Papers

Expansion planning models usually fix storage duration before optimization, setting how much storage power to build but not for how long it can discharge. We present a generation-transmission-storage expansion framework in which candidates of many durations compete on annualized cost, making the duration mix an optimization outcome. It is a rolling-horizon, two-stage stochastic linear program, each five-year stage is a full year at hourly resolution under ten coherent inflow, wind, and solar scenarios. We apply it to the Brazilian Interconnected System over 2030-2050, where demand roughly doubles to 1,716 TWh/year and variable renewable energy (VRE) supplies most new capacity. Two cases are compared: 36 pumped-storage hydropower (PSH) candidates over four subsystems and nine durations (4-144 h) alongside 4-h battery energy storage systems (BESS), and BESS alone. With PSH available, the model builds 31.2 GW / 755 GWh of PSH and no BESS, dominated by 12-h capacity with 4.9 GW of 72-h units. Without PSH it builds 38 GW / 152 GWh of BESS and 7.8 GW more gas-fired capacity, mostly open-cycle peakers. PSH lowers 2050 annualized system cost by US$ 5.0 billion/year (6.8%), operating cost by 23.5%, thermal generation by 34 TWh/year, and long-run marginal cost by 20%; 2035 VRE curtailment falls from 8.4% to 3.2%. The magnitudes are specific to Brazil, but the underlying mechanism is general: limiting storage candidates to a single duration understates the system's optimal energy-storage requirement and overstates its residual need for thermal capacity.

We prove that mapping class groups of surfaces and of locally finite connected graphs support a unique Polish group structure.

In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schrödinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time. As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.

In Abner J. Salgado and Ignacio Tomas. Diagonally implicit Runge-Kutta schemes: discrete energy-balance laws and compactness properties. J. Number. Math., 31(4):313-341, 2023, the notion of $U$-stability for Diagonally Implicit Runge-Kutta (DIRK) schemes was introduced. Here we establish the equivalence between $U$- and $B$- (algebraic) stability for two- and three-stage DIRK schemes, which then} provides suitable Bochner norm a priori estimates for $B$-stable methods. As applications, we first prove the convergence of $U$-stable DIRK schemes for linear coercive evolution problems under minimal regularity via energy estimates and compactness. Second, we apply these discretizations to gradient flows, which allow us to derive discrete local energy dissipation inequalities and provide counterexamples that demonstrate the limitations of stagewise energy monotonicity.

The quotient-affine metric gives an intrinsic Riemannian geometry to full-rank correlation matrices, but its geodesic distance has no closed form and we are not aware of an analytic asymptotic null distribution for it. We connect this geometry, introduced in 2019, with Jennrich's 1970 asymptotic test for equality of correlation matrices. The quadratic form underlying Jennrich's statistic is exactly one half of the quotient-affine metric tensor. The identity arises because eliminating marginal standard deviations from Gaussian Fisher information performs the same projection as quotienting out diagonal rescalings. Jennrich's statistic therefore evaluates the local quotient-affine quadratic form directly. Moreover, for two independent Gaussian samples with a common population correlation matrix, the squared geodesic distance, scaled by effective sample size, converges in distribution to $4\chi^2_d$, where $d = p(p-1)/2$. For $p=2$, the result reduces to the two-sample Fisher $z$ test.

In this paper, we compute the (shifted) Hankel determinants of Catalan-like sequences, which arise naturally from the weighted enumerations of nonintersecting Motzkin meanders. Among these determinant evaluations, one and a half are newly discovered, featuring generic shifted Hankel determinants; two were formulated earlier by Cigler and Krattenthaler in an equivalent combinatorial form; and the rest were conjectured by Cigler. As an application, we further confirm a conjectural binomial determinant identity proposed by Cigler and Krattenthaler.

We determine the strong converse exponent for the entanglement cost of quantum state merging, showing that it is characterized by the optimized $\alpha$-$z$ conditional Rényi entropies with $z=\alpha/2\in[1/2,1]$. This contrasts with the sandwiched conditional Rényi entropies that typically govern strong converse exponents in quantum information theory. As a consequence, we derive the strong converse exponent of the partially smoothed conditional min-entropy in purified distance. This exponent is governed by club-sandwiched conditional entropies, whereas global smoothing leads to a sandwiched expression.

Biological reaction networks often exhibit complex transient behavior that cannot be explained solely by the analysis of steady states or long-term persistence. Existing structural approaches identify persistent system properties and have considered transitions between organizations, but do not provide a general criterion for admissible transitions between arbitrary species configurations. We introduce admissible transitions, a new structural concept that describes feasible changes between species subsets using only reaction network structure and feasible reaction fluxes, independently of kinetic parameters. We prove that solutions of reaction-based ordinary differential equation systems induce canonical sequences of admissible transitions with a fundamental asymmetry: closure-building transitions are uniquely determined by network structure, whereas downward transitions are generally non-unique and depend on the realized trajectory. This establishes a structural layer linking network topology to transient system evolution. The framework is illustrated using a classical HIV immune-response model. By extending structural reaction network analysis from persistent states to transient dynamics, the proposed theory provides a general, kinetics-independent framework for analyzing reachability, organization formation, and transient behavior in biological systems.

To achieve sustainable intelligent mobility, 6G-empowered robotic vehicles (RVs) require high-fidelity visual perception under stringent bandwidth and energy constraints. Semantic communication offers a spectral-efficient solution but suffers from severe interference in uplink non-orthogonal multiple access (NOMA) RV networks. To address this, we propose a knowledge distillation-driven and generative models-enhanced NOMA framework for robust and green RV communications, named KDG-SemNOMA. First, we develop a ConvNeXt-based deep joint source-channel coding (DeepJSCC) architecture with an enhanced attention feature (AF) module for dynamic channel adaptation. Second, to mitigate interference without inference overhead, an orthogonal transmission teacher model guides the NOMA student model via a two-stage knowledge distillation strategy. Finally, to address the over-smoothing artifacts of pixel-wise optimization, we introduce a channel-conditional GAN (cGAN). By explicitly taking the Stage-I initial reconstruction and channel states as conditional inputs, this module refines coarse outputs into high-fidelity images with realistic textures. Experiments on FFHQ-256 demonstrate that KDG-SemNOMA significantly outperforms state-of-the-art methods in both pixel-level accuracy and perceptual fidelity.

We study versions of results in combinatorial geometry related to families of convex sets in $\mathbb{R}^d$ whose intersection contains a $k$-dimensional affine space. We prove generalizations of the colorful Radon theorem, the fractional Helly theorem, the colorful Helly theorem, and the selection-structure Helly theorem. When $k=0$, our arguments give new proofs of the corresponding versions for points.

We consider the $\phi^4_2$ model. It has been called the simplest non- trivial super-renormalisable model. The method we use is the multiscale loop vertex expansion, an improvement of constructive field theory. We prove analyticity and Borel summability of the cumulants up to a finite order.

In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $\Delta$, we show that the second-smallest eigenvalue of its Laplacian matrix satisfies $\lambda_2(L_G)\lesssim\Delta\frac g n$. We also obtain an improved eigenvalue bound for $K_h$-minor-free graphs of $\lambda_2(L_G)\lesssim\Delta\frac{h^2(\log h)^2}n$. In fact, our results directly prove much stronger results for reweighted eigenvalues, including higher reweighted eigenvalues. As a consequence, we obtain bounds not just on Laplacian eigenvalues, but also on normalized Laplacian eigenvalues and Steklov eigenvalues. Our results for genus-$g$ graphs are optimal for all of these kinds of eigenvalues, while our results for $K_h$-minor-free graphs are optimal up to $\log(h)$ factors. Our techniques for genus-$g$ graphs bootstrap bounded-degree bounds of normalized eigenvalues for entire classes to bounds for reweighted eigenvalues for the same classes without the bounded-degree limitation, while our techniques for $K_h$-minor-free graphs generalize an argument of Korhonen and Lokshtanov, making use of the Lovász local lemma.

We present a simple combinatorial method for computing the socle of representations of $\mathrm{GL}_n(F)$, where $F$ is a local non-Archimedean field, parabolically induced from two ladder representations satisfying a permissibility condition. We adapt recent approaches to such problems using the classical Robinson–Schensted–Knuth correspondence (RSK) and introduce a new variant of RSK that we call mixed-RSK. Some combinatorial properties of this map are investigated and then used to resolve a conjecture of Erez Lapid.

Contraction coefficients quantify the loss of distinguishability induced by a channel and provide a strong form of the data-processing inequality. While standard contraction coefficients ignore auxiliary quantum systems, existing extensions based on complete contraction coefficients require the compared states to have identical reference marginals. In this work, we introduce conditional contraction coefficients, a novel family that incorporates arbitrary quantum reference systems by subtracting the distinguishability already present in the reference system. We develop a general framework for contraction coefficients with such quantum side information, including the corresponding strong-data-processing-inequality (SDPI) constants, expansion coefficients, and relative contraction coefficients. For the trace distance, we show that the optimization can be restricted to orthogonal input states. For the quantum relative entropy, we prove that its conditional contraction coefficient is exactly equal to the contraction coefficient of the conditional mutual information, extending the classical correspondence between relative-entropy contraction and mutual-information contraction to the setting with quantum side information. More generally, we identify structural properties of divergences required for these results and discuss extensions beyond the relative entropy. These results establish a unified framework for analyzing information contraction in quantum network settings, where quantum side information and distributed correlations are intrinsic features of the information-processing task. Applications include an extension of the Polyanskiy-Wu bounds on mutual information contraction, new perspectives on mixing times, and fundamental limits on quantum memories.

In this paper we investigate stochastic heat equations driven by multiplicative noise on moving domains. We establish the well-posedness within a nonhomogeneous variational framework. Furthermore, by combining stochastic De Giorgi iteration with Dirichlet parabolic estimates, we obtain the Hölder regularity of the solutions.

This paper introduces a multi-class extension of a discrete-velocity kinetic traffic flow model based on a non-local Prigogine-Herman framework. We derive a hyperbolically scaled system of equations from a continuous kinetic formulation describing interactions between different vehicle classes through braking and relaxation terms. The model is then discretized with respect to the velocity variable for an arbitrary number of vehicle classes, and the structural properties of the resulting formulation are analyzed. In particular, we prove hyperbolicity and total linear degeneracy. Due to the non-conservative structure of the model, we employ a path-conservative finite volume scheme for the numerical approximation of the system. Finally, we derive the corresponding diffusively-corrected macroscopic multi-class model, investigate its stability and present numerical simulations on a single-lane road to illustrate the theoretical findings.

We model equity markets using geometric Brownian particles entering and exiting at rank-dependent intensities. In the many-firm limit, the capital distribution converges to the solution of a reaction-diffusion equation with reaction term built from the intensities. Calibrated on CRSP data, the reaction term is bistable, and the long-run distribution is a traveling wave: we prove existence, uniqueness, and, for constant coefficients, exponential relaxation. Turnover, not drift, stabilizes the calibrated market. With measured volatility, the wave tracks the empirical capital distribution in every decade, determines the capitalization growth of diversity-weighted portfolios, and places the market just inside the boundary of the diverse phase. Turnover reclaims most of what rebalancing gains.

A functional digraph is a directed graph where each vertex has an out-degree of at most 1. We study the number of successor-closed subsets of a functional digraph, that is, subsets from which no edge leaves. We show that functional digraphs have a simple recursive formula for their corresponding generating function. Using this formula, we determine, among all functional digraphs with a fixed number of vertices and edges, the one that maximizes and the one that minimizes the number of successor-closed subsets of every size simultaneously. Somewhat unexpectedly, this extremal result yields a quantitative strengthening of the set-pairs inequality of Bollobas: rather than merely guaranteeing that some pair of a large enough family must violate the hypothesis of the theorem, we show that a uniformly random subset of the family witnesses a violation with high probability, quantitatively in terms of how far the family size exceeds the classical threshold. We further show that the same approach applies to the skew variant of Bollobas's inequality due to Hegedus and Frankl, yielding an analogous probabilistic strengthening.

We develop $L^1$ bounds for the difference between a test function of a random sum and a standard normal random variable, where the summands are assumed to be independent but not necessarily identically distributed. The bounds are obtained through a new version of the approximate zero bias transformation specifically developed for random sums. Although the identical distribution assumption is relaxed, the bounds are of order $1/\sqrt{n}$, matching the order of existing bounds in the literature under the same distributional assumption on the number of summands. The main results are then applied to three real-world settings: random sums obtained from simple random sampling with outliers, total insurance claims, and generative AI response times.

In several papers, Stochel and Szafraniec studied moment problems on algebraic sets from an operator-theoretic perspective, investigating when positive definite sequences satisfying polynomial relations admit representing measures. Within this framework, Stochel introduced type A sets, and Bisgaard classified the plane curves defined by relations between two monomials that have this property. Curto and Fialkow introduced a stronger, truncated version of the type A property, requiring that the existence of a positive semidefinite extension of prescribed degree guarantees the existence of a representing measure. Motivated by Bisgaard's classification, we determine which plane curves defined by relations between two monomials satisfy this extension property. In the affirmative cases, we obtain explicit bounds on the required extension degree. In the negative cases, we construct truncated sequences that admit positive semidefinite extensions of arbitrarily high order but have no representing measure supported on the curve. These constructions yield explicit polynomials that are nonnegative on the corresponding curves but are not sums of squares in their coordinate rings. In the affirmative cases, we also derive explicit degree bounds for sums-of-squares certificates of strictly positive polynomials.

Deterministic source-sink models are widely used to assess the long-term evolution, capacity, and sustainability of the low Earth orbit (LEO) environment. These models propagate shell-averaged populations through ordinary differential equations (ODEs), relying on individual collision, disposal, and decay events to average out within sufficiently large altitude shells. As constellation traffic is increasingly organized into kilometer- and sub-kilometer-scale shells, this averaging assumption becomes strained. We formulate the multi-shell, multi-species LEO environment as a Markov jump process and, from this formulation, recover the conventional source-sink ODE as a large-volume limit and derive a stochastic differential equation (SDE) approximation whose fluctuations scale as V^{-1/2} in the per-shell volume V. The diffusion approximation is validated against an exact discrete-event simulation of the underlying jump process. Sweeping V at fixed spatial density over the 450-800 km band, we find that the two descriptions agree at shell volumes comparable to those used in established source-sink models but diverge as shells narrow. At the finest shell volume considered, the mean debris population reaches roughly 4.5 times the ODE prediction, with several realizations undergoing runaway growth absent from the ODE trajectory. This departure is driven by nonlinear collision terms, through which population variance and covariance raise expected collision rates, generating further debris and reinforcing the collision-debris feedback. Narrow-shell stochasticity thus both widens the outcome distribution and shifts the expected trajectory. Because the stochastic model shares its parameterization with the deterministic one, it provides a scale-consistent extension of existing source-sink models for evaluating shell configuration, collision risk, and long-term LEO sustainability.

In this paper, we study a Riesz-type weighted sum of the Liouville function, $f(x):=-\sum_{n\le x}\frac{\lambda(n)\log n}{\sqrt n}\log\frac{x}{n}$ for sufficiently large $x$. Motivated by a recent research on the sign criteria for the Riemann Hypothesis arising from weighted prime-counting functions, we investigate the sign behavior of $f(x)$ and its relation to the zeros of the Riemann zeta function. We first prove that if $f(x)$ is non-negative for all sufficiently large $x$, then the Riemann Hypothesis holds. The proof is based on the Mellin transform of $f$ and the analytic properties of $\frac{\zeta(2s)}{\zeta(s)}$. Conversely, assuming the Riemann Hypothesis, the Simple Zero Conjecture, and an absolute convergence condition involving the nontrivial zeta zeros, we derive an explicit formula for $f(x)$ in terms of these zeros. In particular, we show that $f(x)\sim \frac{(\log x)^3}{12|\zeta(\frac{1}{2})|}$ as $x\to \infty$. Consequently, we show that under these hypotheses, $f(x)$ is eventually positive.

We say that an antichain in a lattice $L$ is Boolean if it generates a Boolean sublattice in $L$ in a particularly nice way. These purely combinatorial objects play a role in the representation theory of the incidence algebra of the lattice $L$, as indicated by recent results of Rognerud, Y{\i}ld{\i}r{\i}m and of the last author with Klász, Kleinau and Marczinzik. Given these motivations, it is natural to ask: Can we count and construct Boolean antichains in certain lattices? In this paper we give a construction of Boolean antichains in Boolean lattices, partition lattices and Tamari lattices. Furthermore, we share the surprisingly elegant formulas we found for the number of these Boolean antichains.

Motivated by the recent introduction of chromatic persistent homology, we generalize the Euclidean minimum spanning tree (EMST) for $n$ points in $\mathbb{R}^2$ to the lunar EMST for the case in which the points come in $s+1$ colors. Calling the intersection of $s+1$ disks of radius $r$ centered at points with pairwise different colors a lune, the generalized EMST reflects the history of the union of lunes as $r$ goes from $0$ to $\infty$, and its cost is twice the difference between the radii when the arcs and nodes of the tree are formed. If the points are chosen uniformly at random in $[0,1]^2$ and colored randomly, the expected cost converges to some constant (that depends on $s$) times $\sqrt{n}$, as $n$ goes to infinity. The main contribution of this paper is a proof that this constant exists, however similar to the case of the classic EMST, its precise value remains elusive.

This paper studies the real-time reconstruction and remote actuation of two binary Markov sources over a shared wireless channel with multi-packet reception (MPR). Unlike many existing collision-based formulations that discard simulta- neous transmissions, we exploit MPR and evaluate communica- tion through reconstruction and actuation errors rather than raw delivery rates. We consider two sensors observing the sources and aim to find sampling policies that minimize the weighted real- time reconstruction error (RTE), or equivalently the weighted cost of actuation error (CAE) for the considered binary sources, under per-sensor sampling constraints. We first obtain closed- form expressions for the RTE and CAE in terms of the effective update probabilities. When the sensors randomize independently, the MPR-induced update-rate map becomes bilinear, and the constrained optimization is nonconvex. We exploit the geometry of the achievable update-rate region to show that the search over Pareto-efficient independently randomized policies reduces to a finite set of one-dimensional boundary-branch searches with closed-form candidates. As a benchmark, we allow time sharing among joint sensor actions, and we prove that at most two Pareto- extreme modes are sufficient, and use this benchmark to quantify the loss caused by independent randomization. Numerical results reveal that MPR capability alone is not sufficient; simultaneous decoding improves the task-oriented objective only when con- current reception is reliable for both sources; otherwise, policies that avoid simultaneous transmissions can be equally effective.

We study nearest and farthest Voronoi diagrams of lines in three-dimensional Euclidean space, which partition space according to the closest and farthest input lines, respectively. For arbitrary lines, a gap remains between the known quadratic lower bound and near-cubic upper bounds for the complexity of the nearest diagram. We consider input lines chosen from one of two continuous families of straight lines on a smooth surface defined by a quadratic equation, with one line from each family passing through every point of the surface. Under the geometric assumptions stated in the paper, we prove quadratic upper bounds on the total number of vertices, edges, faces, and connected three-dimensional regions in both diagrams. We also construct quadratically many nearest vertices using lines from a single family on one fixed surface, establishing a tight quadratic worst-case bound. The farthest diagram has quadratic total complexity for every input satisfying our assumptions. Our proof uses a parametrization that arranges the input lines in a cyclic order and expresses tangency to a sphere by a polynomial equation of degree four. At a Voronoi vertex, the four tangent input lines give four distinct roots. The sign changes between these roots, together with the nearest or farthest condition, force two intervening parameter intervals to contain no other input parameters. Thus, only quadratically many choices of four lines can define a vertex, and each choice yields at most eight isolated candidate centers.

Motivated by theoretical problems in deep learning, we conjecture that post-composing a fixed number of pairwise distinct nonconstant polynomials with a generic polynomial of sufficiently large degree yields linearly independent polynomials. This generalizes Newman–Slater's theorem on powers of polynomials. We establish several cases of this conjecture and its origin-passing variant: We prove the result for two polynomials, and for an arbitrary number of polynomials when their degrees are bounded. Furthermore, we show how the conjecture implies a complete understanding of the identifiability (i.e., parameter symmetries) of deep fully connected neural network architectures with generic polynomial activation functions. In particular, for network architectures with layer-specific activations of increasing degree, our established versions of the conjecture fully characterize the set of parameters yielding the same end-to-end network function. As a special case, we fully resolve the identifiability of shallow polynomial networks.

In this paper, we study the structures of single connected $k$-leaky $(r+1)$-completed Hurwitz numbers. We first prove that, for fixed genus, after extracting an explicit product of Pochhammer type factors, the stable connected leaky Hurwitz numbers for partition $\mu$ are polynomials in the quotients $[\mu_i]$ modulo $k+r$. We then give a closed formula for the generating function of single connected leaky Hurwitz numbers, which can be used to study the genus dependence of leaky Hurwitz numbers.

This work develops a regularization analysis of non-accelerated and accelerated primal-dual methods for solving linear inverse problems in the presence of noisy data. We investigate a Condat-Vũ algorithm and an accelerated primal-dual hybrid gradient method in Hilbert spaces, with focus on quantifying the effect of data perturbations on the reconstruction error. For the non-accelerated scheme, we derive error estimates in terms of Bregman distances, whereas for the accelerated scheme we establish error estimates in norm. The study accommodates a general class of convex data fidelities satisfying suitable perturbation conditions, which are verified explicitly for equality constrained and Morozov regularization. For the non-accelerated method, the analysis is further extended to Banach spaces, taking into account non-Euclidean geometries and including a particular non-reflexive setting tailored to nonnegative solution reconstruction. The results recover known behavior in classical settings while extending the regularization analysis to these more general frameworks. Numerical experiments with representative regularizers, including sparsity and entropic models, support the theoretical findings and illustrate practical performance under noise.

Building on the stochastic variational framework established in the companion paper, we investigate here the linearized stochastic water-wave system, consisting of a large-scale stochastic wave dynamics coupled to transport dynamics for the small-scale correlation modes. Within this framework, we develop, in the deep-water regime, a kinetic theory for surface gravity waves interacting with unresolved stochastic velocity fields. An energy analysis yields a wave-action kinetic equation exhibiting two distinct regimes: a diffusive scattering regime and a quartic interaction regime with structural similarities to Hasselmann–Zakharov theory. In the present framework, these effective quartic interactions arise through stochastic transport of unresolved fluctuations by the large-scale flow rather than through classical intrinsic resonant nonlinearity. Scaling laws are derived for the diffusion tensor and the effective growth rate, revealing a Miles-type production–dissipation mechanism. Using JONSWAP spectra, we then compare the strength of stochastic transport and classical Hasselmann interactions. For realistic oceanic values of unresolved velocity variance ($\sigma_u \approx 0.1\,\mathrm{m\,s^{-1}}$) and decorrelation time ($\tau_c \approx 10\,\mathrm{s}$), stochastic transport is found to compete with, and often exceed, classical four-wave interaction rates over broad spectral ranges. The transport intensity $S=\sigma_u^2\tau_c$ emerges as a key parameter controlling the transition between interaction regimes. These results suggest that unresolved stochastic transport may play a substantially larger role in spectral evolution than is commonly represented in operational wave models, and motivate the inclusion of transport-induced source terms alongside standard resonant interaction closures.