Papers

We develop a theory of Kubo-Ando type means for positive operators on anti-dual pairs. This framework extends the classical theory of operator means beyond bounded Hilbert space operators and provides a common setting for several concrete classes of positive objects, including nonnegative sesquilinear forms and states on $C^*$-algebras. We also introduce a Busch-Gudder type strength function for positive operators on anti-dual pairs and investigate its interaction with operator means. Applications are given to Hilbert space operators, nonnegative forms, and representable positive functionals.

We show how, associated with any decision problem, we may derive related functions measuring uncertainty, discrepancy and dependence of distributions. Such "coherent" functions have special properties, which we characterise, and each function essentially determines the others. The theory is applied to the Bayesian formulation of the problem of choosing an experiment in order to make a subsequent prediction. It is shown that coherent choice criteria may be based on any of the coherent functions, with related functions yielding identical solutions.

Scheduling $n$ jobs with deadlines and processing times on a single resource to minimize late jobs equates to finding a maximum-cardinality feasible subset. Moore (1968) proposed a natural greedy algorithm for this: process jobs in nondecreasing order of processing time, adding each if the set remains feasible. While Zhao and Yuan recently matched the $O(n\log n)$ time of the classic Moore-Hodgson algorithm via amortized $O(\log n)$ feasibility queries, our first contribution is a simpler augmented BST data structure that achieves worst-case $O(\log n)$ query time. Moore's greedy algorithm is known to produce a maximum-cardinality feasible subset with minimum total processing time. Though the system of feasible subsets does not form a matroid, the algorithm behaves like a matroid greedy algorithm. Our second contribution explains this phenomenon: if all jobs have equal processing times, the problem corresponds to a nested matroid; with distinct processing time "tiers," Moore's algorithm effectively solves a sequence of nested matroid problems for each tier. Finally, we present an explicit linear-size flow network defining a polymatroid rank. A set is feasible exactly when its rank equals its total processing time, and Moore's algorithm accepts a job precisely when it yields its full processing-time marginal. By contracting shorter accepted jobs and measuring residual rank, we recover the tier matroid's complete rank function, offering a deep structural explanation for the greedy algorithm's correctness.

In this paper, we investigate the arithmetic properties of the Fourier coefficients of weakly holomorphic modular forms of weight $k$ for the congruence subgroup $\Gamma_0(2)$, which have poles exclusively at the cusp $\infty$. Using the structure of canonical bases and the explicit duality between the coefficients of spaces of weakly holomorphic modular forms, which have a pole only at $\infty$, we prove a general family of congruences modulo primes $p \ge 5$ and derive several corollaries as examples of the theorem's application.

In this article, we study the family of elliptic curves $E_{-2pq}: y^2=x^3-2pqx$, where $p$ and $q$ are distinct odd primes. Using a $2$-isogeny methods and some elementary techniques, we obtain explicit possibilities for the Mordell–Weil ranks, conditional on the Parity Conjecture. Moreover, in the rank-one case, we are also able to derive explicit conditions that are independent of the parity conjecture. Moreover, the main results depend only on the residue classes of $(p,q)$ modulo $8$ and the Legendre symbols $\legendre{p}{q}$.

We introduce a Tutte polynomial for hypergraphs, $T_{\mathrm{HG}}$, together with $T_k$, a related Tutte polynomial for $k$-polymatroids. Both invariants admit deletion–contraction recursions that remain within their respective classes, and they are linked by the fact that $T_{\mathrm{HG}}$ specializes to $T_k$ on the associated polymatroid of any $(k+1)$-uniform hypergraph. We show that $T_{\mathrm{HG}}$ satisfies several desirable Tutte type properties, including multiplicativity and duality, while $T_k$ further admits a universality theorem, as well as a convolution product formula. In the uniform hypergraph setting, these latter results specialize back to $T_{\mathrm{HG}}$. We also relate $T_{\mathrm{HG}}$ to hypergraph extensions of the Potts and random cluster models. In particular, we study degree dependent random cluster and Potts partition functions, as well as Grimmett's many body Potts model, and compare their relationship with $T_{\mathrm{HG}}$ in both the general and uniform settings. Finally, we compare $T_{\mathrm{HG}}$ with the polymatroid Tutte polynomial $\mathcal{T}_P$ of Bernardi, K'alm'an, and Postnikov, showing that the two are incomparable in distinguishing power. As a consequence, we answer negatively a question raised by these authors by proving that the characteristic polynomial is not, in general, a specialization of $\mathcal{T}_P$.

We study the vector Dyson equation $$-\frac{1}{m(z)}=z\mathbf{1}-\mathbf{a}+Sm(z),$$ with parameter $z$ in the complex upper half-plane $\mathbb{C}_+$, where $\mathbf{a}\in\mathbb R^d$ and $S$ is a nonnegative matrix, not necessarily symmetric. This equation has a unique vector solution $m(z)\in\mathbb{C}_+^d$, for which we establish a complete measure decomposition and prove regularity. We then develop a graph-theoretic approach to the singularity and stability problem for non-symmetric matrices $S$. The graph structure of $S$ identifies the possible degeneracies of the stability operator as $z$ approaches the real axis. In particular, for non-backtracking matrices, we prove square-root growth at regular edges, cubic-root growth at regular cusps, and complete stability estimates. We also obtain the corresponding estimates for symmetric matrices in the periodic setting.

The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) has a zero term. This problem, whose decidability has been open for many decades, arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and probabilistic model checking, amongst many others. In the present paper, we introduce a notion of "large" zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a double exponential of the magnitude of the data defining the given LRS. We establish two main results. First, we define an infinite set of prime numbers, termed "good", having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conclude, conditionally on a strengthening of the classical Cramér conjecture, that large zeros do not exist, which would entail decidability of the Skolem Problem. Second, we show unconditionally that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. This in turn immediately yields a Universal Skolem Set of density one, answering a question left open in the literature.

Fractional gradient descent (FGD) incorporates long-range memory through Caputo-type operators and has been shown to improve stability in ill-conditioned and nonconvex optimization problems. Despite these advantages, its practical use remains limited, mainly due to the high computational cost of evaluating history-dependent convolutions, which scales quadratically with the number of iterations. In this paper, we focus on making Caputo-based optimization computationally viable without sacrificing its intrinsic memory structure. We begin by expressing the fractional descent direction as a discrete convolution over past gradients, which provides a unified view of the method. Based on this formulation, we introduce two complementary mechanisms to reduce the cost of the memory term. The first uses a sum-of-exponentials (SOE) approximation of the power-law kernel, leading to efficient recursive updates. The second approach, newly proposed in this paper as dyadic hierarchical discrete convolution (DHDC), compresses the gradient history through a multiscale aggregation strategy. Rather than treating these approximations as purely numerical accelerations, we interpret them as perturbations of the ideal Caputo operator. This viewpoint allows us to analyze how the compressed memory affects the optimization dynamics. Under standard $\mu$-strong convexity and $L$-smoothness assumptions, we show that the resulting method still exhibits monotone descent and linear convergence, provided that the approximation error remains controlled.

We study a class of positive matrix Hamiltonians arising from the canonical differential expressions of Melik–Adamyan and appearing in the appendix of Alpay–Gohberg. Let $J$ and $B$ be self-adjoint involutions on $\mathbb C^{2n}$ satisfying $JB=-BJ$, and let $H>0$ satisfy $HJH=J$. For $m>0$ we consider $$ \mathcal A_{m,H}=H^{-1}\left(-iJ\frac{d}{dt}+mB\right) $$ in the weighted space $L^2_H$. A locally absolutely continuous $J$-unitary gauge $\Theta$ representing $H$ reduces this expression to the free massive Dirac operator plus the Hermitian coefficient $$ P_{m,\Theta}=-i\Theta^*J\Theta'+m(\Theta^*B\Theta-B). $$ Whenever this coefficient belongs to $L^2$, the corresponding self-adjoint realization, including its operator domain, is independent of the chosen representing gauge. Minimizing $\int\mathrm{Tr}|P_{m,\Theta}|^2$ over the gauge fibre defines an intrinsic energy. A two-sided Birman–Schwinger decoupling, combined with a truncated pseudo-relativistic estimate proved here, gives a $3/2$-moment bound for all eigenvalues in the gap $(-m,m)$ in terms of this energy. The Dirac estimate applies to arbitrary Hermitian matrix coefficients in $L^2$ and requires no sign condition. On the half-line we treat every self-adjoint Lagrangian boundary condition. Two reflection-compatible conditions require no endpoint correction, while an arbitrary condition contributes at most $2nm^{3/2}$. At zero mass, the optimal-gauge energy is computed explicitly in terms of $H^{-1/2}H'H^{-1/2}$. For a scalar hyperbolic-rotation family the massive gauge minimization reduces exactly to a one-dimensional phase functional. We prove existence of a minimizer in the principal phase sector and give an explicit trial phase that strictly and quantitatively improves the positive lift whenever the corresponding first variation is nonzero.

For $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is a sequence $(\beta_1,\dots,\beta_n)$ of positive integers whose nondecreasing rearrangement $\beta_1'\le\beta_2'\le\cdots\le\beta_n'$ satisfies $\beta_i'\le b_1+\cdots+b_i$. The $\mathbf{b}$-parking-function polytope $\mathfrak{X}_n(\mathbf{b})$ is the convex hull of all $\mathbf{b}$-parking functions of length $n$ in $\mathbb{R}^n$. We prove that every lattice slice of $\mathfrak{X}_n(\mathbf{b})$, obtained by fixing one coordinate at an integer value, is itself a $\mathbf{b}'$-parking-function polytope of one dimension less, with an explicit parameter vector $\mathbf{b}'$; this yields a recursion for the number of lattice points of $\mathfrak{X}_n(\mathbf{b})$. We further show that every dilate of a $\mathbf{b}$-parking-function polytope is a translate of another such polytope, that the number of lattice points is a polynomial function of $\mathbf{b}$, and we deduce an explicit formula for the Ehrhart polynomial of $\mathfrak{X}_n(\mathbf{b})$ for arbitrary $\mathbf{b}$ as a finite sum indexed by draconian sequences, resolving a problem of Hanada, Lentfer, and Vindas-Meléndez; an equivalent formula was recently obtained, independently, by Liu and Thawinrak in a closely related setting. In the special case $\mathbf{b}=(a,b,\dots,b)$, we obtain an explicit closed form and a generating function for the Ehrhart polynomial. As an application, we classify magic positivity in the two-parameter family $\mathfrak{X}_n(a,b)=\mathfrak{X}_n(a,b,\dots,b)$: the polytope $\mathfrak{X}_n(a,b)$ is magic positive if and only if $(n,a,b)\ne(2,1,1)$. Thus, we answer a problem posed by Ferroni and Higashitani for $\mathfrak{X}_n(a,b)$. Our result extends recent work of Liu and Zhang on partial permutahedra and leads us to conjecture that magic positivity holds for every $\mathfrak{X}_n(\mathbf{b})$ with $n\ge3$.

Fix integers $0\leq k<n$. Let $F_{n,k}(N)$ be the least size of a finite set $B\subset\Z^n$ that contains a filled axis-parallel cube $k$-skeleton centered at each point of some $N$-point set. We prove that $F_{n,k}(N)$ has order $N^{1-(n-k)/(2n^2)}$, with constants depending only on $n$ and $k$. Thornton proved the upper bound and lower bounds with every smaller exponent; the endpoint lower bound was open for $k\geq1$. For square boundaries in $\Z^2$, the exponent is $7/8$. A midpoint count and Shearer's inequality handle large radii; induction in lattice cells handles small radii.

We mainly consider three problems on cycle lengths and cycles with chords in graphs: (a) Gao, Huo, and Ma \cite[Question~1.5]{GaoHuoMa2021} asked whether, for every fixed $k\ge3$, there is a function $f_k(n)\to\infty$ such that every $n$-vertex $(k+1)$-critical graph contains $f_k(n)$ consecutive cycle lengths. (b) Let $g_k(n)$ be the maximum integer $t$ such that every $n$-vertex $k$-critical graph with $k\ge4$ contains an odd cycle with at least $t$ chords. Voss conjectured (see \cite[pp.~168]{VossBook}) that $g_k(n)\to\infty$ as $n\to\infty$ for each $k\ge4$, which extends a 1976 conjecture of Erdős (see also Erdős Problem~1091 \cite{Bloom1091}). (c) Kára and Král \cite{KaraKral2003} conjectured that every graph on $31$ vertices with minimum degree at least $8$ contains a cycle with at least $31$ chords. We answer question (a) in the negative for $k=3$, and disprove conjecture (b) for all $k\ge5$. We point out the work of Alexeev-Putterman-Sawhney-Sellke-Valiant (2026) on Erdős Problem 1901 disproves the case $k=4$ for conjecture (b). We prove conjecture (c). We also discuss two other related problems in the part of concluding remark.

For any $\alpha \in (0,1)$ there is a Hölder continuous function $h:\mathbb{R} \rightarrow \mathbb{R}$ with nonporous graph $Gr(h) \subset \mathbb{R}^2$ which is thin for doubling measures. Construction of these functions requires analysis of a known example with nonporous graph and bounded variation on the unit interval.

In this paper, we establish sufficient conditions for which a class of random circle maps, the so-called random Blaschke products, exhibits either synchronization or chaotic dynamics. We also establish necessary and sufficient conditions for such systems to have a unique random fixed point attractor in the unit disc. As an example, for a class of two map cocycles, we identify the random physical measures as a parameter is varied and investigate the critical value where the dynamics transitions from order to chaos.

Fix a constant $c$ with $0<c<1$. We say a graph $G$ on $n$ vertices is $c$-hollow if every maximal independent set of $G$ has size at least $cn$. Denote by $\tau(G)$ the size of a smallest set of vertices $T\subseteq V(G)$ such that every maximal independent set in $G$ intersects $T$, i.e., $T$ is a transversal for the family of maximal independent sets. In 1991, Bollobás, Erdős, and Tuza conjectured that if $G$ is $c$-hollow, then $\tau(G)=o(n)$. Using a random construction, we show there exist $c$-hollow graphs with $\tau(G)=\Omega\left(\frac{n^{1/3}}{\log n }\right)$, establishing the first nontrivial lower bound constraining the conjecture and complementing a closely related lower bound due to Alon for maximum independent sets. We also show the conjecture holds in a strong form for the class of cographs and split graphs.

Score-based generative models exhibit a puzzling behavior: they often appear to cover all modes of a target multimodal distribution and yet may fail to learn the correct relative mode amplitudes, which can be interpreted as mixture weights. We resolve this apparent paradox by relating the diffusion score matching (DSM) loss to the error in estimating mixture weights from generated samples. We show that, even when the target score is insensitive to mixture weights, generated samples can recover the weights accurately if the scores at intermediate noise levels are informative about the weights. Accordingly, we define the diffusion score sensitivity index (DSSI) as the variation in the DSM loss relative to changes in a parameter. We then show that the DSSI governs the accuracy with which the parameter of the target distribution can be estimated from generated samples. For Gaussian mixtures in arbitrary dimensions, we prove that the mixture weight estimation errors are on the same order as the DSM loss under mild conditions. Empirically, we show the emergence of sensitivity during the noising process of benchmark data distributions under typical noise schedules, and that these sensitivity values predict how well a well-trained model recovers mixture weights. Furthermore, we show that the choice of noise schedule can reduce diffusion sensitivity, leading to mode amplification. Although we focus on mixture weights, the proposed sensitivity framework governs the recovery of any qualitative parameter of the target distribution.

We extend the parametric mean-payoff game framework of Parsons et al. to bi-objective tropical pseudolinear optimization with general two-sided constraints. The problem is to simultaneously minimize two tropical pseudolinear objectives over the feasible set of a general two-sided system U otimes x oplus b is less than or equal to V otimes x oplus d, we characterize the Pareto front via a parametric mean-payoff game in two parameters ( lambda 1, lambda 2). The feasibility region R is convex and the Pareto front P is a convex piecewise-linear curve with finitely many breakpoints, these properties are natural extensions of the single-parameter case to two parameters. In addition, we give as a new result, the joint denominator bound: the cycle coefficients satisfy k 1( gamma ) + k 2( gamma ) is less than or equal to 2 for any elementary cycle gamma, yielding | Delta | is less than or equal to 2 for every 2 times 2 Newton system, except in the fully decoupled case, and implying that all breakpoints have half-integer coordinates for integer data. Optimality and infeasibility certificates are given in terms of the cycle structure of the parametric game. Two algorithms are developed, a directional bisection algorithm (O(n squared (n+m) log M) per direction) and a Newton scheme tracing the complete Pareto front via 2 times 2 linear solves in at most | S | steps, independent of M. The directional bisection algorithm is pseudo-polynomial in n, m and M. The Newton scheme is independent of M but requires up to |S| steps, where |S| is exponential in n. Lastly, we give numerical experiments on random instances to confirm the directional bisection complexity bound exactly; the Newton scheme's worst-case bound is not attained by random instances but is shown to be tight via explicit adversarial constructions.

Let ${\rm Poly}_D(\vec\mu^*)$ be the ramification stratum in the parameter space of degree $D \geq 2$ complex polynomials consisting of polynomials with ramification profile $\vec\mu^*$. In this paper, we introduce a space $\mathcal{T}_D(\vec\mu^*)$ of framed decorated polynomial trees and identify it with the dynamical tropicalization of ${\rm Poly}_D(\vec\mu^*)$. A non-trivial point is that the tropicalization of ${\rm Poly}_D(\vec\mu^*)$ is not available a priori, as the stratum does not come with a previously known proper toroidal compactification. We resolve this issue by identifying ${\rm Poly}_D(\vec\mu^*)$ with a rigidified framed Hurwitz space. Using the twisted admissible cover compactification, together with additional rigidifying data, we construct a proper toroidal compactification and hence an associated Berkovich skeleton. We prove that $\mathcal{T}_D(\vec\mu^*)$ is isomorphic to this skeleton. We further show that the projectivized tree space $\mathbb P\mathcal{T}_D(\vec\mu^*)$ compactifies ${\rm Poly}_D(\vec\mu^*)$. Finally, we compare this compactification with the DeMarco–McMullen compactification of polynomial moduli by projectivized space of polynomial trees.

Oscillatory dynamics arise ubiquitously in nonlinear systems, yet identifying a physically interpretable phase and phase dynamics in nonlinear, high-dimensional oscillations remains a central unresolved problem. Here we establish the principle of a universal dynamical clock, a physical perspective in which oscillations of arbitrary dimensionality and geometry are equivalently represented as uniform rotation through an equant-induced nonlinear viewing coordinate, inspired by Ptolemy's equant and formalised through an areal-uniformity principle reminiscent of Kepler's second law. Using a machine-learning framework, we demonstrate the existence of such an equant for a broad class of oscillatory dynamics and construct the associated dynamical clock and phase dynamics under additive forces, including noise, periodic perturbations, and coupling. Its value in uncovering new physical rules and phenomena is demonstrated by four findings: (i) collective oscillations in Escherichia coli populations obey a previously unexplained superlinear scaling law, resolving a long-standing open problem posed in 2004; (ii) the response mechanisms of engineered genetic circuits to changes in gene expression and environmental conditions; (iii) a classical-mechanics counterpart of the Berry geometric phase emerges naturally from the phase of the dynamical clock; and (iv) optimal equant non-uniformity provides a geometric early-warning signal for critical transitions and enables prediction of critical parameters. By providing operational and system-agnostic phase dynamics that can be constructed directly from data, the dynamical clock enables principled classification, comparison, and control of oscillatory systems, and offers a new route to understanding how specific dynamical regimes support distinct functional behaviours in networked systems.

We investigate optimal transport distances induced by general Lagrangian action functionals. Extending the classical Monge - Kantorovich and Benamou - Brenier theories, we derive a unified variational framework that connects several equivalent formulations of the induced transport distance, including Lagrangian, Eulerian, convex optimization, Hamilton - Jacobi dual, and Hamiltonian flow formulations. Under standard convexity assumptions on the Lagrangian, we establish the equivalence of these formulations through variational arguments and convex duality. The resulting optimality system reveals a natural Hamiltonian structure on the Wasserstein space, providing a direct link between optimal transport, Hamiltonian dynamics, and optimal control. The proposed framework extends the classical quadratic-cost theory to general Lagrangian costs and offers a unified perspective for the analysis of transport metrics generated by action functionals.

Optimization models are built in a variety of modeling languages and solved by a variety of solvers, but once a solution exists, the information needed to understand it is fragmented: each solver exposes a partial, differently named set of native diagnostics, and the modeling language has already canonicalized the formulation the user wrote. We present pyoptexplain, a practitioner-first Python library for post-optimality analysis of optimization models that sits above this layer. It adapts a model authored in any of five modeling front ends, namely cvxpy, Pyomo, gurobipy, docplex, and OR-Tools, into a normalized internal representation, solves it through a choice of backends, and answers the why and what-if questions of an optimization decision through one uniform interface. The design rests on two observations. First, a post-optimality quantity requested from different backends for the same problem can come back as an exception, a structurally meaningless zero or a basis-dependent value that disagrees across solvers, so reporting whatever one solver returns is unreliable. pyoptexplain reports a quantity only when both the representation and the chosen backend can justify it, and does not approximate unavailable information. Second, repeated scenario analysis can amortize its cost by extracting the model once and reusing a warm solver session across a batch of scenarios. pyoptexplain builds a single scalable what-if interface, uniform across its modeling languages and backends and returning a certified report for every scenario, at a cost within a small constant factor of the bare solver. A reproducible computational study substantiates both claims. Source code is available at https://github.com/h-fellahi/pyoptexplain and installation can be done through the Python Package Index https://pypi.org/project/pyoptexplain/.

Let $K$ be a field of characteristic $p>0$. For a Pascal finite automorphism $F$ of the affine plane we show that its order is determined by its Pascal depth, $|F|=p^{\lceil\log_p\tau_K(F)\rceil}$, and that, combined with Dolgachev's theorem on the plane Cremona group, this pins the order spectrum of Pascal finite plane automorphisms to $\{1,p,p^2\}$ and bounds the Pascal depth by $\tau_K(F)\le p^2$. For the polynomial group $\text{GA}_2(K)$ we give a second, independent proof of the order-$p^2$ ceiling, a purely group-theoretic argument from the Jung–van der Kulk amalgam and Serre's tree theorem, using no birational geometry. We prove that the bound is sharp in two independent senses. Order~$p^2$ is attained by the length-two Witt vectors, and Pascal depth $p^2$ is attained by an explicit tame automorphism $G_p$, for which we give a characteristic-free proof that $\tau_K(G_p)=p^2$. We contrast the plane with higher dimensions, where both order and depth are unbounded.

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same is true for equilibria obtained as limits of finite approximations. Techniques developed in this paper show that infinite games long treated as intractable become amenable to direct equilibrium analysis.

A curve over a number field has infinitely many degree-$d$ points if and only if there exists a degree-$d$ $\mathbb{P}^1$- or $AV$-parametrized point. We show that a $\mathbb{P}^1$-isolated $AV$-parametrized point arises only from an abelian translate satisfying a certain property, which we term “indecomposable". We apply our results to a family of genus-7 curves for which we can show that every effective abelian translate in degree 5 is decomposable over the algebraic closure, giving a negative answer to a question of Viray and Vogt.

Self-distillation (SD) is typically studied when the student is retrained on the teacher's original training inputs. In many practical deployments, however, the labeled training data are no longer available, and one has access only to the trained predictor and fresh unlabeled covariates. We study SD in this prediction-only regime through a fresh-X prediction-mixed scheme: a pure-distilled student is trained on fresh covariates pseudo-labeled by the teacher, and the final predictor is an affine combination of the teacher and student predictions. For ridge regression under proportional asymptotics, we derive deterministic equivalents for the optimally mixed prediction risk under general anisotropic covariance and deterministic signal. We show that this risk is strictly smaller than the teacher risk for almost every pair of teacher and student regularization levels, including when the fresh covariates are out-of-distribution and even when their covariance is isotropic. We further show that the optimal mixing weight cannot be identified from unlabeled data alone, but can be consistently estimated in a single post-training step using a small independent labeled calibration set, without additional model fitting. Finally, for binary logistic regression, we show that prediction mixing can outperform both the teacher and the pure-distilled classifier.

Emergency department (ED) boarding occurs when admitted patients remain in the ED while awaiting inpatient beds. Boarding is a major driver of ED crowding and has been associated with poor patient outcomes. We propose a framework to help EDs reduce boarding time and length of stay by using information about current patients and bed availability to proactively request inpatient beds before admission decisions are finalized. We formulate the problem as a Markov decision process in which predictions of each patient's admission probability and time to disposition are aggregated to guide early inpatient bed requests. This formulation leads to three data-driven policies based on approximate dynamic programming, reinforcement learning, and a newsvendor-type approach. Using a simulation model based on data from a large ED, we evaluate these policies across a wide range of settings. The simulation study shows that proactive aggregate bed requests can reduce average boarding times for admitted patients by 30-70% and average length of stay for all ED patients by 6-15%, while creating only modest idle time for prepared inpatient beds. The newsvendor heuristic provides the most attractive tradeoff between ED performance and inpatient bed idle time, whereas the reinforcement learning heuristic produces smoother bed-request patterns when stability in downstream hospital processes is especially important. Our work shows how EDs can use prediction tools to make proactive bed-request decisions that improve ED operations while helping managers balance reductions in ED delays against inpatient bed idle time. Our findings also illustrate the value of evaluating both simple myopic heuristics and more sophisticated reinforcement learning-based approaches, since each can offer distinct advantages depending on the performance measures and implementation constraints most important to managers.

We derive new upper bounds for the real degree of the contour of the amoeba of algebraic curves and hypersurfaces. Our approach refines the Pfaffian method of Lang–Shapiro–Shustin by replacing large total-degree estimates with logarithmic-conormal elimination and sparse mixed-volume techniques based on the transformed Newton polytopes. This yields universal bounds together with significantly sharper sparse estimates for broad classes of Laurent polynomials. Several comparisons and explicit examples illustrate the improvement of the new bounds over the previously known universal estimates.

The existence of a Dirac complement for a given Dirac structure is a central question in the structure theory of Courant algebroids and the deformation theory of Dirac structures. We study this problem in detail, proving the unobstructedness of lagrangian or local Dirac complements and providing examples that show the complexity of this question. We introduce a cohomology class whose nonvanishing prevents the existence of a Dirac complement and apply it to several families of examples. On the other hand, by using Lie-theoretical techniques, we prove that, for a Lie algebra $\mathfrak{g}$ endowed with a definite form, the diagonal in $\mathfrak{g} \oplus \bar{\mathfrak{g}}$ does not admit a complement unless $\mathfrak{g}$ is abelian. This includes real compact semisimple Lie algebras with their Killing form.

We study the parabolic maximal operator $M_{\textrm{par}}$ along the moment curve $(t,t^2)$. In 1988, Christ proved that $M_{\textrm{par}}$ maps the parabolic Hardy space $H_{\textrm{par}}^1(\mathbb R^2)$ into $L^{1,\infty}(\mathbb R^2)$. We show that this result is sharp at $p=1$: for every $0<p<1$, the natural extension from $H_{\textrm{par}}^p(\mathbb R^2)$ to $L^{p,\infty}(\mathbb R^2)$ fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space $H_{\textrm{par}}^{p,*}(\mathbb R^2)$ and a weak tendril space $\mathcal T^{p,\infty}(\mathbb R^2)$, and prove that $$ M_{\textrm{par}}: H_{\textrm{par}}^{p,*}(\mathbb R^2) \longrightarrow \mathcal T^{p,\infty}(\mathbb R^2), \qquad 0<p<1, $$ is bounded. At $p=1$, these spaces recover those in Christ's theorem: $H_{\textrm{par}}^{1,*}(\mathbb R^2)=H_{\textrm{par}}^1(\mathbb R^2)$ and $\mathcal T^{1,\infty}(\mathbb R^2)=L^{1,\infty}(\mathbb R^2)$. Thus, our result provides a natural extension of Christ's work to the range $0<p<1$.