type spectral sequence converging strongly to the homotopy of $p$-completed $TP(MU)$.

The stabilizing effect of a background magnetic field on electrically conducting fluids has been rigorously established for the standard MHD equations. This paper extends this theory to the more physically accurate damped wave-type MHD equations, where the induction equation is hyperbolic-parabolic and the velocity field has only vertical damping with no dissipation. These two features make the stability analysis harder than in the standard MHD setting. To overcome these difficulties, we design an energy functional exploiting the anisotropic structure, and discover a remarkable cancellation between the two most dangerous nonlinear terms by exploiting the full algebraic structure of the coupled system. As a consequence, we prove that any small perturbation near the background magnetic field is globally stable and establish optimal decay rates consistent with the 2D heat equation. To the best of our knowledge, this is the first rigorous stability result for the damped wave-type MHD equations near a background magnetic field.

In this paper, we study the congruence $\binom{qn}{n} \equiv q^n \pmod n$ for a prime base $q$. Motivated by the OEIS sequence \seqnum{A080469} and the conjectural existence of infinitely many ternary solutions of the form $n=3^t p$, we analyze the more general family $n=q^t p$, where $p\neq q$ is prime. Our main result shows that, in this family, the congruence is equivalent to two independent conditions: a congruence modulo $p$ and an inequality in the sum of the digits. This reduces the search for such solutions to factoring an explicit integer and applying a base-$q$ digit-sum filter. We use this criterion to produce new large solutions for $q\in\{2,3,5,7,11\}$. We also prove that square solutions $n=p^2$ are exactly governed by Wieferich primes in base $q$.

Quantum fields exhibit a rich entanglement structure which is still not fully understood. In this work, we study the entanglement structure of the vacuum state of a massless scalar field in (1+1)-dimensions – a paradigmatic case for both high energy and condensed matter physics. We fully characterize the entanglement negativity between two arbitrary compact spacelike-separated regions of the field by calculating the logarithmic negativity along with the modes carrying it, called negativity cores. We achieve this using a framework based on the Kähler structure of Gaussian states, wherein we calculate the diagonalization of the operator associated with the partially-transposed restricted linear complex structure. In doing so, we extend the methods of this framework by proposing a basis-independent definition of the transpose operation. The explicit diagonalization we perform is enabled by a reformulation of the eigenvalue problem as a boundary value problem in the complex plane. Our results also suggest extensions to higher dimensions and fermionic fields.

Learned reconstruction operators for inverse problems are typically trained under a fixed noise model, and generalize poorly when the distribution during testing differs from the one assumed during training. Distributionally robust optimization (DRO) addresses this by optimizing against the worst-case distribution within a prescribed ambiguity set, but standard Wasserstein DRO perturbs the full joint distribution uniformly, which can be overly conservative and ignores the physics of the measurement process. We develop a structured DRO framework in which the ambiguity set is restricted to structured perturbations aligned with the data-acquisition process. This allows us to learn data-driven reconstruction operators that remain robust to distributional shifts. By constraining perturbations to subsets such as $P(Y|X)$, our framework models uncertainty in the forward operator and noise model more faithfully, accommodating any noise model expressible as a stochastic forward operator. We establish strong duality for this general formulation and derive explicit finite-dimensional dual representations for perturbations in the joint, marginal, and conditional distributions. A central result is an explicit worst-case risk bound that induces Tikhonov regularization on the Lipschitz constant of the reconstruction operator, and is less conservative relative to standard DRO for well-posed problems. Numerical experiments on deblurring and sinogram-to-CT reconstruction demonstrate improved robustness, stability, and interpretability over standard DRO and MSE baselines. In the linear setting, the learned operator becomes effectively low-rank, truncating at the intrinsic dimension of the data and recovering a data-driven analogue of truncated-SVD regularization.

Valid possibilistic inferential models provide exact finite-sample calibration, but validity alone does not determine which valid procedure results in the most informative inferential summary. This paper proposes Choquet risk as a decision-theoretic criterion for comparing valid possibility measures in finite samples. Given a non-negative penalty functional, Choquet loss is defined as the Choquet integral of that penalty with respect to the data-dependent possibility measure, and Choquet risk as its sampling expectation. A key reduction expresses this risk through the nested $\alpha$-cuts of the contour, linking procedure-level efficiency to the expected performance of calibrated confidence sets. For concentration penalties, the criterion reduces to integrated expected set size, equivalently expected contour volume, so levelwise optimal confidence sets induce Choquet-risk optimal valid contours. The framework is developed along two classical routes to optimality. First, a possibilistic notion of unbiasedness is introduced and shown, under validity, to coincide with unbiasedness of the induced confidence sets and tests, allowing UMPU and most-accurate-unbiased results to be transferred to valid contours. Second, an equivariant minimax theory is developed, including a Gaussian-location result in which the Gaussian possibility contour is Choquet-risk minimax for radial distance-to-truth losses. The construction also extends confidence risk from additive confidence distributions to non-additive calibrated inferential-model output, with Choquet loss acting as a least-favourable confidence loss. Finally, the paper clarifies the penalty-dependence of efficiency comparisons and motivates invariant size criteria and divergence-based intrinsic losses connected locally to Fisher–Rao geometry.

Sudoku is a compact and familiar setting for teaching a surprisingly deep lesson in integer linear programming, namely that the most natural decision variables are not always enough to produce an effective or convenient linear model. This paper compares two formulations of Sudoku. The first uses binary assignment variables indicating whether a particular digit is assigned to a particular cell. This formulation is less natural from the perspective of the puzzle board itself, but it encodes the puzzle rules through simple assignment constraints and extends easily to variants such as Killer Sudoku. The second formulation uses the more natural approach of assigning one integer variable to represent the value in each cell, but then the central requirement is that the values in each row, column, and block must be all different. When this all-different requirement must be expressed using linear constraints while retaining the cell-value variables, the formulation becomes a large collection of pairwise disjunctions linearised by big-M inequalities. The paper includes AMPL code for both the assignment model and the natural all-different model, and derives a second integer program for certifying uniqueness. The paper also uses generalised Sudoku as a careful entry point to computational complexity, while emphasising that the standard nine-by-nine puzzle is not itself an asymptotic problem class.

A new, fast adaptive regularization methods is proposed and analyzed under local Lipschitz smoothness of the $p$-th order tensor. For nonconvex problems, it achieves the optimal $\mathcal{O}\!\left(|\log(\epsilon)|\epsilon^{-(p+1)/p}\right)$ complexity to obtain first-order $\epsilon$-stationary points and in the convex case, it yields $\mathcal{O}\!\left(|\log(\epsilon)|\epsilon^{-1/p}\right)$ iterations to drive the optimality gap below $\epsilon$, thus matching the complexity bounds of standard tensor methods under global Lipschitz smoothness yp to logarithmic terms. The proposed algorithm follows the line of standard tensor methods with an appropriately chosen regularization and suitable modifications. Initial numerical experiments and comparisons for some nonconvex regression problems are made with the standard adaptive cubic regularization where we showcase some potential of the proposed method.

We consider the defocusing Wick-ordered cubic fractional nonlinear Schrödinger equation on the two-dimensional torus with dispersion relation $\omega(k)=|k|^\alpha$. In the weakly dispersive regime $\frac{29}{15}<\alpha<2$, we construct global dynamics for almost every initial datum with respect to the associated Gibbs measure as the limit of the finite-dimensional truncated flows and prove invariance of the Gibbs measure. The core of the proof is an almost sure local theory based on the method of random averaging operators (arXiv:1910.08492v2). The main new ingredients are fractional lattice counting estimates and localized random tensor bounds, which exploit the geometric structure of the fractional phase in place of the classical number-theoretic tools available for quadratic dispersion.

Connectivity degradation in transport networks under structural disturbance is a central problem in network resilience research. Existing methods rely mainly on percolation theory and topological connectivity measures. They focus on whether paths exist and whether connected components fragment. These approaches cannot capture functional degradation where network topology remains intact but propagation ability has already declined substantially. This paper introduces quantum percolation theory into transport network connectivity analysis and proposes Dynamic Propagation Connectivity (DPC) as a new measure that characterises network propagation ability under disturbance. By mapping a transport network under disturbance into a propagation operator system, this paper establishes a spectral analysis framework for DPC and defines the time-averaged participation index as its core quantification. This paper provides a series of rigorous theoretical results. DPC remains constant under homogeneous disturbance and degrades under heterogeneous disturbance. This paper establishes a quantitative relationship between the degradation rate, the minimum eigenvalue spacing of the propagation operator, and heterogeneous deviation strength. This paper proves a separation theorem between DPC and algebraic connectivity. It derives an analytical expression for DPC and a second-order perturbation approximation on the ring graph. Numerical experiments on three transport benchmark networks verify all theoretical conclusions and confirm degradation monotonicity, separation from algebraic connectivity, and degradation amplification by network size. This paper provides a theoretical framework for transport network resilience assessment that goes beyond topological connectivity.

In this work, we study the nonlinear orbital stability of solitary-wave solutions for a class of higher-order Boussinesq systems with Hamiltonian structure. Using variational methods and the asymptotic connection with generalized fifth-order KdV equations, we establish orbital stability results for a broad family of homogeneous and nonhomogeneous nonlinearities satisfying suitable scaling assumptions. We also perform numerical simulations to investigate the stability criterion associated with the solitary waves. The numerical results suggest that the range of wave velocities leading to orbital stability may be larger than that predicted by the theoretical analysis.

This paper completely determines the symbol-pair distance distributions of all repeated-root $\Delta$-constacyclic codes of length $4p^{s}$ over the finite commutative chain ring $R_{3}=\mathbb{F}_{p^{m}}[u]/\langle u^{3}\rangle$, where $p^{m}\equiv1 \pmod 4$. The distance characterization is explicitly classified according to the quadratic character of the shift unit $\Delta \in R_{3}^{*}$. When $\Delta$ is a non-square unit, the exact symbol-pair distances are established across all eight distinct ideal classifications of the ambient ring. Conversely, when $\Delta$ is a square unit, the distance profiles are derived by evaluating direct sum decompositions and local ring reductions. By evaluating the symbol-pair singleton bound, we prove that only the trivial ideal $\mathcal{C}=\langle1\rangle$ achieves maximum distance separability (MDS) , as structural constraints rule out any non-trivial MDS configurations. Finally, computational examples of length 20 over $\mathbb{F}_{5}+u\mathbb{F}_{5}+u^{2}\mathbb{F}_{5}$ are provided to validate the derived distance formulas.

We study the two-color distinct-part series \(S_1(q)\), equivalently Andrews' generating function \(v_d(q)\) for strictly concave compositions, and its odd and even companions \(T_o(q)\) and \(T_e(q)\). We determine the coefficients of \(S_1(q)\) modulo \(4\) and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion \(C(q)=(q;q)_\infty T_o(q)\), we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that \(c(n)\) can be nonzero only when \(24n+28\) is represented by \(x^2+3y^2\).