Papers

The classical theorems of Kolmogorov–Nagumo–de Finetti and of Aczel–Maksa characterize quasi-arithmetic means from two complementary directions: the former for compatible families of means satisfying the replacement axiom, and the latter for bisymmetric means of fixed arity. We refine both representation results by showing that the required continuity follows automatically. Our main result states that every reflexive, symmetric, bisymmetric and partially strictly increasing $n$-variable operation on a real interval is continuous and hence quasi-arithmetic. The proof is based on a recursive construction on $n$-adic rationals given by bisymmetry, and a dense-domain continuity argument. The same method also yields the regularity-free Kolmogorov–Nagumo–de Finetti theorem for compatible families of strictly increasing symmetric means.

The Analytic Hierarchy Process (AHP) and other classic Multi-Criteria Decision Making (MCDM) techniques excel when decision makers can provide consistent pair wise judgments. Real world problems, however, often involve inconsistent, n-wise, or non-linear preference structures that render traditional methods inadequate. The {\alpha}-Discounting MCDM ({\alpha}-D MCDM) extends AHP by embedding a global discounting parameter {\alpha} that transforms an inconsistent system of preference equations into a solvable algebraic system.

A multiset is a 'set' in which elements may occur multiple times. These structures are ideal for expressing the outcome of an election, for instance of the form 60 'yes' and 40 'no'. Moreover, multisets are a useful datatype in vote counting algorithms. This will be illustrated in three different forms of vote counting, known as: 'instant-runoff', 'De Borda', and 'single transferrable vote'. The relevant abstract properties of multisets are: (1) they form a (free) commutative monoid, and (2) they form a functor, and (3) also a monad. This paper illustrates how such categorical properties can be put to good use in deriving and expressing election outcomes. The emphasis is not on the (elementary) category theory involved, but on its application in voting systems.

In this work, we study the symmetric hit problem in four variables by analyzing the relative cohit space, defined as the kernel of the inclusion-induced map from symmetric to ordinary cohits. We establish that this relative space is isomorphic to a kernel in the first group cohomology of the permutation group and, equivalently, to an $\operatorname{Ext}^1$ kernel for the general linear group, equipped with a canonical action of the normalizer quotient. We use explicit Lucas–Cartan identities to construct unconditional fixed Steenrod preimages for nontrivial stabilizers, resolving infinite families of monomials strictly above the minimal-spike weight. Using the ordinary and symmetric Kameko isomorphisms, we reduce the determination of complete bases in two generic degree families to exact calculations in degrees one, three, six, ten, and sixteen. The relative space first becomes nonzero in degree fifteen, where it is one-dimensional, and it is an indecomposable two-dimensional normalizer module in degree eighteen. In the family $5\cdot2^t-4$, the symmetric cohit dimension stabilizes at six and the relative dimension at one for $t\geq2$; in the family $7\cdot2^t-4$, they are four and zero for $t\geq1$. We give complete bases by explicit exponent formulas and prove Janfada's monomial implication [7, Conjecture 1] in every degree of both families. We extend the known optimal bound for ordinary Kameko isomorphisms to every permutation-invariant cohit space, prove its optimality there, and transport the entire symmetric hit basis by a direct-sum formula.

Clustering is a central tool for discovering latent structure in unlabeled data; yet modern clustering pipelines often end with a hard assignment of each observation to a cluster without rigorous measures of assignment uncertainty. We propose a novel weighted conformal approach for constructing valid confidence sets for cluster labels. The key difficulty is that the labels available for calibration are not observed ground-truth labels, but synthetic labels produced by a data-dependent clustering algorithm. Our method develops a conformal inference algorithm that corrects the resulting mismatch with the latent target labels through weights by formulating conformal clustering as a conditional label-distribution shift problem. We first derive an oracle procedure that attains finite-sample marginal coverage and then develop a computationally tractable and implementable version using estimated conditional label probabilities and novel augmented calibration. We show that the coverage of the estimated-weight procedure depends on the estimator, giving an explicit bound on the loss relative to the nominal level. Empirical studies demonstrate that the proposed weighted approach offers improvements over the recently proposed split conformal clustering procedure in terms of informative confidence set size, especially in nonlinear and high-dimensional clustering applications.

We construct explicit isomorphisms between three models for the $B(\infty)$ crystal in type ${\tt A}_1^{(1)}$: affine Mirković–Vilonen polytopes, Kleshchev multipartitions, and a new model we call upper ledge diagrams. We also present some clarifying results on these crystals, giving a direct method for completing an affine MV polytope from the data of one of its boundary root partitions, and a non-iterative recognition theorem which characterizes Kleshchev multipartitions in type ${\tt A}_1^{(1)}$. We apply these results to the representation theory of KLR algebras, where they yield a combinatorial dictionary between cuspidal- and cellular-theoretic frameworks, along with some augmented branching rules for real root functors of induction and restriction.

Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $\lambda$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\lambda$. They conjectured that pre$_j$ is injective on the set of partitions of $n$ with length $\ell \geq j$. The $\ell = j$ case was disproved by Devnani and Eyyunni; they instead conjectured the statement to be true for $\ell > j$. In this article, we answer this refined conjecture in the negative by proving that pre$_j$ is not injective on partitions of $n$ with length $2j$ for $j \geq 3$. We also prove that the analogous map prh$_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions.

Global parameter identifiability is a property of a parametric ODE model to recover the parameter values uniquely from the input-output data. Not all parametric ODE models have this property, and checking for parameter identifiability is a prerequisite to perform numerical parameter estimation. There are many algorithms and software packages for global parameter identifiability, and frequently the runtime is large. However, the computational complexity for this problem has not been analyzed yet, though there are complexity results for local (finitely many values fit the data) parameter identifiability. In this paper, we estimate the complexity of checking global parameter identifiability over real fields for ODE models that depend linearly on the state variables and rationally on the parameters. In particular, we prove that it is equivalent to the injectivity problem.

We prove that the set of positive integers having between $1$ and $n$ prime factors, counted with multiplicity, has degree of accessibility $2^n$. In particular, the case $n=1$ answers a question of Landman and Robertson asking whether the set of prime numbers is $2$-accessible.

This paper introduces a new framework for quantum simultaneous information and power transfer (QSIPT), enabling the joint use of quantum states for classical information and energy transfer in quantum communication systems. We propose a model in which quantum states simultaneously transmit classical information through a quantum channel and transfer energy to an energy harvesting (EH) receiver. The trade-off between communication rate and harvested energy is characterized by the capacity-power function, defined for both discrete-variable (DV) and continuous-variable (CV) quantum channels. For DV channels, we derive properties of the capacity-power function, providing analytical upper and lower bounds for the amplitude damping channel and an exact closed-form characterization for the quantum erasure channel. For CV channels, we extend the framework by introducing a generalized beam-splitter (BS) receiver with adjustable transmissivity, jointly optimized with a transmitter mean-photon-number budget, splitting the channel output between the information decoder and the EH receiver. We analyze the capacity-power and achievable-rate-power trade-off under various Gaussian encoding schemes, including coherent, squeezed, and thermal states, for both lossy bosonic and additive Gaussian noise channels. Closed-form expressions are derived for coherent-state encoding under the joint photon-number-budget and adjustable-transmissivity formulation; squeezed-state inputs are evaluated numerically. Our results show that, within the considered displaced Gaussian encoding class, coherent states achieve the best capacity-power trade-off, squeezed states do not outperform coherent-state encoding under the phase-insensitive channel and passive receiver architecture, and thermal states enable energy transfer without supporting reliable communication.

Simulating large molecular systems comprising thousands of atoms requires highly scalable methodologies. While modern Density Functional Theory (DFT) codes exhibit linear scaling, solving the associated large, sparse generalized eigenproblems remains a critical computational bottleneck on exascale architectures. In the context of the LimitX project, we propose a data-driven framework to accelerate these calculations. By shifting the machine learning target from discrete eigenvalues to the coefficients of an interpolating Chebyshev polynomial, and by comparing both all-atom and fragment-based structural representations, we successfully overcome the dimensionality constraints of large-scale spectral prediction. We investigate three machine learning models (Kernel Ridge Regression, Graph Neural Networks, and Random Forests) trained on a novel 2 TB dataset of protein dimers. The predicted spectra provide initial guesses that effectively bypass early Self-Consistent Field (SCF) iterations in BigDFT. Ultimately, these spectral predictors will be deployed to dynamically optimize upcoming rational filter-based eigensolvers, such as FrASE, which is currently in initial development.

We establish basic properties of the deformed mass and phase functions on the space of stability conditions. We prove that these functions are continuous and deduce that the space of stability conditions admits a homeomorphic embedding into a product space of finite measures. Subsequently, we give a proof of the triangle inequality for deformed mass functions and provide estimates for the deformed mass of truncations of objects with respect to a slicing.

We consider the periodic quintic nonlinear Schrödinger and prove small-mass global well-posedness in $H^s(\mathbb{T})$ for $s>0$. The proof relies on a new derivative-loss-free $L^6_{t,x}$ Strichartz estimate which is established using the high-low method, an asymmetric superlevel set estimate and a new refined broad-narrow argument. Although our $L^6_{t,x}$ Strichartz estimate is not sharp, being valid on slightly shorter time scales than the optimal logarithmic scale, combining it with the $I$-method enables the extension of local solutions to arbitrary times.

High-resolution energy system capacity expansion models (CEMs) for energy transition planning often result in large-scale mixed-integer linear programming (MILP) formulations. Benders decomposition (BD) offers a scalable solution approach by iteratively solving a master problem (MP) for investment decisions and multiple subproblems (SPs) for operational decisions. However, accumulated Benders cuts generated by the SPs can make MP solution a major computational bottleneck. Incomplete SP parallelization can also introduce further bottlenecks when SPs exceed available CPUs. We develop clustering-enhanced BD methods to address these challenges, by using clustering to group similar SPs for: a) aggregated Benders cut construction and b) identification of representative SPs to be solved most frequently. For grouped-cuts, we examine two adaptive formulations based on dual variables and a fixed-grouping formulation based on exogenous time-series inputs. We evaluate these methods in an electricity-sector CEM across varying system sizes, temporal SP lengths, inter-SP coupling strengths represented by CO2 policy, computational resources, and stochastic settings. Relative to a benchmark regularized multi-cut formulation, adaptive grouped cuts outperform fixed grouping and provide substantial benefits under weak inter-temporal coupling. The largest gains occur in larger systems with shorter SP horizons, where the MP accounts for a greater share of runtime. Their effectiveness declines under strong inter-temporal coupling, such as annual CO2 emissions limits, where the benchmark multi-cut performs best. The representative-SP method outperforms the benchmark under limited parallelization when SP solution dominates runtime. Overall, the preferred BD strategy depends on inter-SP coupling strength and whether computational burden lies in the MP or the SPs.

Let $\Bbbk$ be a field, $H$ a colour Hopf algebra and $A$ a graded $H$-comodule colour algebra. We give a sufficient condition for a colour $(A,H)$-Hopf module to be injective as a graded $H$-comodule and we deduce relative projectivity in the category of colour $(A,H)$-Hopf modules. We generalize the Fundamental Theorem of $(A,H)$-Hopf modules to the context of colour $(A,H)$-Hopf modules. Using this result, we show that the categories of graded $A^{coH}$-modules and of colour $(A,H)$-Hopf modules are equivalent, $A$ is faithfully flat as a graded right $A^{coH}$-module and is a graded Hopf-Galois extension of $A^{coH}$. Under some assumptions, we show that $M^{coH}$ is a graded $A$-module and we prove that the graded global dimension of $A$ is equal to the graded projective dimension of the graded $A$-module $A^{coH}$.

The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely $1$-unrectifiable. We extend the two-projection theorem to certain families nonlinear projections and consider applications to pinned distance sets, radial projections, and curve projection operators. Further, we use a multiscale framework to obtain a quantitative version of our nonlinear two-projection theorem. Our arguments utilize methods introduced by Tao, who provided a quantitative treatment of the classic linear two-projection theorem.

Many researchers have proposed their own potential-energy functions for the study of diatomic molecules, some of which have been proved to be equivalent. In this paper we suggest the application of a few simple rules to determine the equivalence between molecular potentials. The resulting systematic procedure is illustrated by means of several examples.

Complementing a theorem of Škrekovski, we characterize the $(h-1)$-critical graphs embeddable in surfaces of Euler genus at least $5$, where $h$ denotes the Heawood number of the surface. Outside of a few small cases, the bulk of our proof is determining the genus of the join of a complete graph and the 5-cycle. As a byproduct of our proof, we also provide a simpler solution to the minimum triangulations problem for nonorientable surfaces using the theory of current graphs.

We consider the problem of generating a large collection of initial guesses for local minima of multimodal non-convex continuous optimization problems. The goal is for these initial guesses to be high-quality (i.e., a numerical solver converges quickly) and diverse (i.e., represent many different local minima). Identifying multiple locally optimal solutions enables flexible downstream decision-making, but typically requires expensive global search. Existing data-driven methods predict initial guesses using only the final converged optima from offline solver runs, which discards information about the local neighborhoods of solutions and limits the available training data. We propose GLENS (Global Search via Learning from Solver Iterates), a data-efficient global search method that leverages intermediate solver iterates as free data augmentation. GLENS consists of two components: a neighborhood structure model that uses diffusion models to learn the local geometry around optima conditioned on problem parameters, and a solver behavior model that learns refinement directions to further guide samples towards nearby optima during diffusion sampling. Experiments on modified non-convex benchmark problems and a two-robot obstacle-avoidance navigation problem show that GLENS generates high-quality initial guesses while preserving the multimodal distribution of diverse local optima. The resulting initial guesses lead to faster solver convergence across different problem settings and solvers. We also analyze how key hyperparameter choices affect the performance.

The task of inductive logic is to develop a formal framework to analyze inductive reasoning. Historically this was accomplished by assigning probabilities to sentences of a logical language. Two natural criteria for such a system are: (i) the underlying language should be rich enough to express scientific hypotheses, and (ii) the probabilities should be, in some sense, accessible. The first criterion suggests that the language should at least contain the language of arithmetic, while the second suggests that probabilities should be computable. We show that these two criteria are in tension with one another. Various natural proposals for an inductive logic result in probabilities that are not arithmetically definable, much less computable. We isolate the assumptions responsible for this result, and search for a weaker inductive logic with more accessible probabilities. The most natural weakening results in probabilities that are arithmetically definable but still are not computable.

We demonstrate that the three standard notions of completeness for Cartan geometries are not actually equivalent, modernizing and extending results due to Yeaton Clifton. In an effort to rectify the folklore on the topic, we also provide detailed explanations of how specious proofs in the literature that would claim that these types of completeness are equivalent break down. In the affine case, we construct geodesically complete homogeneous examples with an inextensible exponential curve for each connected Lie group of dimension at least 2, as well as an infinite-dimensional family of examples whose exponential curves are defined for all time but whose overlying frame bundle is not complete as a metric space with respect to a natural metric for the Cartan connection.

We study empirical $L_p$ moments of a random vector $\pmb\varphi$ based on its i.i.d.\ copies $\pmb\varphi^1,\ldots,\pmb\varphi^m$, that is, $\frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p$. Our main result is a new estimate for the expected uniform deviation \[ \mathbb{E}\sup_{y\in D}\biggl| \frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p -\mathbb{E}|\langle \pmb\varphi,y\rangle|^p \biggr| \] over an arbitrary index set $D$. The proof is based on a new bound for Talagrand's $\gamma$-functional, sharper than the standard Dudley-type entropy estimate. We then apply this estimate to the following two problems. First, for $p>2$, we study Marcinkiewicz-type discretization of $L_p$ norms on an $N$-dimensional subspace $X_N\subset B(\Omega)$ of bounded functions on a probability space $(\Omega,\mu)$. We obtain bounds in terms of the norm of the embedding $ (X_N,\|\cdot\|_{L_p(\mu)})\hookrightarrow B(\Omega). $ In particular, we prove that when this norm is of order $N^{1/p}$ and \[ m \ge C(p)\, N\log N\,(\log\log N)^{p-1}, \] then $m$ random samples suffice to approximate the $L_p(\mu)$ norm uniformly on $X_N$ by the sampled discrete $L_p$ norm. This substantially improves the previously known bound in this setting $ m \ge C(p)\, N(\log N)^{\min\{p,3\}}, $ and is optimal up to the factor $(\log\log N)^{p-1}$ in the random-sampling setting. Second, for $1\le p<2$, we obtain an $L_p$ analogue of the restricted isometry property via random sampling for bounded orthogonal systems and, more generally, for $N$-element systems $\mathcal D_N$ satisfying a Riesz-type condition. We prove that when \[ m \ge C(p)\, s\log N\,(\log s)^2\,\log\log s, \] then $m$ random samples suffice to guarantee an $L_p$ restricted isometry-type property uniformly over the class of all $s$-sparse functions generated by $\mathcal D_N$.

Motivated by the analysis of the behaviour of extremes from multivariate heavy-tailed distributions, we introduce a novel notion of statistical depth, referred to as Polar Depth. The polar depth function is naturally expressed in polar coordinates, as is the limiting distribution of a regularly varying random variable, beyond asymptotically large thresholds, once its marginals have been appropriately normalized. Not only does the polar depth function make it easy to order the extreme values taken by a heavy-tailed random variable X and finds natural applications in anomaly detection, but it is also possible to show, as we prove it under appropriate assumptions in this article, that the polar depth of the largest observations, i.e. observations X which norm is larger than t>0, converges to the polar depth of the limiting distribution as t converges to infinity. Although designed to quantify the depth of multivariate extremes, the polar depth is interesting in its own right, insofar as this notion is more relevant for distributions whose support is included in a halfspace than the alternatives proposed in the literature, the halfspace depth in particular. Here, we demonstrate its properties and analyze statistical issues related to its estimation from both finite-sample and asymptotic points of view. We present numerical results to empirically demonstrate its relevance, particularly for the statistical analysis of extreme observations and more specifically for the identification of anomalies among them.

Perelman's proof of the Poincare conjecture shows that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. The fundamental groups of 3-manifolds attract lots of interest from mathematicians of different fields. As it was stated in a famous survey of Allen Hatcher "The classification of 3-manifolds", one would want to know exactly which groups occur as fundamental groups of these manifolds. The Stallings-Jaco-Hempel reformulation of the Poincare conjecture inspired several connections between low-dimensional topology, equations over free groups, and combinatorial group theory. The reformulation reduces the problem to study epimorphisms from the fundamental group of a closed orientable surface onto the direct product of two free groups (they correspond to Heegaard splittings of 3-manifolds and were named splitting homomorphisms). Olshankii (1989) constructed (in non-explicit form) first non-trivial examples of such splitting epimorphisms and verified the standardness of some of them. We construct up to equivalence all the splitting coordinate-surjective homomorphisms (among them, the genuine splitting epimorphisms are exactly those for which our constructed associated group balanced presentation is trivial). We give generators and relations of the corresponding balanced presentation (so all closed orientable 3-manifold groups) that can be studied by algebraic methods. We analyse a big class of such homomorphisms/presentations (including all Olshanskii's epimorphisms) and show that splitting epimorphisms are rare, in this case the corresponding balanced presentation of the trivial group can be reduced to the standard one by Andrews-Curtis transformations and the epimorphisms are standard.

We study the fluctuations of logarithmic Green singularities in the spherical ensemble, viewed as a random discretization of the two-sphere. Smooth observables exhibit the usual Sobolev or Gaussian free field fluctuations, whereas logarithmic singularities live on a larger logarithmic scale and asymptotically decouple in high-dimension, producing an explicit white-noise limit. The result gives precise asymptotics for logarithmic potentials and characteristic polynomials, with constants expressed through chordal geometry on the sphere.

This work deals with the ergodicity of the stochastic dissipative Hasegawa–Mima equations in a bounded domain driven by additive noise. The uniqueness of the invariant measure is obtained by the asymptotic coupling technique.

Model Predictive Path Integral (MPPI) control is a powerful sampling-based method for solving stochastic optimal control problems and has enabled real-time control in complex robotic systems. Despite its empirical success, its theoretical understanding remains limited. In this work, we show that MPPI can be interpreted as a special case of the Expectation-Maximization (EM) algorithm applied to a probabilistic inference formulation of optimal control. This perspective leads to a generalized EM-MPPI framework that extends MPPI beyond the commonly used Gaussian parameterization. We analyze the convergence behavior of this algorithm and characterize the local convergence rate in terms of the covariance of the posterior trajectory distribution and the exploration distribution. For exponential-family distributions, we establish a sufficient increase property of the log-likelihood when the log-partition function is strongly convex. Specializing the analysis to Gaussian MPPI yields explicit global and local convergence characterizations. The code for the experiments will be available upon acceptance.

We study smooth functions on Euclidean space whose Hessian is positive definite outside a bounded set, with emphasis on products of squared distance functions. More precisely, we first prove a simple convexity principle: if the superlevel region $f^{-1}([c,\infty))$ is contained in the Hessian-positive region of $f$, then the sublevel set $\{f\le c\}$ is convex. We apply this to finite products $F_P(x)=\prod_{p\in P}\|x-p\|^2$, proving that their Hessian-positive complements are bounded. For the two-centre product $F_{p,q}(x)=\|x-p\|^2\|x-q\|^2$ in dimension $n\ge2$, we compute the Hessian-positive region and the exact value \[ h_{max}(F_{p,q})=\frac{\|p-q\|^4}{4}. \] This value is sharp for convexity of sublevel sets in the following sense: we prove convexity above it and nonconvexity below it. This also gives the exact convexity and quasiconvexity truncation levels for the two-centre model.

Over the past half-dozen years, stochastic rounding (SR) has regained significant attention as a quantization scheme for low-precision floating-point arithmetic, with applications spanning numerical analysis and modern machine learning systems. Recent work has shown that SR acts as an implicit regularizer by increasing the smallest singular value of extremely tall-and-thin (or, symmetrically, short-and-fat) matrices. In this work, we substantially sharpen and extend this understanding in two directions. First, we show that the regularization effect of SR is not restricted to extreme aspect ratio regimes: it persists for matrices with constant aspect ratio. Second, we demonstrate that SR does not merely regularize the smallest singular value, but instead lifts entire clusters of singular values at the tail of the spectrum. Together, these results provide a more general characterization of stochastic rounding as a spectral regularizer, revealing that its effects extend beyond extremal aspect ratios and act on a broader portion of the singular value spectrum.

The causal propagator (or Pauli-Jordan function), which multiplied by $i$ is the spacetime commutator of the field $[\phi(x),\phi(x')]$, plays an essential role in scalar quantum field theory. We discuss the role of the causal propagator and its spectrum in recent developments in defining quantum field theory in a more explicitly covariant manner, as well as in causal set theory. We then present a conjecture for its asymptotic spectral density in a free theory, and give examples that lend evidence to the conjectured scaling. Our work has implications for Lorentzian spectral geometry in much the same way as Weyl's asymptotic law has for Riemannian spectral geometry.