arXiv subjects

Kansuke Nunota

Publications and source records attributed to Kansuke Nunota.

At least 19 recordsLinked to original sources

arXiv2026

Harmonic Angular Mapping Method for Binary Microlensing Grid Search

Searching for binary-lens microlensing solutions is computationally demanding because of the complex parameter space and the large number of trial models required. We present an algorithm that accelerates binary-lens grid searches by expanding the angular dependence of precomputed magnification maps in Fourier modes. By contracting the observation-dependent weighted sums in Fourier space, the $χ^2$ values for all trial source-trajectory angles can be evaluated efficiently using inverse fast Fourier transforms (FFTs), avoiding repeated calculations over individual observations. We tested the method on 200 Roman-like synthetic binary-lens events. For 166 events (83%), the recovered fits agreed with Levenberg--Marquardt fits initialized at the injected parameters to within $|Δχ^2|\leq1$. The FFT map scan required an average of 5.05~min per event. For the same discrete search volume, the Fourier angular evaluation was estimated to be approximately 900 times faster than conventional magnification-map sampling. These results demonstrate that the method can efficiently identify useful initial conditions for binary-lens light-curve modeling.

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arXiv2026

Two Isolated Microlensing Events Near the Upper Shore of Einstein Desert

The Einstein desert is a deficit in the angular Einstein-radius ($θ_{\rm E}$) distribution of isolated objects over the range $9 {\rm μas} < θ_{\rm E}<26 {\rm μas}$. Although this interval contained no events when the desert was first identified, several subsequent detections have fallen within it, raising questions about its detailed structure and requiring a larger detection sample. In this paper, we report the observations and analyses of two microlensing events possibly caused by isolated giant planets or brown dwarfs whose $ θ_{\rm E}$ lie near the upper shore of the desert. MOA-2021-BLG-145 has $ θ_{\rm E} = 23.1\pm2.0 ~{\rm μas}$, placing it just inside the upper boundary, and KMT-2026-BLG-0357 has $ θ_{\rm E} = 36.5\pm 3.7~{\rm μas}$, placing it above the desert. We also compare the $ θ_{\rm E}$ distributions of isolated objects and bound planetary events. The bound population shows no corresponding deficit over the same range, suggesting that the Einstein desert may be associated with the formation or subsequent evolution of the isolated population. Furthermore, some FFPs may lie above the upper shore of the Einstein desert, indicating that a more complete understanding of the FFP population requires direct lens-mass measurements.

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arXiv2026

OGLE-2012-BLG-0563: Relative Proper Motion Constraint from Keck Follow-up finds the correct Lens

We present Keck/NIRC2 adaptive optics imaging results of planetary microlensing event OGLE-2012-BLG-0563. We find that the OGLE-2012-BLG-0563L planetary system consists of a 1.12 +/- 0.09 Jupiter-mass planet orbiting a 0.81 +/- 0.04 solar-mass host star at a distance of 5.49 +/- 0.56 kpc from the Sun. This event has a lens-source separation of 23.28 +/- 0.48 mas, which is the smallest lens-source separation measured to date with Keck AO data. We find that the incorrect solution - which identifies the Northeast (NE) star as the source star - is favored if one only considers the discovery paper, the Fukui et al. (2015) solution, or an unconstrained model. However, by performing an image-constrained light curve model fit using the relative proper motion direction constraint from the Keck follow-up data, we demonstrate that the correct solution identifies the NE star as the lens star. The discrepancy in the unconstrained models is driven by systematics in the FTS and MOA B & C data, alongside the impact of the relative proper motion direction constraints. This outcome offers two important lessons for future Roman data analysis: (1) thoroughly verify systematics across all passbands and identify the precise origin of chi-squared differences, and (2) utilize image-constrained light curve modeling incorporating relative proper motion direction constraints to identify the correct lens.

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arXiv2026

Two Low-Mass-Ratio Microlensing Planets from High-Magnification Events: KMT-2021-BLG-0247 and MOA-2023-BLG-169

We present an analysis of two planetary microlensing events, KMT-2021-BLG-0247 and MOA-2023-BLG-169, both of which exhibit high magnifications and low planet--host mass ratios, but with markedly different levels of parameter constraint. For KMT-2021-BLG-0247, the finite-source effect and microlens parallax are both clearly detected, leading to unusually tight constraints on the physical properties of the lens system. The host mass and lens distance are approximately $M_{\rm L}=0.9\pm0.1\,M_\odot$ and $D_{\rm L}=6.7^{+1.0}_{-0.3}\,\mathrm{kpc}$, respectively, and the planet mass is $M_{\rm p}=41\pm5\,M_\oplus$. The projected planet--host separations are $a_\perp=3.4^{+0.7}_{-0.3}\,\mathrm{au}$ and $3.0^{+0.6}_{-0.3}\,\mathrm{au}$ for the wide and close solutions, respectively. In contrast, MOA-2023-BLG-169 involves an extremely faint source, with $I_{S,{\rm OGLE}}=26.73$, such that the light curve permits a broad family of strongly correlated solutions. We therefore construct the physical-parameter likelihood using approximate invariant combinations of the light-curve parameters and supplement it with post-event Euclid/VIS imaging from the Euclid Q2 Galactic Bulge Survey (Beaulieu et al. 2026). The Euclid data reveal a component at the event position with ${\rm VIS}_{\rm AB}=23.05\pm0.09\,{\rm mag}$. Interpreting this component as the combined light from the source and any luminous lens, the resulting posterior gives $M_{\rm L}=0.48^{+0.23}_{-0.17}\,M_\odot$, $D_{\rm L}=3.50^{+2.82}_{-1.20}\,\mathrm{kpc}$, $a_\perp=1.62^{+0.77}_{-0.43}\,\mathrm{au}$, and $M_{\rm p}=143^{+103}_{-63}\,M_\oplus$. MOA-2023-BLG-169 demonstrates that, for an extreme faint-source event, even the event timescale and angular Einstein radius can remain strongly dependent on the Galactic prior and independent flux constraints.

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arXiv2026

First Direct Identification of a Multi-Star Microlens System Hosting a Planet

Nearly 25 years have passed since gravitational microlensing delivered its first exoplanet discoveries. The relative proper motion between lens and source systems allows them to be resolved as they separate on the sky over many years. This enables scientists to study light coming directly from the lenses and place tight constraints on the physical properties of these planetary systems. In this paper we present an analysis of high angular resolution images of the microlensing target OGLE-2006-BLG-284 using Keck adaptive optics and HST data taken more than 15 years after the event. We clearly detect the lens system and measure its motion across the Keck and HST epochs. We measure a lens-source relative proper motion of $5.80 \pm 0.15$ mas yr$^{-1}$. A combination of the light curve parameters and the host star(s) brightnesses in $V,\, I$, and $K$ passbands gives primary and secondary host star masses of $M_{\rm 1} = 0.43 \pm 0.05\,M_{\odot}$ and $M_{\rm 2} = 0.12 \pm 0.02\,M_{\odot}$, and a planetary companion mass of $m_p = 165^{+40}_{-33}\,M_{\oplus}$. Although we are unable to determine if the planet orbits one or both stars in the system, our analysis motivates next-generation high-precision RV measurements that can distinguish these two possibilities. This is the first direct detection of flux coming from a two-star microlensing system which hosts a planet.

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arXiv2026

A Bayesian Inference Framework for Binary Lens Events Fully Incorporating Higher-Order Effects

Higher-order effects, such as microlensing parallax and lens orbital motion, are essential for characterizing the physical properties of microlensing planetary systems including their masses and distances. In practice, higher-order effects are often included in light-curve modeling only when their signals are strong, because their inclusion in weakly constrained cases tends to drive the inferred parameters toward regions of parameter space that are disfavored by standard Galactic models. This binary choice over physically continuous effects introduces an event-dependent selection function that complicates population-level interpretation and prevents strong and weak detections from being combined within a single uniform framework. In this work, we investigate the origin of instabilities associated with higher-order effects, and show that such instabilities can arise from a mismatch between commonly adopted uninformative priors in the light-curve parameter space and the distributions predicted by Galactic models. Motivated by this, we develop a new Bayesian framework that enables higher-order effects to be incorporated in a stable and uniform manner. As part of this framework, we introduce Galactic Prior Modeling Engine (gapmoe), a dedicated tool that enables efficient evaluation of the Galactic prior density and allows it to be incorporated directly into the light-curve inference. Using simulated events, we demonstrate that our framework robustly recovers the true physical parameters while fully accounting for microlensing parallax and lens orbital motion. This framework eliminates the need for ad-hoc model selection and provides a scalable and statistically consistent pathway for analyzing the large samples expected from next-generation surveys such as the Roman Space Telescope.

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arXiv2026

Testing KMTNet--PRIME Optical--Near-Infrared Source-Color Constraints in KMT-2024-BLG-0211 and KMT-2024-BLG-1522

We present analyses of two 2024 microlensing events, KMT-2024-BLG-0211 and KMT-2024-BLG-1522, jointly observed by KMTNet in the optical and PRIME in the near-infrared. For KMT-2024-BLG-0211, the KMTNet--PRIME $I-H$ color provides a useful constraint on the angular source radius for a short-timescale finite-source event with a giant source. The event is consistent with a low-mass stellar lens, although the weak parallax constraint leaves the lens mass and distance uncertain. For KMT-2024-BLG-1522, we compare the results obtained from $V-I$, $I-H$, $V-H$, and $J-H$ color constraints. The microlensing parameters are nearly identical among the four analyses, yielding a robust binary-lens solution with nearly equal masses. However, the inferred source properties and lens physical parameters depend on the adopted source color because the different color estimates imply different angular source radii. Comparing the $(V-I)_{\rm KMT}$ versus $(I-H)_{\rm KMT,PRIME}$ plane shows that the inferred source colors lie off empirical color--color relations but within the scatter of observed stars. The prevalence of such deviations should be tested with a larger sample of KMTNet--PRIME events and this specific case can be further tested with future adaptive-optics follow-up, which can directly measure the lens--source relative proper motion and lens flux.

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arXiv2026

A Search For Stellar-mass Black Holes Via Astrometric Microlensing II: 2012-2015 Keck Candidates

The Milky Way is expected to host $\sim$10$^8$ stellar-mass black holes with an uncertain binary fraction. The only proven method to detect isolated stellar mass black holes is gravitational microlensing. Here we report the results of a microlensing search for black holes with photometry and astrometry. By combining 10 years of seeing-limited photometry from OGLE and MOA with diffraction-limited photometry and astrometry from adaptive optics imagers at the W.~M.~Keck Observatory, we constrain lens masses for OGLE-2012-BLG-0169, OGLE-2014-BLG-0613/MOA-2015-BLG-041, OGLE-2015-BLG-0029/MOA-2015-BLG-170, and OGLE-2015-BLG-0211. Of the four long-duration microlensing events monitored, we ruled out black hole lenses in 3 events, which likely have stellar or white dwarf lenses. OGLE-2015-BLG-0211 remains a black hole candidate with a poorly constrained lens mass with a 1$σ$ upper mass limit of 3.2$M_\odot$ and a 3$σ$ upper mass limit of 21.6$M_\odot$. This event suffered from poor observing conditions and significant astrometric reference frame uncertainties, but its analysis may benefit from additional astrometric data in the upcoming Gaia Data Release 4. Of the six long-timescale ($t_E>100$ days) microlensing events from this work and previous studies, one black hole has been confirmed with a second not ruled out. We briefly examine Galactic model simulations and find that our result agrees with current expectations. Ultimately, we need a larger sample of isolated black holes to constrain their formation processes. This will be possible in the coming years with Rubin and Roman, as well as improved astrometry from JWST and large, ground-based telescopes equipped with adaptive optics.

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arXiv2026

KMT-2026-BLG-0083L: A Two-Jovian-Planet System Orbiting an M Dwarf Discovered by Microlensing

We present the analysis of the microlensing event KMT-2026-BLG-0083, which was independently detected by the KMTNet, OGLE, and PRIME surveys and also monitored by the DREAMS survey. The combined data provide dense coverage of two distinct short-duration anomalies in the light curve that cannot be reproduced by a standard binary-lens model. Independent analyses of the two anomalies indicate that each is produced by a planetary companion to the lens, motivating a triple-lens interpretation. The modeling yields two pairs of degenerate solutions arising from the well-known inner--outer degeneracy, with each pair exhibiting two local solutions depending on whether the source passes above or below the distant planetary companion. The preferred model indicates two giant planets with mass ratios $q_2=(5.25\pm 0.08)\times 10^{-3}$ and $q_3=(3.04\pm 0.24)\times10^{-3}$ orbiting a common host. Bayesian analysis indicates that the host is an M-dwarf star with a mass of $0.48^{+0.36}_{-0.28}~M_\odot$, hosting two giant planets with masses of $2.65^{+1.96}_{-1.55}~M_{\rm J}$ and $1.54^{+1.14}_{-0.90}~M_{\rm J}$. The projected planet--host separations are $10.6^{+1.7}_{-2.2}$~au and $1.7^{+0.3}_{-0.3}$~au, placing the inner planet near the host's snow line and the outer planet at a substantially larger separation. Thus, KMT-2026-BLG-0083L becomes the seventh confirmed multiple-planet system discovered through gravitational microlensing, providing another example of a cold giant planetary system orbiting a subsolar-mass star.

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arXiv2026

MOA-2020-BLG-108Lb: A Giant Planet Beyond the Snow Line of a Low-Mass Lens Near the Lower Boundary of the Mass-Ratio Desert

We present an analysis of the microlensing event MOA-2020-BLG-108, which was discovered in June 2020 by the MOA collaboration toward the Galactic bulge. The observed light curve shows significant deviations from the standard single-lens single-source model. We find two degenerate binary-lens single-source solutions, corresponding to the wide and close configurations, with a companion-to-host mass ratio of $q\sim0.02$ and projected host--companion separations of $s=1.33\pm0.01$ and $s=0.76\pm0.01$, respectively. These solutions improve the fit by $Δχ^2>4430$ compared to the single-lens model. We detected the finite-source effect in the light curve and obtained the angular Einstein radius of $θ_{\rm E} = 0.7\pm0.1\:\mathrm {mas}$, which provides a mass--distance relation for the lens. We conducted a Bayesian analysis to estimate the physical parameters of the lens system. The results indicate that the lens system consists of a host star with a mass of $M_{\rm L,H} \sim 0.6\:M_\odot$ at a distance of $D_{\rm L}\sim5$ kpc and a giant planet with a mass of $M_{\rm L,C}\sim10\:M_{\rm {Jup}}$ orbiting beyond the snow line. Conventional planet formation theories suggest that giant planets are unlikely to form around low-mass stars. Furthermore, several statistical studies have suggested the existence of a companion-to-host mass-ratio desert in the range $0.02 \lesssim q \lesssim 0.05$, and the companion in the lens system discovered in this work lies near the lower boundary of this desert. Objects near the planet-brown dwarf boundary may form through multiple pathways, and this discovery provides an additional data point for understanding their formation mechanisms.

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arXiv2026

Systematic KMTNet Planetary Anomaly Search. XIII. Complete Sample of 2021 Prime Field Planets

The Systematic KMTNet Planetary Anomaly Search series was conducted using the KMTNet data archived from $2016$ to $2019$. From this first phase of the series, we reported a total of $50$ planetary systems hidden in the data archive, which represent about $35\%$ of the total microlensing planets discovered from $2016$ to $2019$, demonstrating that this semi-machine-based search is a crucial channel for building a complete microlensing planet sample. We continue this series for $2021$ and beyond to expand the microlensing planet sample. In this work for the $2021$ KMTNet high-cadence fields (Prime fields), we find seven hidden planetary systems and three planet candidates. These new planets represent about $33\%$ of the total microlensing planets discovered within the Prime fields observed during the $2021$ bulge season. While the by-eye search is the primary channel for detecting microlensing planets (i.e., two-thirds of microlensing planet discoveries), this work clearly shows that a systematic search series is still necessary for constructing a complete microlensing planet sample. Such a sample is essential for conducting unbiased statistical studies of planet demographics in our Galaxy. Datasets for all the events used for analyses in this work are publicly available: doi:10.5281/zenodo.21472225.

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arXiv2026

Characterizing Microlensing Planetary System OGLE-2014-BLG-0676L with High-Resolution Image Constrained Light Curve Modeling

We present an analysis that incorporates high-resolution Keck adaptive optics (AO) imaging into microlensing light-curve modeling for the planetary microlensing event OGLE-2014-BLG-0676. Using Keck AO observations obtained 6.3 years after the event, we directly resolved the lens and source. The Keck images reveal a tension, in that the $K$-band source flux is $0.52 \pm 0.22$ magnitudes brighter than predicted by previously reported light-curve models. By incorporating the Keck imaging constraints into the light-curve modeling, we find a host star mass of $M_{\rm host} = 0.60^{+0.17}_{-0.14}\,M_{\odot}$, a lens distance of $D_{\rm L} = 1.88^{+0.63}_{-0.35}$ kpc, a planet mass of $m_{\rm p} = 3.11^{+1.11}_{-0.63}\,M_{\rm J}$, and a projected separation of $a_{\perp} = 2.04^{+0.44}_{-0.35}$ au and $a_{\perp} = 3.72^{+0.92}_{-0.72}$ au for the close and wide solution, respectively. These results demonstrate the power of combining high-angular-resolution imaging with microlensing light-curve modeling to mitigate potential systematic effects and modeling degeneracies, enabling robust determinations of the physical properties of microlensing planetary systems. The results presented here can be confirmed by future observations from the \textit{Roman}'s Galactic Plane Survey.

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arXiv2026

Four Cold Giant Planets Discovered by High-Cadence Microlensing Surveys

We report the discovery of four cold giant planets identified through the analysis of microlensing events detected by high-cadence surveys: OGLE-2016-BLG-0261, KMT-2025-BLG-0026, KMT-2025-BLG-0030, and KMT-2025-BLG-2272. The planetary signals appear as short-duration anomalies in the light curves and are well described by binary-lens single-source models with mass ratios between the lens components of order $q \sim 10^{-3}$. Finite-source effects are securely measured in three out of four events, enabling determinations of the angular Einstein radius. A Bayesian analysis incorporating the measured event timescale and angular Einstein radius yields host masses of $\sim 0.07$--$0.6~M_\odot$ and companion masses of $\sim 0.2$--$2.5~M_{\rm J}$, confirming that all companions lie in the giant-planet regime. The projected separations are ~ 0.7--6 au, placing all planets at or beyond the snow lines of their host stars. The inferred lens distances span $\sim 6.6$--$7.9$ kpc, with all systems consistent with bulge lenses. These detections expand the sample of cold giant planets from homogeneous high-cadence surveys and highlight the sensitivity of microlensing to planetary systems beyond the snow line, providing further constraints on the occurrence and properties of giant planets around low-mass stars.

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arXiv2026

On the Reparameterization Between Cartesian Position-Velocity Vectors and Orbital Elements in the Kepler Problem

Reparameterization from the standard set of orbital elements to Cartesian position-velocity vectors can be computationally advantageous for orbit inference problems, particularly when orbital elements are weakly constrained. Here we present compact analytic expressions for the Jacobian determinants of this transformation and its variants, which enable consistent transformation of prior probability densities under reparameterization and are therefore useful for a Bayesian treatment of such problems. We then use these results to clarify the application of this reparameterization in microlensing and astrometric contexts. We first revisit the widely used formulation of lens orbital motion during binary microlensing events presented by Skowron et al (2011). We show that their parameterization inadvertently adopts an incorrect definition of the longitude of the ascending node with respect to the sky-projected binary axis at a reference epoch, which renders the intermediate Jacobian formally singular. Using our closed-form expression, we provide a corrected analytic derivation of the Jacobian for this transformation and show that the resulting formula remains effectively unchanged when the longitude of the ascending node is properly defined with respect to an axis independent of the binary orbit. We also perform an explicit quantitative comparison of astrometric orbit fitting using a gradient-based Markov Chain Monte Carlo algorithm under the two parameterizations, and find that reparameterizing to Cartesian state vectors improves sampling efficiency and robustness relative to orbital-element sampling.

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arXiv2026

Mass Production of 2023 KMTNet Microlensing Planets. III: Three Planets from the Subprime Field

To complete the analysis of the 2023 KMTNet subprime-field microlensing planetary events identified by its AlertFinder system, we present the analysis of six events, KMT-2023-BLG-(1810, 0084, 1118, 0584, 1697, 2218). We find that the first three events are securely confirmed as planetary, with inferred mass ratios of $\log q \sim -1.9$, $-2.0$, and $-2.6$, respectively. The remaining three events exhibit the well-known degeneracy between binary-lens/single-source (2L1S) and single-lens/binary-source (1L2S) models, and two of these also admit viable stellar binary solutions. A Bayesian analysis indicates that the companions in the confirmed planetary events are likely either super-Jupiters orbiting beyond the snow line of M- or K-dwarf hosts or, for two degenerate solutions of KMT-2023-BLG-1118, Saturn-mass planets orbiting late-type M dwarfs. To date, the 2023 KMTNet sample contains 25 unambiguous planetary events, and its mass-ratio distribution is consistent with that of the KMTNet planetary sample from 2016--2019.

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arXiv2026

Candidate Microlensing Brown Dwarfs in Binary Lens Systems from the 2023--2025 Observing Seasons

We present detailed light-curve analyses of ten binary-lens microlensing events observed during the 2023--2025 seasons and selected as candidates for hosting brown-dwarf companions. The sample includes OGLE-2023-BLG-0249, KMT-2023-BLG-1246, OGLE-2023-BLG-0079, KMT-2024-BLG-0072, KMT-2024-BLG-0897, KMT-2024-BLG-1876, KMT-2024-BLG-2379, KMT-2025-BLG-0922, KMT-2025-BLG-1056, and KMT-2025-BLG-2427. For each event, we carry out modeling of the light curve, explore relevant degeneracies, and, when finite-source effects are present, determine the angular Einstein radius. For OGLE-2023-BLG-0249, we additionally measure the microlens parallax, which allows a direct determination of the lens masses and distance. For the remaining events, we estimate the physical lens properties via Bayesian analyses incorporating Galactic priors. The resulting posteriors show that the lens companions in all systems have median masses in the brown-dwarf regime, and the lenses of two events (KMT-2025-BLG-0922 and KMT-2025-BLG-1056) are consistent with binaries in which both lens components fall within the brown-dwarf mass range. Spanning a wide range of projected separations and distances, these detections illustrate the power of high-cadence microlensing surveys to build a census of brown-dwarf companions, including faint and distant systems beyond the reach of flux-limited methods.

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arXiv2026

Mass Production of 2023 KMTNet Microlensing Planets. II: Two Planets and A Brown Dwarf

To expand the homogeneous microlensing planetary sample of the Korea Microlensing Telescope Network (KMTNet), we investigate six planetary candidates identified by the AnomalyFinder search in the 2023 prime-field data, namely KMT-2023-BLG-1592, OGLE-2023-BLG-0766, KMT-2023-BLG-0332, KMT-2023-BLG-0486, KMT-2023-BLG-0792, and OGLE-2023-BLG-1043. Light-curve modeling indicates that the first two events have planetary mass ratios of $\log q \sim -3.0$ and $-2.6$, while the third exhibits a brown dwarf mass ratio of $\log q \sim -1.4$. The remaining three events show the well-known degeneracy between the binary-lens single-source (2L1S) and single-lens binary-source (1L2S) interpretations. A Bayesian analysis yields companion masses of about 0.6 and 1.2 Jupiter masses for the two planetary systems, likely orbiting beyond the snow lines of M- or K-dwarf hosts. A review of the KMTNet planetary sample shows that candidates discovered by AnomalyFinder are significantly more likely to exhibit the 2L1S/1L2S degeneracy, consistent with the tendency of AnomalyFinder to detect subtler planetary signals.

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arXiv2026

Three Saturn-mass Microlensing Planets Identified through Signals from Peripheral-caustic Perturbations

We present the discovery and analysis of three microlensing planets identified through brief positive anomalies on the wings of their light curves. The events, KMT-2021-BLG-0852, KMT-2024-BLG-2005, and KMT-2025-BLG-0481, were detected in high-cadence survey data from the KMTNet, OGLE, MOA, and PRIME collaborations. The anomaly morphologies are consistent with major-image perturbations induced by planetary-mass companions located near the peripheral caustic. A systematic exploration of model degeneracies, including binary-source scenarios, higher mass-ratio binary lenses, and the inner--outer caustic degeneracy, firmly establishes the planetary origin of each signal. Measurements of the angular Einstein radius and event timescale, combined with Bayesian priors from a Galactic model, yield the physical parameters of each system. The hosts are low-mass stars (0.12--0.75~$M_\odot$), while the companions are Saturn-mass planets (0.16--0.59 $M_{\rm J}$) projected at separations of 1.1--7.8 au, placing them beyond the snowline of their hosts. These results demonstrate the capability of microlensing to detect and characterize cold giant planets around low-mass stars at kpc distances, populating the critical transition region between ice giants and gas giants.

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