In this work, we study Restricted Assignment scheduling on multiple machines, where each job can be processed only on a specified subset of machines and the objective is to minimize the makespan. We introduce a learning-augmented setting in which a possibly infeasible predicted assignment is provided. The prediction error (moved-load) is measured by the total processing volume that must be reassigned in order to obtain an optimal feasible schedule. Using a single prediction, we obtain two types of guarantees. First, we design an algorithm whose approximation ratio degrades smoothly with the prediction error while retaining a worst-case guarantee independent of the prediction quality. More precisely, for any fixed constant, we can make the additive dependence on the prediction error arbitrarily small, at the cost of increasing the polynomial running time. This guarantee can also be combined with any approximation algorithm for the problem without predictions to obtain robustness. Second, given a makespan estimate, we provide a repair procedure that returns a schedule matching this estimate in time parameterized by the prediction error. This allows the algorithm to exploit the separation between estimation and approximation algorithms for Restricted Assignment. Finally, we complement the repair algorithm with a parameterized hardness result, showing that exact moved-load repair with a given target makespan is W[1]-hard when parameterized by the amount of moved-load.

Technological societies are separated in time, not just space -- that is the lesson of the Drake equation. Might the best way to seek them be to find technosignatures that persist long after their creators? I present work I and my collaborators have done on the idea of passive technosignatures, requiring no upkeep from an active society. These range from microscopic to galactic in scale, including specular reflections from shiny artifacts in the Solar System, lens flares from X-ray binaries, and the survivability of Dyson swarms. I discuss prospects for detecting these technosignatures. In the end, what we may be left with are the end products of collisional cascades: dust.

Transit deserts are areas where public transportation is inadequate despite evidence of travel demand, a condition that affects tens of millions of residents across the Americas. Planning for these areas is difficult because the usual demand signal is missing: ridership cannot be observed before service exists. To address that setting, we formulate risk-aware transit desert remediation as a partially observable Markov decision process with Conditional Value-at-Risk constraints for financial tail risk. The model uses demographic, land-use, and employment data to set a prior over latent demand, then updates that prior as new service deployments produce ridership observations. A myopic belief-aware planner is evaluated on 25 cities using a unified financial model for operating cost, capital expenditure, fare revenue, and net subsidy. After five years, the planner remediates a median of 53.6% of transit-desert tracts and improves on static optimization by 5.0 percentage points on average, with gains in 16 of 25 cities. Gains are largest at moderate budgets (+9.9 points at baseline) and persist under 50% prior-demand miscalibration, while population density and existing transit density are the strongest structural predictors of remediation cost ($R^2\!=\!0.41$ on per-tract cost)

Recent analyses of high-redshift Lyman-$α$ forest observations have revealed strong correlations on scales exceeding 200 cMpc at redshift z = 6. Reproducing these large-scale correlations has proven challenging for current large-volume reionization simulations. In this work, we investigate these large-scale correlations using mock spectra generated from the extended SCRIPT semi-numerical reionization model. We find that while the fiducial model ensemble systematically predicts smaller correlation lengths than those inferred from the 67 sightlines in the extended XQR-30 sample, a small fraction of individual mock realizations can naturally reproduce the observed signal. Using a delete-2 jackknife analysis, we demonstrate that the observed large-scale correlation length is disproportionately driven by a rare pair of highly transmissive sightlines associated with high-redshift transmission spikes. By inserting two such highly transmissive sightlines into our mock realizations, the fraction of models consistent with the observed redshift evolution and correlation length increases significantly from 17.5% to 74.1%. Furthermore, we show that spatial fluctuations in the ionizing mean free path remain an essential physical ingredient for reproducing the observed correlation structure. Our results suggest that the unexpectedly large Lyman-$α$ correlations can be reconciled with existing reionization models when accounting for cosmic variance and the outsized statistical impact of rare, highly transmissive sightlines.

The Mader model is the most widely used mathematical framework for muscular energy metabolism in German-language sport science, underpinning lactate diagnostics, maximal lactate steady state (MLSS) estimation and training prescription. Despite decades of use, neither its dynamic ODE formulation nor its steady-state equations have been available as open code, leaving results based on the model impossible to reproduce independently. We close this gap with MetaboliSim, an open-source Python implementation of both formulations: a dynamic model that integrates the five-variable ODE system (phosphate potential, $\dot{V}\mathrm{O}_2$, muscle and blood lactate, and glycogen) with a fourth-order Runge-Kutta scheme, and a steady-state model that computes MLSS power and the lactate-power relationship in one- and two-compartment variants. We verified implementation correctness against published reference values and assessed physiological plausibility across constant-load, step-test, sprint and running protocols. The implementation reproduces the published reference output within stated tolerances and remains numerically stable throughout (halving the time step changes blood lactate by less than 0.01 mmol/L), with both formulations yielding congruent MLSS estimates. Key physiological behaviour ($\dot{V}\mathrm{O}_2$ on-kinetics, lactate accumulation, PCr dynamics and the sub/supra-MLSS separation) emerges directly from the model equations without protocol-specific tuning, and a sensitivity analysis shows MLSS power varying approximately linearly with $\dot{V}\mathrm{O}_{2\max}$ and nonlinearly with $\dot{V}\mathrm{La}_{\max}$. As the first openly available implementation of the complete Mader model (AGPL-3.0), MetaboliSim lets independent groups reproduce, verify and build on published model-based results. Source code: https://codeberg.org/3phos/metabolisim; Platform: https://metabolisim.org

We observe that the Kronecker product of tensors is the operation that converts the determinant polynomial into Cayley's first hyperdeterminant. We apply the Kronecker product to iterated matrix multiplication, which results in the hypercomputant, a VNP-complete and VW[1]-complete polynomial whose hardness we prove via the equivariance of the Kronecker product. The construction works over arbitrary commutative semirings and also for the tensor algebra and the exterior algebra. For the tensor algebra this gives a version of "noncommutative VNP", and for polynomials over the nonnegative real numbers this gives a version of "monotone VNP", each with the hypercomputant as the complete object. We take a parameterized complexity viewpoint and compare the noncommutative setting and the monotone setting. Using standard techniques we obtain optimal algebraic branching program width lower bounds in both settings, and these are notably not always the same. We also prove the polystability of the hypercomputant and that its isotypic components are characterized by their stabilizer.

Existing scientific relation extraction benchmarks mainly target domains such as computer science, where entities are tasks, methods, datasets, materials, or metrics. This leaves a gap in variable-oriented empirical fields such as psychology, where findings are expressed as relations among constructs, measurements, interventions, and outcomes. We introduce variable-centered empirical graph extraction, the task of mapping scientific abstracts to typed graphs whose nodes are normalized variables and whose edges represent empirical and hierarchical relations. To support this task, we construct EmpiriGraph-Psy, a benchmark of 210 psychology abstracts annotated by domain-trained annotators with normalized variables, concept hierarchies, empirical relation types, and validation states. We evaluate frontier and open-weight LLMs using both direct extraction and a staged graph-construction pipeline that separates variable extraction, normalization, hierarchy construction, evidence selection, relation extraction, and edge validation. The staged pipeline substantially outperforms direct extraction, with the best configuration achieving a macro-F1 of 0.74. Error analysis shows that moderation relations and concept hierarchies remain the most challenging cases, highlighting the difficulty of extracting higher-order empirical claims and implicit abstraction structure from scientific abstracts.

During the century of existence of the notion of coherent states, either linear or nonlinear, several schemes for their construction, theoretical or experimental, have been developed. Generally, the mathematical structure of coherent states depends on the choice of ladder operators, and consequently on the structure constants. In this paper, we propose a way to construct generalized coherent states for anharmonic oscillators that is based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions, that is, on some of the most general special functions. These states are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system. In addition, the action of these operators is easy to find if the expression for the dimensionless energy eigenvalues is known.

We develop a covariate-assisted approach to partially identified parameters that are solutions to an under-identified system of linear equations with known coefficients. Examples include bounds on treatment effects, models of unemployment with state dependence, choice-theoretic models of IV, and random utility models. The boundary (i.e., support function) of the proposed identified set is represented as an average of intersections of regression functions, aggregated over the covariate distribution. We show that the boundary is a regular parameter, propose asymptotic theory, and demonstrate using an empirical application to Jobs First.

Recently, it has been shown that the combinatorial complexity function $N_c(n)$ of a typical triangular billiard has weakly exponential growth, i.e., for almost any triangle and any $\varepsilon>0$ there is a constant $C$ such that $N_c(n)<Ce^{n^\varepsilon}$. We give the first example, an infinite family of explicitly described triangles with weakly exponential complexity growth. The simplest example in this family is a right triangle with integer side lengths. Moreover, we find one-parameter families of triangular tables with weakly exponential growth for generic parameters.

This paper proposes BPCOL+, an exact branch-and-price algorithm for the Graph Coloring Problem. The algorithm integrates enhanced pricing strategies based on Zero-Suppressed Binary Decision Diagrams (ZDDs) for solving the pricing problem associated with the maximal-stable-set-based set-covering formulation. After computing upper and lower bounds at the root node using heuristic procedures and column generation, BPCOL+ reduces the size of the ZDD by applying maximal stable set reduction techniques that exploit alternative dual vectors. Computational experiments on the 137 DIMACS benchmark instances and on 5,000 recently proposed Erdős--Rényi instances show that BPCOL+ outperforms existing exact branch-and-price algorithms and remains highly competitive with state-of-the-art SAT-based exact solvers. In particular, BPCOL+ solves 96 DIMACS instances within one hour and proves optimality for 4,641 of the 5,000 Erdős--Rényi instances.

Old pulsar wind nebulae are among the foremost galactic high-energy gamma-ray sources. However we still lack a robust approach to their modeling, especially in the light of forthcoming high-energy observatories like Astri or CTAO. Part of the problem is due to the complex interaction that characterizes these systems. Understanding this complexity has then become mandatory for further advancements. We aim to develop a new approach to investigate the role the thermal thick layer of massive filaments, seen in objects like the Crab nebula and 3C 58, but likely present in all pulsar wind nebulae, can exert on the dynamics of the late reverberation phase, and compare results with standard approaches that neglect such a layer. A new formulation of the one-zone thin-shell plus Lagrangian formalism, developed in a series of previous papers, is here extended to the case of a thick layer of filamentary ejecta, complementing our former work, mostly focused on the initial free-expansion phase. We compare the dynamics of reverberation with and without the presence of filaments, and show that in the former case reverberation may be substantially anticipated and the following compression takes much longer. The total compression of the system does not seem to change much, and the qualitative behavior is preserved. Our results suggest that the presence or absence of an extended filamentary layer might affect the duration of both the free-expansion phase (shortening it) and the following compression phase during reverberation (lengthening it), but it does not change much the overall compression of the nebula. While this changes the relative number of systems in these two phases, and their contribution to high-energy emission, some peculiar radiative effects associated with the level of compression in old systems, like the "super-efficiency", might not be much affected.

We prove that a simple unital Jordan superalgebra of arbitrary dimension belongs to the list of known simple unital superalgebras or lies in a certain proper subvariety.

We study murmurations in the depth aspect for holomorphic cusp forms of conductor $\ell^{2a}$ and fixed weight, where $\ell$ is an odd prime. For both $\mathrm{GL}_2$ and the definite quaternion algebra ramified at $\{\infty,\ell\}$, we determine the murmuration density as $a\to\infty$ with $\ell$ fixed. The resulting density agrees with the one previously obtained for odd conductor exponents, and hence gives a uniform density for cusp forms of conductor $\ell^n$ as $n\to\infty$. We also consider the case of Maass forms of conductor $\ell^n$. Finally, we compute the murmuration density in conductor $\ell^n$ as $\ell\to\infty$ with $n\geq3$ fixed.

We study a finite-scale one-component regularity mechanism for suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. The results are organized in three layers. The first layer is unconditional. Under a fixed scale-invariant local bound Phi(1)=A(1)+E(1)+C(1)+D(1) <= M, smallness of the critical vertical-component quantity C_3(1)=int_{Q_1} |u_3|^3 dx dt yields a positive lower bound, depending only on M, for the local regularity radius at the origin. The proof converts one-component smallness into approximation by the two-and-a-half-dimensional limiting class and then into Caffarelli--Kohn--Nirenberg smallness at a smaller scale. The pressure approximation is measured in a quotient by spatially harmonic functions. This pressure topology reflects a genuine obstruction: time-dependent harmonic pressures may have bounded scale-invariant L^{3/2}-oscillation while their pointwise gradients lie beyond the control provided by the available scale-invariant quantities. The second layer is a conditional logarithmic refinement. A prepared two-shadow comparison package replaces the abstract compactness modulus by a logarithmic modulus and gives a logarithmic finite-scale decay. The third layer is a conditional relaxed-shadowing refinement. The comparison class is enlarged to smooth no-stretching horizontal flows V=(v_h,0), with the comparison pressure allowed to have partial_3 pi not equal to 0. The resulting vertical residual pairs with the small component u_3 in the relative-energy identity. Under the buffered strong-flow and localized relaxed stability inputs stated below, this gives a power-type relaxed harmonic approximation and a power-type finite-scale decay. The unconditional theorem is separated from the logarithmic and power-type assumptions; the latter two layers identify the quantitative stability mechanisms needed to upgrade the compactness modulus.

We revisit Hawking radiation from two black-hole families that can approach macroscopic wormhole configurations: the Simpson--Visser black-bounce geometry and the Casadio--Fabbri--Mazzacurati braneworld geometry. The earlier analysis of these backgrounds relied on WKB greybody factors. Here we replace that approximation by direct numerical scattering for the photon and massless Dirac channels and then recompute the emission spectra and integrated luminosities. The qualitative picture remains the same: as the wormhole endpoint is approached the black holes cool, the total flux is strongly suppressed, and the residual emission becomes increasingly fermion dominated. The quantitative picture, however, changes substantially. Close to the Schwarzschild limit the WKB estimates are reasonably accurate, but far from that limit the error can be large, and near the cold endpoint it can reach orders of magnitude. In particular, the WKB calculation can substantially overestimate the remaining luminosity precisely in the regime where the evaporation rate is most sensitive to the low-frequency tail of the greybody factors. In a fixed-parameter Simpson--Visser half-decay estimate, the direct-greybody luminosities increase the WKB lifetime coefficient by a factor of about 85. These results show that reliable evaporation rates for black holes evolving toward wormhole-like endpoints require direct numerical greybody factors rather than barrier-top WKB estimates alone.

Solving Bethe-Salpeter equation (BSE) for the two-particle Green's function is the most widely used approach for taking into account the particle-hole (exciton) interaction in electronic excitation in the context of the many-body theory based on Green's function. In BSE calculations, the static approximation to the screened Coulomb interaction kernel is commonly employed. However, when the excitonic character is significant as typically indicated by a large exciton binding energy, dynamical screening effects become non-negligible, rendering the static approximation questionable. Because of the large computational cost due to the dense Brillouin zone integration necessary for convergence, solving the dynamical BSE for extended systems remains a significant challenge, especially when combined with GW calculation for the calculation of quasi-particle energies. In this work, we formulate the plane-wave based effective dielectric function method [Zhang, et al., Phys. Rev. B 107, 235205 (2023)] for the dynamical BSE calculation using atom-centered orbitals as basis functions. We implement this approach in our recently developed all-electron numerical atom-centered orbital (NAO) implementation of BSE@GW [Zhou, et. al. J. Chem. Theory Comput. 21, 291 (2025)] for extended systems. We validate our all-electron NAO-based implementation of the dynamical BSE method, and we then discuss its realistic application to molecular crystal of naphthalene by performing the dynamical BSE@G0W0 calculation.

Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of $N$-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in $\log{(N)}$ and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of $N$ interacting quantum harmonic oscillators with cost polylogarithmic in $N$. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.

Large language models (LLMs) have shown strong performance in financial analysis and surface-level factual error detection, yet their ability to identify fraudulent financial misinformation in audited corporate reporting remains underexplored. Existing financial and audit benchmarks mainly focus on factual verification, numerical reasoning, rule compliance, or audit workflows, but rarely evaluate misleading disclosure narratives or management explanations that obscure the true drivers of reported performance. We introduce AuditFraudBench, an enforcement-grounded benchmark constructed from authentic company filings and regulatory materials, including original and restated 10-K and 10-Q filings, structured financial statements, MD&A disclosures, and SEC Accounting and Auditing Enforcement Releases (AAERs). AuditFraudBench contains three tasks: Profit Source Attribution, Misleading Narrative Detection, and Fraud Pattern Classification, which evaluate whether models can identify the true source of reported performance, detect misleading disclosure framing, and classify misconduct mechanisms into known manipulation patterns. We evaluate GPT, DeepSeek, and Qwen series LLMs on the benchmark. Results show that both proprietary and open models still struggle to jointly reason over financial figures, disclosure framing, restatement evidence, and enforcement-grounded fraud mechanisms. AuditFraudBench provides a challenging testbed for audit-relevant, evidence-grounded evaluation of LLMs in financial reporting.

This work expands the Nash equilibrium algorithm to those systems that have a state delay within the deterministic case, a Linear-Quadratic (LQ) type performance index with generalized cross terms is used, following the constructive approach on optimal control: first, propose the strategies form, close the loop and find the Cauchy solution; secondly, express the cost function according to those expressions; and finally use the Bellman equation as restrictions to find the strategies parameters. The stability of the closed loop is demonstrated. The algorithm is applied to a thermal prototype with an ESP32 micro-controller and its performance is compared with optimal PI controls inside two commercial/industrial PID controllers REX-C100.

We present a first-principles spectral mechanism for the emergence of nonextensive $q$-exponential dilution and power-law relaxation in non-ideal transport systems. By modeling an incompletely mixed reactor as a layered diffusion matrix with an absorbing boundary, we demonstrate that macroscopic power-law tails depend on the geometric interaction between the initial tracer placement and the domain's boundary configuration. For a one-dimensional system, an asymmetric, volumetrically distributed initial concentration profile projects onto the low-wavenumber eigenmodes, generating an emergent Gamma distribution of relaxation rates; at an infinitesimal boundary layer thickness ($Δz \to 0$), this profile yields the nonextensive $q$-exponential decay function exactly across the entire temporal domain with $q = 5/3$. Extended to $d$ dimensions under a highly localized, boundary-adjacent singular initial condition, the resulting scaling exponents and corresponding $q$ values depend explicitly on the spatial configuration of the absorbing boundaries. However, in the one-dimensional limit ($d=1$), these distinct initial states and boundary formulations intersect, rendering the $q=5/3$ exponent geometrically invariant. Our approach establishes a clear connection between linear diffusion transport and nonextensive statistical mechanics, showing how heavy-tailed transport can be derived from boundary geometry and spectral dimensionality.

The discrete-time treatment number of a graph $H$, denoted by $τ(H)$, was introduced in arXiv:2408.0531(3) and arises from a deterministic process in which each vertex is assigned a color at each time-step. The pathwidth upper bound $τ(H)\leq \lceil\frac{1+pw(H)}{2}\rceil$, is shown in arXiv:2408.0531(3), where $pw(H)$ denotes the pathwidth of graph $H$. Equality holds when $H$ is the complete binary tree of depth $d$ (denoted by $BT(d)$) and $1 \le d \le 6$. In this paper, we characterize the sizes of all subsets of vertices of $BT(d)$ whose boundary has $3$ or fewer vertices and use this result to prove that $τ(BT(d))= 3$ for $8\leq d\leq 10$; in these cases, equality also holds in the pathwidth upper bound. By the hereditary property of the treatment number, all larger complete binary trees have treatment number at least $3$. In contrast, we provide an explicit construction to show that $τ(BT(7))=2$, while the pathwidth upper bound only shows $τ(BT(7))\le 3.$ We construct an infinite family of graphs, each with a cut-vertex, whose treatment number depends on the number of components when the cut-vertex is removed. We use a combination of pathwidth and vertex cuts to prove another upper bound on the treatment number and use this to construct an infinite family of graphs whose boundary size is limited, but whose treatment number is unlimited.

Let $F$ be a family of $n$ unit disks in $\mathbb{R}^2$ with $n$ being odd. We use $\mbox{OA}(F)$ to denote the area of the set of points that is covered by an odd number of disks. The purpose of this note is to disprove the conjecture $\mbox{OA}(F) \geq π$ which was suggested in the literature and to present some examples.

The central challenge in optimizing green ammonia systems is satisfying the minimum-load requirements of the Haber-Bosch (HB) process under renewable uncertainty. We develop a robust optimization framework consisting of a strategic capacity planning model and an operational flow model under solar and wind uncertainty. The strategic model is a mixed-integer optimization (MIO) problem with flexible HB operating modes, namely hot-idling and shutdowns. To address the resulting computational challenges, we propose a robust scenario-reduction framework that combines k-means clustering with robust optimization to generate adversarial renewable trajectories. For the operational model, we develop adaptive robust rolling-horizon formulations under forecast uncertainty. Computational results show that the proposed framework produces feasible capacity plans under out-of-sample simulation, whereas existing approaches based on constraint aggregation fail to satisfy HB minimum-load requirements. Adaptive policies achieve higher ammonia production than static robust policies for a given robustness level, but provide weaker protection against realizations outside the uncertainty set.

We present molecular dynamics simulations of the random walk of a dumbbell - two beads connected by a spring - in a one-dimensional periodic potential and compare our results in limiting cases to theoretical analytical equations. The relevant parameters in this system are the spring constant, the equilibrium distance of the spring (relative to the periodicity of the potential), and the amplitude of the potential. Dumbbells with equilibrium distances incommensurate with the potential periodicity and with a sufficiently large spring constant exhibit enhanced diffusion. The diffusion constant can exceed that of a single bead in the same potential landscape. In this case, the dumbbell resembles a traction elevator, with the two connected beads acting as the elevator car and counterweight: the ''elevator effect''.

Superconducting weak links with high transparency offer an appealing approach for designing compact magnetic-field sensors, as their phase-dependent Andreev bound-state (ABS) spectrum produces a strong flux-to-signal response with minimal dissipation. One way to achieve ballistic transport in the weak link is to use the edge states of topological insulators, since these states resist backscattering and provide a unique path to developing topological insulator-based weak-link devices for highly efficient magnetic-field sensing. Building on this, we propose a weak-link device using superconducting Nb electrodes and a nanoribbon of $\text{Bi}_4\text{Br}_4$ as the normal region, forming a Nb-1D (one-dimensional) $\text{Bi}_4\text{Br}_4$-Nb Josephson junction. We develop first-principles tight-binding Hamiltonians and orbital-resolved interface couplings in the Wannier basis, including spin-orbit coupling, based on density functional theory (DFT) calculations. The Eliashberg spectral function of bulk Nb, obtained via density functional perturbation theory (DFPT), indicates an electron-phonon coupling strength of 1.19 and a transition temperature of about 9 K, aligning well with conventional superconductivity in Nb. The subgap conductance is primarily influenced by Andreev processes. The ABS spectrum leads to a non-sinusoidal current-phase relation (CPR) with high forward skewness ($+$1.74) and phase sensitivity to the magnetic field. Overall, our findings suggest that the Nb-1D $\text{Bi}_4\text{Br}_4$-Nb weak link is a promising platform for on-chip superconducting magnetic sensors, compatible with scalable nanofabrication and broader development of topological-superconductor hybrid electronics.

Plasmas in space and astrophysical environments are frequently found in (quasi-)stationary states not in thermal equilibrium, indicated by velocity distribution functions (VDFs) featuring nonthermal characteristics, such as Lorentzian high-energy tails and kinetic (temperature and strahl/beam) anisotropies. For decades, in-situ observations of electrons and ions in the solar corona, solar wind or planetary magnetospheres have been modelled using so-called Kappa VDFs, whose standard form, however - depending on the considered VDF moment - imposes stringent constraints on the minimum value of the Kappa power-law parameter. Such limitations have posed difficulties on the derivation of a fully-consistent hydrodynamic formulation of suprathermal plasmas. To improve this situation, the recently proposed regularized Kappa VDF avoids strict constraints on the Kappa parameter by introducing an exponential, regularizing cut-off. In this work we assume that the plasma particles are described by regularized-Kappa VDFs that feature anisotropies not only in their thermal velocity spread parameters but also in their exponential cut-offs. The dielectric tensor components for waves propagating along a magnetic field are derived, in terms of new plasma dispersion functions. The resulting dispersion equations are solved with two complementary techniques, namely by using the analytical expressions derived here and by employing the ALPS code. This is done for physical parameters that give rise to electromagnetic and electrostatic instabilities. Both approaches show excellent agreement and the results demonstrate how an anisotropy in the regularizing cut-off of VDFs can serve, depending on an instability's nature, either as a stabilization factor or as an additional free energy source.

We discuss the relationship between the chain rule for the divergence operator and the renormalization property for weak solutions of the continuity equation. We construct an example of bounded divergence-free vector field on the plane, which demonstrates that in general the first property is not sufficient for the second one.

Equitable allocation of indivisible goods to agents in online settings is an algorithmic primitive with applications for load balancing, network routing, online marketplaces, and multi-agent systems. We consider a general setting in which allocations are constrained to be bases of discrete polymatroids that arrive online. Our work demonstrates that a simple, myopic algorithm called Brick-Laying, which greedily minimizes the sum of squared loads on agents, achieves a universal and objective-free notion of optimality called majorization minimax-optimality [BDK26] for this setting. As a consequence, Brick-Laying simultaneously guarantees minimax optimal competitive ratios and regret for all Schur-concave and Schur-convex objectives, and for any number of agents and resources (despite being agnostic to problem scale). Departing from popular primal-dual analysis, we employ majorization to compare allocations. We leverage the conjugates of integer partitions -- which act as a discrete dual to majorization -- to characterize worst-case instances for the Brick-Laying algorithm. Our approach reveals a novel structural connection between the geometry of partitions and online equitable allocation.

The recent development of small-scale quantum networks poses the question of whether such a technology could also operate at scale in the futuristic Quantum Internet. The question can be answered with a classical approach where an arbitrary quantum network is represented as a classical graph, and communication reliability is assessed using methods proper of network theory. Unfortunately, sufficient conditions for viable network-wide communication have been established only for special topologies like regular lattices. No practical communication protocols have been developed so far for real network topologies, if not for relatively small networks. Here, we overcome these limitations by devising a family of quantum communication protocols that can be applied to networks with arbitrary topology, composed of even hundreds of millions nodes. By performing a systematic analysis on both real and synthetic graphs, we show that the proposed protocols are sustainable on heterogeneous networks. For random scale-free graphs, we analytically prove that viable quantum communication persists in the thermodynamic limit. Our findings provide evidence that the Quantum Internet will be capable of sustaining a ultra-large-scale growth comparable to one already experienced by its classical predecessor.