Double Stars Research Articles

Be X-Ray Binaries as Progenitors of Double Neutron Stars: Evidence of Similar bi-modal Classifications

Abstract We explore the evolutional links between Be X-ray binaries (BeXBs) and double neutron stars (DNSs) by analyzing their physical parameters and classifications. We find that both BeXBs and DNSs show positive correlation trends between the neutron star (NS) spin period–P and binary orbital period–P orb, which may relate to the influence of binary accretion. In addition, BeXBs show a bi-modal classification divided by P ∼ 40 s/P orb ∼ 60 days, where BeXBs with P < 40 s/P orb < 60 days exhibit stronger accretion-induced properties, e.g., the lower average NS magnetic field strength, than those with P > 40 s/P orb > 60 days. Similarly, DNSs exhibit a bi-modal classification divided by P orb ∼ 1 day, where DNSs with P orb < 1 day share the stronger accretion-induced properties, e.g., the higher average accretion rate of the recycled NSs, than those with P orb > 1 day. The multi-parameter analysis with the new aggregated data set improves the knowledge of the bi-modal classifications of BeXBs/DNSs. In addition, we consider the influence of the common envelope phase/supernova kick of the donor star on the evolution from BeXBs to DNSs, and further discuss their compatibility with the observed bi-modal classifications of BeXBs/DNSs. Last, we propose a potential evolutional relation between the bi-modal BeXBs and DNSs.

On the Formation of the Double Neutron Star Binary PSR J1846-0513

Abstract The Double Neutron Star binary PSR J1846-0513 is discovered by the Five hundred-meter Aperture Spherical radio Telescope (FAST) in Commensal Radio Astronomy FAST Survey. The pulsar is revealed to be harbored in an eccentric orbit with e = 0.208 and orbital period of 0.613 day. The total mass of the binary is constrained to be 2.6287(35)M⊙, with the upper mass limit of the pulsar at 1.3455 M⊙ and the lower mass limit of the com panion at 1.2845 M⊙. To reproduce its evolution history, we simulated the 1D model of its progenitor from neutron star - helium (He) star binary system via MESA code. Since the large eccentricity is widely believed to originate from an asymmetric supernova explosion, we examined the dynamical effects of the supernova. Our results show that the binary may
stem from a helium star with initial mass of 3.3 − 4.0 M⊙ accompanied by a neutron star in a circle orbit with initial period of ∼ 0.5 day.

On Radio k – Chromatic Number for Mycielski of Some Graphs

Objectives: In radio networks, the main difficulty is managing the radio spectrum, assigning radio frequencies to transmitters optimally without any interferences. This study aims to find the smallest span of Mycielski of some graphs, the maximum color assigned to any node is called span. Methods : This study focused on the problem of reducing interference by modeling it with a radio k - coloring problem on graphs. Where transmitters are modeled as nodes in a graph, with edges connecting nodes that represent transmitters in close proximity to one another. For a graph , with node set , edge set and an integer , a radio k - coloring of is a function satisfying the condition for any two nodes and , where is the distance between and in . The radio k - coloring of , is the maximum color assigned to any node of and it is denoted by . The radio k -chromatic number of is the minimum value of taken over all radio k - coloring of and it is denoted by . Findings: This study obtained the radio k-chromatic number for Mycielski of some graphs for and 3 Novelty: To solve the channel assignment problem in radio transmitters, the interference graph is developed, and the channel assignment has been converted into a graph coloring. Reducing the interference by a radio k - coloring problem will motivate many researchers to find the radio k - coloring in various graphs. Keywords: Radio k – Coloring, Mycielski graph, Double Star graph, Triple Star graph, Sunlet graph, Helm graph and Closed Helm graph

MeerKAT observations of pair-plasma induced birefringence in the double pulsar eclipses

ABSTRACT PSR J0737−3039A/B is unique among double neutron star systems. Its near-perfect edge-on orbit causes the fast spinning pulsar A to be eclipsed by the magnetic field of the slow spinning pulsar B. Using high-sensitivity MeerKAT radio observations combined with updated constraints on the system geometry, we studied the impact of these eclipses on the incident polarization properties of pulsar A. Averaging light curves together after correcting for the rotation of pulsar B revealed enormous amounts of circular polarization and rapid changes in the linear polarization position angle, which occur at phases where emission from pulsar A is partially transmitted through the magnetosphere of pulsar B. These behaviours confirm that the eclipse mechanism is the result of synchrotron absorption in a relativistic pair-plasma confined to the closed-field region of pulsar B’s truncated dipolar magnetic field. We demonstrate that changes in circular polarization handedness throughout the eclipses are directly tied to the average line of sight magnetic field direction of pulsar B, from which we unambiguously determine the complete magnetic and viewing geometry of the pulsar.

KM UMa: An Active Short-period Detached Eclipsing Binary in a Hierarchical Quadruple System

Abstract The first detailed photometric and spectroscopic analysis of the G-type eclipsing binary KM UMa is presented, which indicates that the system is a short-period detached eclipsing binary. The radial velocity curves were calculated using the cross-correlation function method based on Large Sky Area Multi-Object Fiber Spectroscopic Telescope, Sloan Digital Sky Survey, and our observations, which determined the mass ratio as q = 0.45 (±0.04). Based on the light curves from the Transiting Exoplanet Survey Satellite, other survey data, and our multiband observations, the positive and negative O’Connell effects have been detected evolving gradually and alternately over the last 20 yr, which can be explained by the presence of spots on the primary component. A superflare event was detected in the SuperWASP data on 2007 February 28, further indicating that KM UMa is a very active system. We calculated its energy to be 5 × 1034 erg by assuming it occurred on the primary star. Utilizing hundreds of medium-resolution spectra and one low-resolution spectrum, the equivalent width variations of the H α line were calculated, indicating the presence of a 5.21 (±0.67) yr magnetic activity cycle. The orbital period variations were analyzed using the O–C method, detecting a long-term decrease superimposed with a periodic variation. The amplitude of the cyclic variation is 0.01124 (±0.00004) day, with a period of 33.66 (±0.0012) yr, which exceeds the 5.21 yr activity cycle, suggesting that this is more likely attributable to the light travel time effect of a third body. Simultaneously, a visual companion has been detected based on the Gaia astrometric data, indicating that KM UMa is actually in a 2+1+1 hierarchical quadruple system.

VELOcities of CEpheids (VELOCE)

Classical Cepheids provide valuable insights into the evolution of stellar multiplicity among intermediate-mass stars. Here, we present a systematic investigation of single-lined spectroscopic binaries (SB1s) based on high-precision velocities measured by the VELOcities of CEpheids (VELOCE) project. We detected 76 (29%) SB1 systems among the 258 Milky Way Cepheids in the first VELOCE data release, 32 (43%) of which were not previously known to be SB1 systems. We determined 30 precise and three tentative orbital solutions, 18 (53%) of which are reported for the first time. This large set of Cepheid orbits provides a detailed view of the eccentricity e and orbital period Porb distribution among evolved intermediate-mass stars, ranging from e ∈ [0.0, 0.8] and Porb ∈ [240, 9000] d. The orbital motion on timescales exceeding the 11 yr VELOCE baseline was investigated using a template-fitting technique applied to literature data. Particularly interesting objects include (a) R Cru, the Cepheid with the shortest orbital period in the Milky Way (∼238 d); (b) ASAS J103158−5814.7, a short-period overtone Cepheid exhibiting time-dependent pulsation amplitudes as well as orbital motion; and (c) 17 triple systems with outer visual companions, among other interesting objects. Most VELOCE Cepheids (21/23) that exhibit evidence of a companion based on a Gaia proper motion anomaly are also spectroscopic binaries, whereas the remaining do not exhibit significant (> 3σ) orbital radial velocity variations. Gaia quality flags, notably the renormalized unit weight error (RUWE), do not allow Cepheid binaries to be identified reliably although statistically the average RUWE of SB1 Cepheids is slightly higher than that of non-SB1 Cepheids. A comparison with Gaia photometric amplitudes in G-, Bp, and Rp also does not allow one to identify spectroscopic binaries among the full VELOCE sample, indicating that the photometric amplitudes in this wavelength range are not sufficiently informative of companion stars.

CXOU J005245.0-722844: Discovery of a be star / white dwarf binary system in the SMC via a very fast, super-eddington X-ray outburst event

Abstract CXOU J005245.0-722844 is an X-ray source in the Small Magellanic Cloud (SMC) that has long been known as a Be/X-ray binary (BeXRB) star, containing an OBe main sequence star and a compact object. In this paper, we report on a new very fast X-ray outburst from CXOU J005245.0-722844. X-ray observations taken by Swift constrain the duration of the outburst to less than 16 days and find that the source reached super-Eddington X-ray luminosities during the initial phases of the eruption. The XRT spectrum of CXOU J005245.0-722844 during this outburst reveals a super-soft X-ray source, best fit by an absorbed thermal blackbody model. Optical and Ultraviolet follow-up observations from the Optical Gravitational Lensing Experiment (OGLE), Asteroid Terrestrial-impact Last Alert System (ATLAS), and Swift identify a brief ∼0.5 magnitude optical burst coincident with the X-ray outburst that lasted for less than 7 days. Optical photometry additionally identifies the orbital period of the system to be 17.55 days and identifies a shortening of the period to 17.14 days in the years leading up to the outburst. Optical spectroscopy from the Southern African Large Telescope (SALT) confirms that the optical companion is an early-type OBe star. We conclude from our observations that the compact object in this system is a white dwarf (WD), making this the seventh candidate Be/WD X-ray binary. The X-ray outburst is found to be the result of a very-fast, ultra-luminous nova similar to the outburst of MAXI J0158-744.

A MATLAB APP FOR BALANCING A CAPACITOR BANK IN THE DOUBLE STAR CONNECTION - NEUTRAL, NOT CONNECTED

This study presents a methodology for maintaining and rebalancing Double Star configuration capacitor banks, which are critical for managing power consumption peaks. As electricity demand is projected to rise by 21% by 2040, efficient maintenance methods are essential to ensure energy security. The paper introduces a novel approach to rebalance capacitor banks using a Matlab application, which simulates the unbalance current that may occur due to capacitive unit failures. Key aspects of capacitor banks, such as construction, connections, and protection methods, were reviewed to inform the methodology. Capacitor banks are used to control voltage, correct power factor, and reduce losses, but they can experience imbalances due to natural variances in capacitance or failures. The study developed a mathematical model to calculate the unbalance current and validated this model through simulations in LTspice software. The resulting rebalancing flowchart provides a step-by-step guide for technicians to follow during maintenance. The project achieved its goals, demonstrating through 24 simulations that the developed methodology and Matlab application could successfully predict unbalance currents with a negligible margin of error. The paper concludes that the methodology is ready for implementation, and each capacitor bank will require a tailored rebalancing application based on the presented calculations.

Modified load angle direct torque control for dual star motor using fuzzy logic

Space Vector Modulation (SVM) has emerged as a prominent alternative for motor control, particularly within the framework of Direct Torque Control (DTC-SVM). SVM leverages variations in the load angle, which result from changes in motor torque, to generate the appropriate voltage vectors. The conventional approach to this method involves a single Proportional-Integral (PI) regulator, which is characterized by a constant switching frequency and the capability to reduce ripples in torque, flux, and current. Despite these advantages, the PI regulator may face challenges in maintaining optimal performance under varying load conditions. This study seeks to enhance the performance of the DTC-SVM strategy by integrating Fuzzy Logic into the PI regulator. The primary objective is to improve the accuracy and responsiveness of the load angle control, thereby optimizing the motor's operational efficiency. The proposed SVM-Fuzzy control scheme is designed to address the limitations of the traditional PI regulator, offering a more adaptive and robust solution for motor control. To evaluate the effectiveness of the proposed approach, extensive simulations were conducted using Matlab Simulink. The SVM-Fuzzy strategy was compared against the conventional SVM-PI approach to assess its performance in terms of torque, flux, and current regulation. The SVM-Fuzzy control scheme significantly enhances efficiency, stability, and load angle precision, particularly in Double Star Induction Motors (DSIM). It also ensures the durability and reliability of the motor control system under different operating conditions.

Discovery of double stars by Giovanni Battista Hodierna in 1654

ABSTRACT It has been generally accepted that the originators of the double star astronomy were Christian Mayer and William Herschel. We recovered the memory of the poorly known Italian astronomer Giovanni Battista Hodierna, who published the first catalogue of stellar binaries over a century before Mayer and Herschel. We analysed the fourth section of 1654 G. B. Hodierna’s book ‘De systemate orbis cometici deque admirandis coeli characteribus’. There, Hodierna listed a dozen pairs of stars whose identification with modern star names had been lost for centuries. To identify the pairs, we used Hodierna’s Latin descriptions of location in constellations for all primary stars, ecliptic coordinates and angular separations to companions for some, and the Washington Double Star, Hipparcos, and Gaia catalogues. We were able to identify the twelve primaries and eleven multiple systems with companions, of which nine were double and two were triple. Besides, with up-to-date data, we confirmed that four systems are physically bound: Atlas and Pleione, $\alpha ^{1,2}$ Lib, $\nu ^{1,2}$ Dra, and $\theta ^{1}$ Ori A, C, and D. The other seven pairs are alignments of very bright stars at different distances.

The Nature of 500 micron Risers III: a small complete sample

ABSTRACT Herschel surveys have found large numbers of sources with red far-IR colours, and spectral energy distributions (SEDs) rising from 250 to 500 µm: 500 risers. The nature and role of these sources is not fully understood. We here present Submillimeter Array (SMA) interferometric imaging at 200 GHz of a complete sample of five 500 risers with F500 >44 mJy selected within a 4.5 deg2 region of the XMMLSS field. These observations can resolve the separate components of multiple sources and allow cross identification at other wavelengths using the extensive optical-to-IR data in this field. Of our five targets, we find that two are likely gravitationally lensed, two are multiple sources, and one an isolated single source. Photometric redshifts, using optical-to-IR data and far-IR/submm data, suggest they lie at redshifts $z \sim 2.5\!-\!3.5$. Star formation rates and stellar masses estimated from the SEDs show that the majority of our sources lie on the star-formation rate-stellar mass ‘main sequence’, though with outliers both above and below this relation. Of particular interest is our most multiple source, which consists of three submm emitters and one submm-undetected optical companion within a 7 arcsec region, all with photometric redshifts ∼3. One of the submm emitters in this group lies above the ‘main sequence’, while the optical companion lies well below the relation, and has an estimated stellar mass of $3.3 \pm 1.3 \times 10^{11}$ M$_{\odot }$. We suggest this object is a forming brightest cluster galaxy (BCG) in the process of accreting actively star forming companions.

Spatial distribution of plasma density and magnetic field amplitude in the dayside magnetosheath as a function of the IMF orientation

The properties of the magnetosheath are of pivotal importance in determining the coupling between the magnetosphere and interplanetary medium. In particular, the magnetic flux pileup and plasma depletion layer (PDL) modify the boundary conditions of magnetopause reconnection. However, the spatial distribution of the magnetic field strength and plasma density in the magnetosheath and their functional dependence on the interplanetary magnetic field (IMF) orientation remain poorly understood. This study characterizes these aspects in detail through the statistical processing of decades of data from Cluster, Double Star, THEMIS, and Magnetospheric Multiscale (MMS) missions. The first part of this study focuses on the poorly known variations across the magnetosheath, from the shock to the magnetopause. The magnetic pileup and PDL are significantly correlated, with a strong dependence on the IMF cone angle. Their dependence on the IMF clock angle is found only near the magnetopause, consistent with the expected effect of magnetic reconnection. The second part of this study examines the asymmetry in the magnetic field amplitude and density between the quasi-parallel and quasi-perpendicular sides of the equatorial magnetosheath. These asymmetries are characterized for different relative distances to the magnetopause and bow shock boundaries and for different IMF orientation. The magnetic field amplitude, observed to be higher on the quasi-perpendicular side of the magnetosheath, becomes more symmetric as it approaches the magnetopause. The quasi-parallel magnetosheath exhibits a higher plasma density near the magnetopause. However, this asymmetry reverses at approximately the mid-magnetosheath with a decreasing IMF cone angle.

Global Environmental Constraints on Magnetic Reconnection at the Magnetopause From In Situ Measurements

AbstractProgress in locating the X‐line on the magnetopause beyond the atypical due south interplanetary magnetic field (IMF) condition is hampered by the fact that the global plasma and field spatial distributions constraining where reconnection could develop on the magnetopause are poorly known. This work presents global maps of the magnetic shear, current density and reconnection rate, on the global dayside magnetopause, reconstructed from two decades of measurements from Cluster, Double Star, THEMIS and MMS missions. These maps, generated for various IMF and dipole tilt angles, offer a unique comparison point for models and observations. The magnetic shear obtained from vacuum magnetostatic draping is shown to be inconsistent with observed shear maps for IMF cone angles in 12.5° ± 2.5° ≤ |θco| ≤ 45° ± 5°. Modeled maximum magnetic shear lines fail to incline toward the equator as the IMF clock angle increases, in contrast to those from observations and MHD models. Reconnection rate and current density maps are closer together than they are from the shear maps, but this similarity vanishes for increasingly radial IMF orientations. The X‐lines maximizing the magnetic shear are the only ones to sharply turns toward and follow the anti‐parallel ridge at high latitude. We show the behavior of X‐lines with varying IMF clock and dipole tilt angles to be different as the IMF cone angle varies. Finally, we discuss a fundamental disagreement between X‐lines maximizing a given quantity on the magnetopause and predictions of local X‐line orientations.

DTC-SVM strategy of double star induction motor using sliding mode controller enhanced by fuzzy logic

Extensive research and numerous articles on Direct Torque Control (DTC) systems have consistently highlighted various deficiencies associated with the employment of hysteresis control units. Specifically, these issues include significant torque and stator flux ripples, as well as distortions in stator current. Additionally, the sensitivity to parametric variations stemming from Proportional-Integral (PI) controllers poses a challenge due to the complexities involved in their adjustment. To address these drawbacks, this article explores the implementation of Space Vector Modulation DTC (SVM-DTC) technology integrated with a Sliding Mode Controller (SMC). The proposed control scheme for DTC amalgamates the benefits derived from SMC control and the SVM algorithm. The SMC controller is utilized to oversee PI controllers, employing this nonlinear technique ensures superior dynamic performance and robust resistance against external disturbances. Notwithstanding its advantages, one notable issue with SMC is chattering. To mitigate this phenomenon, we adopt an approach that integrates Fuzzy Logic with SMC. By replacing the saturation function (Sat) with a fuzzy inference system, we effectively minimize chattering. The simulation is conducted using MATLAB SIMULINK. Comparative analysis between DTC-SVM strategy and traditional methods including standard SMC and Fuzzy-enhanced Sliding Mode Controller (FSMC), demonstrates that our hybrid nonlinear controller exhibits high efficiency and remarkable robustness.

The DBL Survey I: discovery of 34 double-lined double white dwarf binaries

ABSTRACT We present the first discoveries of the double-lined double white dwarf (DBL) survey that targets overluminous sources with respect to the canonical white dwarf cooling sequence according to a set of well-defined criteria. The primary goal of the DBL survey is to identify compact double white dwarf binary star systems from a unique spectral detection of both stars, which then enables a precise quantification of the atmospheric parameters and radial velocity variability of a system. Our search of 117 candidates that were randomly selected from a magnitude-limited sample of 399 yielded a 29 per cent detection efficiency with 34 systems exhibiting a double-lined signature. A further 38 systems show strong evidence of being single-lined or potentially DBL binaries and seven single-lined sources from the full observed sample are radial velocity variable. The 45 remaining candidates appear as a single WD with no companion or a non-DA white dwarf, bringing the efficiency of detecting binaries to 62 per cent. Atmospheric fitting of all double-lined systems reveals a large fraction that have two similar mass components that combine to a total mass of 1.0–1.3 $\mathrm{M}_\odot$ – a class of double white dwarf binaries that may undergo a sub-Chandrasekhar mass type Ia detonation or merge to form a massive O/Ne WD, although orbital periods are required to infer on which time-scales. One double-lined system located 49 pc away, WDJ181058.67+311940.94, is super-Chandrasekhar mass, making it the second such double white dwarf binary to be discovered.

VLBA Astrometry of the Galactic Double Neutron Stars PSR J0509+3801 and PSR J1930–1852: A Preliminary Transverse Velocity Distribution of Double Neutron Stars and its Implications

The mergers of double neutron star (DNS) systems are believed to drive the majority of short γ-ray bursts (SGRBs), while also serving as production sites of heavy r-process elements. Despite being key to (i) confirming the nature of the extragalactic SGRBs, (ii) addressing the poorly understood r-process enrichment in the ultrafaint dwarf galaxies (UFDGs), and (iii) probing the formation process of DNS systems, the space velocity distribution of DNSs is still poorly constrained, due to the small number of DNSs with well-determined astrometry. In this work, we determine new proper motions and parallaxes of two Galactic DNSs, PSR J0509+3801 and PSR J1930−1852, using the Very Long Baseline Array, and we estimate the transverse velocities v ⊥ of all 11 isolated Galactic DNSs having proper-motion measurements in a consistent manner. Our correlation analysis reveals that the DNS v ⊥ is tentatively correlated with three parameters: spin period, orbital eccentricity, and companion mass. With the preliminary v ⊥ distribution, we obtain the following findings. First, the refined v ⊥ distribution is confirmed to agree with the observed displacements of the localized SGRBs from their host galaxy birth sites. Second, we estimate that around 11% and 25% of DNSs remain gravitationally bound to UFDGs with escape velocities of 15 and 25 km s−1, respectively. Hence, the retained DNSs might indeed be responsible for the r-process enrichment confirmed so far in a few UFDGs. Finally, we discuss how a future ensemble of astrometrically determined DNSs may probe the multimodality of the v ⊥ distribution.

Catalogue of dual-field interferometric binary calibrators

Context. Dual-field interferometric observations with VLTI/GRAVITY sometimes require the use of a binary calibrator. This is a binary star whose individual components remain unresolved by the interferometer, with a separation between 400 and 2000 mas for observations with the Unit Telescopes (UTs), or 1200–3000 mas for the Auxiliary Telescopes (ATs). The separation vector also needs to be predictable to within 10 mas for a proper pointing of the instrument. Aims. No list of properly vetted calibrators was available so far for dual-field observations with VLTI/GRAVITY on the UTs. Our objective is to compile such a list and make it available to the community. Methods. We identified a list of candidates from the Washington Double Star (WDS) catalogue, all with appropriate separations and brightness, scattered over the southern sky. We observed them as part of a dedicated calibration programme, determined whether these objects were true binaries (excluding higher multiplicities resolved interferometrically, but unseen by imaging), and extracted measurements of the separation vectors. We combined these new measurements with those available in the WDS to determine updated orbital parameters for all our vetted calibrators. Results. We compiled a list of 13 vetted binary calibrators for observations with VLTI/GRAVITY on the UTs, and we provide orbital estimates and astrometric predictions for each of them. We show that our list guarantees that there are always two binary calibrators at least at an airmass < 2 in the sky over the Paranal observatory at any point in time. Conclusions. Any principal investigator wishing to use the dual-field mode of VLTI/GRAVITY with the UTs can now refer to this list to select an appropriate calibrator. We encourage the use of whereistheplanet to predict the astrometry of these calibrators, which seamlessly integrates with p2Gravity for VLTI/GRAVITY dual-field observing material preparation.

SN 2023zaw: An Ultrastripped, Nickel-poor Supernova from a Low-mass Progenitor

We present SN 2023zaw—a subluminous (M r = −16.7 mag) and rapidly evolving supernova (t 1/2,r = 4.9 days), with the lowest nickel mass (≈0.002 M ⊙) measured among all stripped-envelope supernovae discovered to date. The photospheric spectra are dominated by broad He i and Ca near-infrared emission lines with velocities of ∼10,000−12,000 km s−1. The late-time spectra show prominent narrow He i emission lines at ∼1000 km s−1, indicative of interaction with He-rich circumstellar material. SN 2023zaw is located in the spiral arm of a star-forming galaxy. We perform radiation-hydrodynamical and analytical modeling of the lightcurve by fitting with a combination of shock-cooling emission and nickel decay. The progenitor has a best-fit envelope mass of ≈0.2 M ☉ and an envelope radius of ≈50 R ⊙. The extremely low nickel mass and low ejecta mass (≈0.5 M ⊙) suggest an ultrastripped SN, which originates from a mass-losing low-mass He-star (zero-age main-sequence mass < 10 M ⊙) in a close binary system. This is a channel to form double neutron star systems, whose merger is detectable with LIGO. SN 2023zaw underscores the existence of a previously undiscovered population of extremely low nickel mass (<0.005 M ☉) stripped-envelope supernovae, which can be explored with deep and high-cadence transient surveys.

Graphs with large (1,2)-rainbow connection numbers

Let G be an edge-colored graph. If every subpath of length at most l+1 within a path P in a graph G consists of uniquely colored edges, then P is called an l-rainbow path. A connected graph G is deemed (1,2)-rainbow connected if there exists at least one 2-rainbow path connecting two distinct vertices within G. The minimum number of colors needed to attain (1,2)-rainbow connectedness in a connected graph G, represented as rc1,2(G), is referred to as the (1,2)-rainbow connection number. If G is a nontrivial connected graph of size m, then rc1,2(G)=m if and only if G is the star or double star of size m. Our main goal is to identify all connected graphs G of size m that satisfy the condition m−3≤rc1,2(G)≤m−1.

On the generation of optical third-harmonic through quasi-phase-matching in quadratic nonlinear crystals

We investigate the simultaneous processes of optical frequency doubling and sum frequency generation in quasi-phase-matched quadratically nonlinear crystals. Specifically, we focus on lattices with domain thicknesses corresponding to the coherence lengths of the two cascaded parametric processes and explore the realistic scenario of a single crystal containing sequential domain segments, each independently satisfying quasi-phase-matching for optical frequency doubling and summing, respectively. We demonstrate that high conversion efficiencies to the third harmonic can be obtained from an arbitrary fundamental wavelength.