Many complex systems, ranging from migrating cells to animal groups, exhibit stochastic dynamics described by the underdamped Langevin equation. Inferring such an equation of motion from experimental data can provide profound insight into the physical laws governing the system. Here, we derive a principled framework to infer the dynamics of underdamped stochastic systems from realistic experimental trajectories, sampled at discrete times and subject to measurement errors. This framework yields an operational method, Underdamped Langevin Inference (ULI), which performs well on experimental trajectories of single migrating cells and in complex high-dimensional systems, including flocks with Viscek-like alignment interactions. Our method is robust to experimental measurement errors, and includes a self-consistent estimate of the inference error.

Functions that are not differentiable in the classical sense have become a central tool in modern mathematical models for imaging, inverse problems, machine learning, and optimal control of differential equations. These models are increasingly formulated in infinite-dimensional function spaces to be independent of problem size and discretization quality. This book presents a unified and rigorous introduction to the infinite-dimensional analysis and algorithmic solution of nonsmooth optimization problems arising from the above-mentioned models, from the necessary theoretical tools of nonsmooth analysis to state-of-the-art algorithms and their convergence and stability analysis.

A graph is diameter-2-critical if its diameter is 2 but the removal of any edge increases the diameter. A well-studied conjecture, known as the Murty-Simon conjecture, states that any diameter-2-critical graph of order n has at most n${}^2$/4 edges, with equality if and only if G is a balanced complete bipartite graph. Many partial results about this conjecture have been obtained, in particular it is known to hold for all sufficiently large graphs, for all triangle-free graphs, and for all graphs with a dominating edge. In this paper, we discuss ways in which this conjecture can be strengthened. Extending previous conjectures in this direction, we conjecture that, when we exclude the class of complete bipartite graphs and one particular graph, the maximum number of edges of a diameter-2-critical graph is at most ((n -- 1)${}^2$/4) + 1. The family of extremal examples is conjectured to consist of certain twin-expansions of the 5-cycle (with the exception of a set of thirteen special small graphs). Our main result is a step towards our conjecture: we show that the Murty-Simon bound is not tight for non-bipartite diameter-2-critical graphs that have a dominating edge, as they have at most (n${}^2$/4) -- 2 edges. Along the way, we give a shorter proof of the Murty-Simon conjecture for this class of graphs, and stronger bounds for more specific cases. We also characterize diameter-2-critical graphs of order n with maximum degree n -- 2: they form an interesting family of graphs with a dominating edge and 2n -- 4 edges.

The present paper deals with connected subtraction games in graphs, which are generalization of takeaway games. In a connected subtraction game, two players alternate removing a connected sub-graph from a given connected game-graph, provided the resulting graph is connected, and provided the number of vertices of the removed subgraph belongs to a prescribed set of integers. We derive general periodicity results on such games, as well as specific results when played on subdivided stars.

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd \left( a^{2}, b \right)$ is squarefree and $X^{2}- \left( a^{2}+b \right) Y^{2}=-4$ has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations.

We introduce CUT, the class of 2-player partition games. These are NIM type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can perform on a single heap. In normal play, the player with the last move wins, and the famous Sprague-Grundy theory provides a solution. We prove that several rulesets have a periodic or an arithmetic periodic Sprague-Grundy sequence (i.e. they can be partitioned into a finite number of arithmetic progressions of the same common difference). This is achieved directly for some infinite classes of games, and moreover we develop a computational testing condition, demonstrated to solve a variety of additional games. Similar results have previously appeared for various classes of games of take-and-break, for example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and Guy (1982). In this context, our contribution consists of a systematic study of the subclass `break-without-take'.

Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum group associated to g is isomorphic as an algebra to the trivial deformation of the universal enveloping algebra of g. In this paper we construct explicitly such an isomorphism when g = sl(n), previously known only for n=2.

The game Arc-Kayles is played on an undirected graph with two players taking turns deleting an edge and its endpoints from the graph. We study a generalization of this game, Weighted Arc Kayles (WAK for short), played on graphs with counters on the vertices. The two players alternate choosing an edge and removing one counter on both endpoints. An edge can no longer be selected if any of its endpoints has no counter left. The last player to play a move wins. We give a winning strategy for WAK on trees of depth 2. Moreover, we show that the Grundy values of WAK and Arc-Kayles are unbounded. We also prove a periodicity result on the outcome of WAK when the number of counters is fixed for all the vertices but one. Finally, we show links between this game and a variation of the non-attacking queens game on a chessboard.

Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore. We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, an octal game on a path P\_n is equivalent to playing the same octal game on a heap of n counters. We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph. We study this game on trees and give a complete resolution of this game on subdivided stars and bistars.

We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the problem of finding a coloring with the minimum number of colors where every vertex receives a distinct label. Finding such a coloring generalizes several other well-known problems of vertex-distinguishing colorings in graphs.We show that for any graph (without connected component reduced to an edge or a single vertex), the minimum number of colors for which such a coloring exists can only take 3possible values depending on the order of the graph. Moreover, we provide the exact value for paths, cycles and complete binary trees.

We perform a thorough study of thermal leptogenesis adding finite temperature effects, RGE corrections, scatterings involving gauge bosons and by properly avoiding overcounting on-shell processes. Assuming hierarchical right-handed neutrinos with arbitrary abundancy, successful leptogenesis can be achieved if left-handed neutrinos are lighter than 0.15 eV and right-handed neutrinos heavier than 2 10^7 GeV (SM case, 3sigma C.L.). MSSM results are similar. Furthermore, we study how reheating after inflation affects thermal leptogenesis. Assuming that the inflaton reheats SM particles but not directly right-handed neutrinos, we derive the lower bound on the reheating temperature to be T_RH > 2 10^9 GeV. This bound conflicts with the cosmological gravitino bound present in supersymmetric theories. We study some scenarios that avoid this conflict: `soft leptogenesis', leptogenesis in presence of a large right-handed (s)neutrino abundancy or of a sneutrino condensate.

Spontaneous decay process of an excited atom placed inside or outside (near the surface) a carbon nanotube is analyzed. Calculations have been performed for various achiral nanotubes. The effect of the nanotube surface has been demonstrated to dramatically increase the atomic spontaneous decay rate -- by 6 to 7 orders of magnitude compared with that of the same atom in vacuum. Such an increase is associated with the nonradiative decay via surface excitations in the nanotube.

This paper studies the data-driven synthesis of linear quadratic integral (LQI) controllers for continuous-time systems. The objective is to achieve optimal state-feedback control with integral action for reference tracking using only measured data. To this end, we derive a data-driven closed-loop parameterization of the augmented dynamics that incorporates the integral state while relying solely on input-state-output measurements of the underlying system. Based on this parameterization, a data-driven convex optimization problem is formulated whose solution yields the optimal linear quadratic regulator (LQR) feedback gain for the augmented system without explicit knowledge of the system matrices. In addition, a policy gradient flow is derived to compute the optimal controller within the space of stabilizing gains. The proposed approach enables data-driven optimal tracking control while avoiding explicit state augmentation in the data collection phase. The effectiveness of the method is demonstrated through a numerical example involving a distributed generation unit (DGU) in a DC microgrid.

The pair-coupled-cluster doubles (pCCD) method has emerged as a viable approach for quantum-chemical studies of strongly correlated systems. Despite its lower formal scaling (O(N$^4$)) compared to other versions of coupled cluster (CC) theory, applications to large chemical structures are still expensive. Fragmentation and embedding strategies offer a viable approach in such cases. In this work, we present a simple and efficient density-embedding scheme based on pCCD electron densities. The main computational benefit arises from the fact that pCCD response $Λ$-equations are much cheaper to compute than those of standard CC methods, providing easy access to one-electron properties. The pCCD densities of the individual subsystems are used to generate static embedding potentials that capture the environment's effect on the embedded system. The individual fragment energies are then iteratively converged in a self-consistent fashion. We demonstrate the reliable performance of this scheme with the estimation of dipole moments of the weakly bound CO2$\cdots$Rg (Rg = He, Ne, Ar, and Kr) complexes and with the modeling of vertical excitations of some microsolvated molecules.

We propose a quantitative notion of permutation stability for finitely generated groups. Our notion is related to, but distinct from, the ``stability rate'' introduced by Becker and Mosheiff (which is valid within the class of finitely presented groups). We construct a family of finitely generated stable groups which exhibit, quantitatively, arbitrarily ``bad'' permutation stability. This means that any application of a ``sample-and-substitute'' algorithm will be very slow in ascertaining whether a given tuple of permutations satisfy the defining relations of our groups.

We present the study of an X-ray selected sample of active galactic nuclei (AGN) in pairs at projected spatial separations 1 <$ r_p$/kpc < 100 at z < 0.1, using XMM-Newton and Chandra data. The pair sample is derived from an initial pool of approximately 2,000 X-ray-selected AGN, and is composed of both AGN-AGN pairs (so called dual AGN) and AGN-galaxy pairs. From this selection, we find that approximately 10% of AGN reside in pairs, and about 4% are paired with another AGN. We performed a detailed X-ray and SDSS optical spectral analysis for AGN in duals and X-ray analysis for AGN in AGN-galaxy pairs, to characterise their absorption properties and investigate the possible triggering mechanisms. We then investigated how obscuration, luminosity, and Eddington ratio depend on projected separation $r_p$. Amongst all AGN in pairs, we found that ~55% are obscured (with hydrogen column density $N_H$ > $10^{22}$ cm$^{-2}$), amongst which ~6% are Compton-thick ($N_H$ > $10^{24}$ cm$^{-2}$). The fraction of absorbed AGN is significantly higher in late-stage mergers ($r_p$ < 30 kpc) compared to early-stage mergers ($r_p$ > 60 kpc). Amongst the AGN in pairs, we also observed an average excess of AGN pairs with respect to a control sample of inactive galaxies in pairs, and that such excess significantly increases with decreasing $r_p$ only for obscured AGN. Finally, in dual-AGN systems, both the bolometric luminosity and the Eddington ratio of the less massive black hole in the pair increase as the separation decreases. These findings suggest that mergers may have an important role in triggering AGN accretion and activity.

This paper addresses distributed consensus optimization problems with mixed-integer variables, with a specific focus on Boolean variables. We introduce a novel distributed algorithm that extends the Consensus Augmented Lagrangian Alternating Direction Inexact Newton (CALADIN) framework by incorporating specialized techniques for handling Boolean variables without relying on local mixed-integer solvers. Under the mild assumption of Lipschitz continuity of the objective functions, we establish rigorous convergence guarantees for both convex and nonconvex mixed-integer programming problems. Numerical experiments demonstrate that the proposed algorithm achieves competitive performance compared to existing approaches while providing rigorous convergence guarantees.

We investigate optically induced phase transitions in the two-dimensional (2D) ferroelectric (FE) material Nb2O2I4 using real-time time-dependent density functional theory (rt-TDDFT). Our results demonstrate that tailored laser pulses can activate specific coherent phonon modes. Specifically, the anharmonic atomic distortions of the A1-1 and A1-2 modes at the Γ-point facilitate the reversal of in-plane polarization. By fine-tuning laser parameters, additional phonon modes at both the Y and Γ points are excited. The resulting nonequilibrium atomic dynamics enable the formation of previously unreported ferroic phases, including three antiferroelectric (AFE) phases and one ferrielectric (FiE) phase. Notably, these optically induced phases can be reverted to the initial FE state using appropriate techniques. This controllable reversibility among multiple ferroic phases positions 2D Nb2O2I4 as a highly promising candidate for next-generation electronic storage applications.

We review the recent progress made with regard to the hadronic light-by-light (HLbL) contribution to the Standard Model prediction of the muon anomalous magnetic moment and how well this compares with predictions from holographic QCD models, which had predicted larger contributions from axial vector mesons and short-distance constraints than the White Paper of 2020. A new holographic prediction concerns tensor-meson contributions, which in holographic QCD play a significant role in short-distance constraints beyond the Melnikov-Vainshtein constraint. When matching also the symmetric longitudinal short-distance constraint, the resulting prediction for the tensor-meson transition form factors agree well with available singly virtual data, but lead to different results than the traditional quark-model ansatz and a sizable positive contribution that could explain the remaining current tension between lattice and data-driven results for the HLbL contribution.

We present a method to construct explicit degenerations of higher-dimensional generalized Kummer varieties. We start with a simple degeneration $f: \mathcal Y \to C$ of abelian surfaces. Then $ \mathcal{Y} \setminus \mathcal{Y}_0$ is an abelian scheme over $C \setminus 0$ and we can form the relative generalized Kummer variety $K^{n-1}_{\circ} = \mathrm{Kum}^{n-1}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. This is naturally a closed subscheme of the relative Hilbert scheme $\mathrm{Hilb}^{n}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. In previous work (joint with Gulbrandsen) we had constructed a compactification $I^n_{\mathcal{Y}/C}$ over $C$ of the latter scheme. The closure $K^{n-1}_{\mathcal{Y}/C}$ of $K^{n-1}_{\circ}$ inside $I^n_{\mathcal{Y}/C}$ yields a canonical way to degenerate the family of generalized Kummer varieties, and is the degeneration we propose. This paper contains a detailed study of the geometry of the scheme $K^{n-1}_{\mathcal{Y}/C}$ and its natural stratification. For $n=2$ we obtain a projective Kulikov model of Kummer surfaces, whereas already for $n=3$ new phenomena occur. We study in detail the dual complex of $K^{2}_{\mathcal{Y}/C}$ and show that this is PL-homeomorphic to the standard $2$-simplex.

Sub-THz bands are promising high bandwidth and data rates, and in the recent years the device technologies made large progress and provided a multitude of transceiver, power amplifier (PA) and phased array devices supporting the frequency bands above 100 GHz. The more painful aspect of sub-THz transmission is the increased power consumption, caused by the large data rates and the related data conversion and processing effort, and on the analog side the low achievable PA efficiency and the reduced achievable output power. When planning a deployment of sub-THz communication systems, the target coverage and throughput can be achieved with a variety of scenarios, which will be different with respect to locations and number of base stations and system architectures. Although leading to similar performance, they will differ significantly in the overall power consumption. With an accurate power consumption model, including also baseband (BB) processing functionality, and system level simulations for different hybrid beamforming and MIMO schemes the related variations in power consumption in relation to a given performance are evaluated. This paper shows the critical design aspects for energy efficient sub-THz deployments by highlighting the sub- THz specific trade-offs between different number of BS with different transmit powers but also changing number of BB units and RF chains.

The gravo-turbulent fragmentation of the interstellar medium is expected to create a hierarchical cascade of cloud structures, crossing the scales from core to disk. We aim to predict how the currently observed top-heavy core mass function (CMF) in the massive protocluster W43-MM1 evolves due to core subfragmentation. We used the getsf algorithm to extract sources in five ALMA images of W43-MM1 at 3 mm, with a spatial resolution ranging from 14 kau to 270 au. Then, we applied FAMILY, a graph-theory-based analysis tool, to create and characterize networks of nested sources in W43-MM1. We compared the hierarchical fragmentation cascade of W43-MM1 to those measured in the NGC 2264 protocluster and in synthetic images of an Orion-like protocluster simulated by magneto-hydrodynamical calculations. Assuming self-similarity, we measure a small fractality index of mathcal F3D =1.19+/-0.10 in W43-MM1, which means that, on average, a cloud structure will fragment into only 1.19 fragments each time the physical scale decreases by a factor of two. We estimate an imbalanced mass partition between siblings, with 2/3 of the mass of siblings at a given scale belonging to the dominant sibling. The mass transfer efficiency, computed from one physical scale to another, is high and corresponds to a core formation efficiency (CFE) from 2400 au to 200 au of ~16%. Based on the fractality and efficiency values measured in W43-MM1, the gravo-turbulent model by Thomasson et al. predicts that its fragmentation below ~14 kau is not driven by turbulence but by gravity. Using these parameters and the measured mass partition, we demonstrate that the fragment mass function, from which the the initial mass function (IMF) emerges, has a high-mass end which remains top-heavy. Therefore, core subfragmentation in W43-MM1, and perhaps more broadly in massive Galactic protoclusters, plays a minimal role in the IMF origin.

Modelling micro- and meso-scopic scale thermodynamic and transport properties of soft condensed matter hinges upon its representation. This is especially relevant for polar solvents such as water, since these require effective representation of their dielectric nature as driven by molecular charge distributions and molecular network structuring. The dielectric nature of a medium leads to complex phenomena such as local polarisability response and restructuring near interfaces in reaction to changes in local charge distributions. Inclusion of such phenomena when using larger-than-atomistic techniques such as coarse-grained molecular dynamics (CG-MD) and dissipative particle dynamics (DPD) is still an open question, to which we provide a novel way to consider and justify the necessary and suitable coarse-graining level, enabling us to compare new polar CG models' performance against that of an underlying atomistic model. We polarise our previous non-polar nDPD water model to prepare it for use in simulations of liquid electrolytes as well as solvated organic membranes and measure its fitness to serve as a dielectric medium by comparing its properties to those of the TIP3P water model, while simultaneously observing changes to properties already represented well by the non-polar model.

LLM-based mobile GUI agents treat every task invocation as an independent reasoning episode, requiring a full LLM inference call at each action step. This per-step dependence makes them stateless: a task completed successfully yesterday is re-derived from scratch today, with no improvement in reliability or speed. We present SkillDroid, a three-layer skill agent that compiles successful LLM-guided GUI trajectories into parameterized skill templates (sequences of UI actions with weighted element locators and typed parameter slots) and replays them on future invocations without any LLM calls. A matching cascade (regex patterns, embedding similarity, and app filtering) routes incoming instructions to stored skills, while a failure-learning layer triggers recompilation when skill reliability degrades. Over a 150-round longitudinal evaluation with systematic instruction variation and controlled perturbations, SkillDroid achieves an 85.3% success rate (23 percentage points above a stateless LLM baseline) while using 49% fewer LLM calls. The skill replay mechanism achieves a perfect 1000% success rate across 79 replay rounds at 2.4 times the speed of full LLM execution. Most critically, the system improves with use: its success rate converges upward from 87% to 91%, while the baseline degrades from 80% to 44%.

We study the coupled cosmological evolution of primordial black holes (PBHs) and radiation in the Arkani-Hamed-Dimopoulos-Dvali (ADD) framework with $n$ large extra dimensions and a fundamental gravity scale $M_\star$ at the TeV scale. For PBHs with horizon radius smaller than the compactification scale, the higher-dimensional geometry implies a larger horizon size at fixed mass and therefore a suppressed Hawking temperature. As a result, radiation accretion can overcome evaporation in the early Universe and drive a ``runaway'' phase of rapid mass growth. By numerically solving the coupled mass and energy-density evolution equations, we show that for $n \geq 2$ initially microscopic PBHs with initial mass $M_i \gtrsim 10^{12}\,$g can grow by many orders of magnitude and potentially reach macroscopic, even solar-mass, scales by matter-radiation equality. We determine the critical initial abundance $β_{\rm crit}$ required for PBHs to account for the observed dark matter density and find that extra dimensions dramatically lower this threshold, allowing viable scenarios with $β_{\rm crit}\sim 10^{-44}$. This identifies a previously unexplored region of parameter space in which the dark matter abundance is achieved through dynamical mass growth rather than large initial collapse fractions.

Sub-Terahertz radio-stripe and distributed MIMO architectures promise extreme spatial reuse and multi-GHz bandwidths, but the cascaded fiber front-haul and RF hardware impairments strongly shape end-to-end performance. This paper presents an open-source, configuration-driven simulator that models the full waveform-level signal chain from CP-OFDM baseband generation in the central unit, through measurement-parameterized polymer microwave fiber and coupler links, to booster/active Radio Units (RUs) with configurable nonlinearity, noise, in-phase and quadrature imbalance, and oscillator phase noise and carrier frequency offset. Wireless propagation is supported via lightweight deterministic and stochastic per-subcarrier channel models as well as site-specific ray-tracing datasets generated with a companion Sionna ray-tracer module. The simulator exports intermediate waveforms and system metrics (e.g., normalised mean square error, signal-to-noise-and-distortion ratio, bit error rate) to enable reproducible studies of impairment accumulation, calibration, and algorithmic choices such as RU selection and beam management.

Interfacial deformation under electric fields is a common phenomenon in many industrial processes. Particularly, we are interested in the dynamics of sessile soap bubbles in a parallel-plate electric field which exhibits a stable deformation regime followed by conical instability. Using side-view imaging, we track the equilibrium shapes, the transition to the unstable regime, and the pre-jet apex dynamics within one experimental system. In the stable regime, the meridional profile is well described by a spheroidal fit, and the aspect ratio collapses across initial bubble sizes onto a single steady-state branch when plotted against the dimensionless field $E^\ast = \sqrt{\mathrm{Bo}_e}$ for data acquired within a fixed ambient session where the electric Bond number $\mathrm{Bo}_e$ is defined as $\varepsilon_0 E_0^2 R_0/(2γ)$. The endpoint of this branch marks the transition to the unstable regime. Above onset of instability, the apex sharpens into a cone with half-angle $30.0^{\circ}$ $\pm$ $0.6^{\circ}$, below the classical Taylor value. To quantify the late pre-jet stage, we define the axial distance $b(t)$ from the instantaneous apex to a fixed reference vertex determined from the terminal cone geometry and measure its evolution. The corresponding rate grows as jetting is approached, and a near-tip inertia-capillary model captures the observed logarithmic trend as an approximation. Together, these measurements establish a single-system experimental benchmark in which stable electrocapillary deformation is organized by a single steady-state branch that leads into conical instability and pre-jet dynamics.

The Caucasus Mountain Observatory of the Sternberg Astronomical Institute of Moscow State University is the only one in Russia and one of the few in the world where is possible to obtain spectral data in the near-infrared (IR) range at $λ$=1-2.5 $μ$m. However, there is a problem of processing the spectra of extended objects, the angular dimensions of which exceed the length of the slit (4.5 arcmin). Obtaining additional spectra of the sky in the immediate vicinity of such objects does not solve the problem, since bright atmospheric hydroxyl lines at $λ$~2 $μ$m change their intensity significantly over a time shorter than the exposure time of a single frame. We have developed a technique that allows us to correctly account for and exclude the contribution of variable atmospheric lines in the spectra of extended objects. This technique has been successfully tested in spectroscopic studies of the star-forming region NGC 7538 (S158) in our Galaxy.

This paper investigates distributed resource allocation optimization over directed graphs with limited communication bandwidth. We develop a novel distributed algorithm that integrates the centralized Proximal Jacobian Alternating Direction Method of Multipliers (PJ-ADMM) with a finite-level quantized consensus scheme, enabling nodes to cooperatively solve the optimization in a distributed fashion. Under the assumption of convex objective functions, we establish that the proposed algorithm achieves sublinear convergence to a neighborhood of the optimal solution, with the convergence accuracy explicitly bounded by the quantization level. Numerical experiments validate that the algorithm achieves competitive performance compared to existing approaches while exhibiting communication efficiency.

Plasmon modes, their dispersion, and the onset of damping when approaching the electron-hole continuum are well understood when electron correlations are weak. However, we know little about how this picture is modified and what additional features emerge in strongly correlated materials. Here, we present a fully ab initio approach to plasmon excitations that combines density functional theory with dynamical mean-field theory, and we use it to reconcile controversial electron energy-loss spectroscopy results in Sr$_2$RuO$_4$. In particular, we show that electronic correlations reproduce the plasmon dispersion, while generating a large intrinsic width already below the electron-hole continuum. An additional high-energy peak reflecting transitions between incoherent features and a sharp increase of the plasmon's energy-momentum dispersion, akin to waterfalls in photoemission spectroscopy, are identified as genuine correlation effects.