Large Degree Research Articles

Design of a Robotic 3D Printing System

Abstract Additive manufacturing enables the production of complex shapes with a large degree of freedom compared to conventional manufacturing techniques such as injection molding, thermoforming or pressing. In particular, the creation of moving parts, complex structures and large-scale printing remains underexplored and research is constantly expanding. Due to design limitations related to the construction of currently available printers, 3D printing technology is increasingly being combined with robotics. Using a robot to 3D print products eliminates the need to design complex printer structures, which ultimately should simplify the process of manufacturing large-scale products and shorten the time required for preparation and production. The paper deals with the description of the designed 3D printing system and the procedure of the steps of creating a robotic simulation model with the generation of a control program for the FDM 3D printing process by an industrial robot.

Differential-geometry-based multi-dimensional joint position-velocity estimation using Crab pulsar profile distortion

The traditional orbit determination method based on pulsar profile distortion can determine the six elements of the orbit. However, the estimation accuracies of these methods are limited and the computational load of a six-dimensional search is huge. To solve this problem, the differential-geometry-based Multi-dimensional Joint Position-Velocity Estimation (MJPVE) using Crab pulsar profile distortion is proposed in this paper. Firstly, through theoretical analysis, it is found that the pulsar profile distortion caused by the initial state error in some joint position-velocity directions is very small. In other words, the accuracies of estimation in these directions are very low. Namely, the search dimension can be reduced, which in turn greatly reduces the computational load. Then, we construct the chi-squared function of the pulsar profile with respect to the estimation error in joint position-velocity direction and use differential geometry to find the joint position-velocity directions corresponding to different degrees of distortion. Finally, we utilize the grid search based on directory folding in these joint position-velocity directions corresponding to large degrees of distortion to obtain the joint position-velocity estimation. The experimental results show that compared with the grouping bi-chi-squared inversion method, MJPVE has high precision and extensive navigation information.

Lessons from Hubble and Spitzer: 1D Self-consistent Model Grids for 19 Hot Jupiter Emission Spectra

We present a population-level analysis of the dayside thermal emission spectra of 19 planets observed with Hubble WFC3 and Spitzer IRAC 3.6 and 4.5 μm, spanning equilibrium temperatures 1200–2700 K and 0.7–10.5 Jupiter masses. We use grids of planet-specific 1D, cloud-free, radiative–convective–thermochemical equilibrium models (1D-RCTE) combined with a Bayesian inference framework to estimate atmospheric metallicity, the carbon-to-oxygen ratio, and day-to-night heat redistribution. In general, we find that the secondary eclipse data cannot reject the physics encapsulated within the 1D-RCTE assumption parameterized with these three variables. We find a large degree of scatter in atmospheric metallicities, with no apparent trend, and carbon-to-oxygen ratios that are mainly consistent with solar or subsolar values but do not exhibit population agreement. Together, these indicate either (1) formation pathways vary over the hot and ultra-hot Jupiter population and/or (2) more accurate composition measurements are needed to identify trends. We also find a broad scatter in derived dayside temperatures that do not demonstrate a trend with equilibrium temperature. Like with composition estimates, this suggests either significant variability in climate drivers over the population and/or more precise dayside temperature measurements are needed to identify a trend. We anticipate that 1D-RCTE models will continue to provide valuable insights into the nature of exoplanet atmospheres in the era of JWST.

Distinct Transcript-Level Expression Profiles and Unique Alternative Splicing in Inflammatory Myopathies.

The pathogenesis of inflammatory myopathies is poorly understood and there is a need to dissect the transcriptome in more granular ways beyond gene expression. We used a set of muscle RNA-sequencing data from different myositis subtypes grouped by their specific autoantibodies (n = 152). We quantified annotated RNA transcripts for each myositis subtype and identified uniquely expressed RNA as well as transcriptional similarities among myositis types. In addition, we quantified event-based alternative splicing with predicted protein changes. And finally, we searched for cryptic exons. We saw considerable overlap in RNA expression among subtypes. In addition, MADCAM1 was previously shown to be uniquely expressed in Mi-2 myositis; we discovered it was two noncanonical transcripts that predominantly contributed to the observed increased expression. At the transcriptional level, dermatomyositis subtypes were least similar to inclusion body myositis (IBM) or Jo1, followed by HMGCR, then SRP and other dermatomyositis subtype. Additionally, we discovered many alternative splicing events that were unique by myositis subgroup, including events in muscle dystrophy genes and one event in SRP72, which was seen uniquely in SRP myositis. Finally, we looked for previously reported cryptic exons in IBM and did not find them. The large degree of transcriptional overlap among myositis subtypes reinforces the need to use disease (in addition to healthy) controls to find unique features of autoimmune disease. Unique alterations in the transcriptome that are seen in one myositis subtype and not others advance our understanding of distinct disease pathology.

Deep, hot,ancient melting recorded by ultralow oxygen fugacity in peridotites.

The oxygen fugacity (fO2) of convecting upper mantle recorded by ridge peridotites varies by more than four orders of magnitude1-3. Although much attention has been given to mechanisms that drive variations in mantle fO2 between tectonic settings1,3,4 and to comparisons of fO2 between modern rocksand ancient-mantle-derived rocks5-10, comparatively little has been done to understand the origins of the high variability in fO2 recorded by peridotites from modern mid-ocean ridge settings. Here we report the petrography and geochemistry of peridotites from the Gakkel Ridge and East Pacific Rise (EPR), including 16 new high-precision determinations of fO2. Refractory peridotites from the Gakkel Ridge record fO2 more than four orders of magnitude below the mantle average. With thermodynamic and mineral partitioning modelling, we show that excursions to ultralow fO2 can be produced by large degrees of melting at high potential temperature (Tp), beginning in the garnet field and continuing into the spinel field-conditions met during the generation of ancient komatiites but not modern basalts. This does not mean that ambient convecting upper mantle had a lower ferric to ferrous ratio in Archaean times than today nor that modern melting in the garnet field at hotspots produce reduced magmas. Instead, it implies that rafts of ancient, refractory, ultrareduced mantle continue to circulate in the modern mantle while contributing little to modern ridge volcanism.

Scalar induced gravitational waves in chiral scalar–tensor theory of gravity

We study the scalar induced gravitational waves (SIGWs) from a chiral scalar–tensor theory of gravity. The parity-violating (PV) Lagrangian contains the Chern–Simons (CS) term and PV scalar–tensor terms, which are built of the quadratic Riemann tensor term and first-order derivatives of a scalar field. We consider SIGWs in two cases, in which the semi-analytic expression to calculate SIGWs can be obtained. Then, we calculate the fractional energy density of SIGWs with a monochromatic power spectrum for the curvature perturbation. We find that the SIGWs in chiral scalar–tensor gravity behave differently from those in GR before and after the peak frequency, which results in a large degree of circular polarization.

Disruptive mood dysregulation disorder, parental stress, and attachment styles

Disruptive mood dysregulation disorder (DMDD) is a relatively new diagnostic entity concerning children with chronic irritability and severe anger outbursts. Currently, there is limited knowledge about the parental factors associated with the disorder. The aim of this study was to compare stress levels and attachment styles in parents of children with DMDD with those of parents of children with other diagnoses. Our sample consisted of 218 children (6–12 years, Mage = 9.68 years) referred to child mental health outpatient clinics. Clinicians used a standardized semi-structured diagnostic interview to identify diagnoses. Parental stress levels and attachment styles were assessed using parent reports. We found that parents of children with DMDD experience significantly higher levels of parenting stress related to factors in the child than parents of children with other diagnoses. Furthermore, parents of children with DMDD show a higher association with insecure adult attachment styles than parents of children without DMDD. Finally, an adult preoccupied-ambivalent attachment style explains the variability in parental stress in the DMDD group to a large degree. We discuss how parental stress and an insecure attachment style can be associated with negative parenting practices. An implication from this study could be that treatment results might be improved by involving parents more in treatment programs for children with DMDD.

Chiral Emission from Optical Metasurfaces and Metacavities

Chiral emission exhibiting a large degree of circular polarization (DCP) is important in diverse applications ranging from displays and optical storage to optical communication, bioimaging, and medical diagnostics. Although chiral luminescent materials can generate chiral emissions directly, they frequently suffer from either low DCP or low quantum efficiencies. Achieving high DCP and quantum efficiencies simultaneously remains extremely challenging. This review introduces an alternative approach to chiral emission. Chiral emission with large DCP can be readily achieved by combining conventional achiral emitters with chiral metasurfaces. Particularly, this article focuses on recent experimental and theoretical studies on perovskite metasurfaces and metacavities that employ achiral perovskite materials. First, chiral photoluminescence from extrinsic and intrinsic perovskite metasurfaces is explained together with theoretical discussions on metasurface design based on reciprocity and critical coupling. Chiral photoluminescence from other achiral materials is also explained. Subsequently, chiral electroluminescence from perovskite metacavities and other achiral materials is discussed. Finally, it is concluded with future perspectives. This review provides physical insights into how ideal chiral emission can be realized by optimizing the design of metasurfaces and metacavities. Compact chiral light sources with both near‐unity DCP and strong emission intensities can have far‐reaching consequences in a wide range of future applications.

Near-field DOA estimation oriented sparse array design via symmetric augmentation of three ULAs

A symmetric sparse array, via symmetric augmentation of three (SAT) uniform linear subarrays, has been designed for direction of arrival (DOA) estimation of uncorrelated near-field or coexisted near-field and far-field narrowband source signals. With an explicit design of the subarray structures and positions, this SAT array 1) involves only two closely spaced antenna pairs (with quarter-wavelength spacing); 2) has a large uniform degree of freedom (uDOF) and difference co-array; 3) can have either even or odd number of antennas; and 4) has closed form expressions for antenna positions, uDOF and weight functions. Simulation results show that the SAT array is superior to the current symmetric sparse arrays in both antenna mutual coupling and consecutive virtual aperture.

Effect of temperature gradient on the chemical structure and physical morphology of polyacrylonitrile fibers during thermal stabilization process

The thermal stabilization process is a crucial stage for microstructure transformation and quality of carbon fibers, which is determined by the stabilization temperature gradient. In this work, the effect of the temperature gradient on the evolution of both chemical structures and physical morphology of polyacrylonitrile (PAN) fibers at multistage of thermal stabilization process was investigated. Combined with functional groups, oxygen element content, and aromatization index, increasing temperature reasonably shifted oxidation and cyclization reaction into early stage and the corresponding samples possessed relatively large degree of cyclization. It was worth noting that high stabilization temperature gradient could accelerate the formation of the conjugated structures at early stage and facilitate the cyclization reaction in the stabilized fibers. As for morphologies, there were aligned microfibrils and transvers lamellae in stabilized fibers. Their radial structure difference increased firstly and then decreased in the later stage. The tensile strength and tensile modulus decreased during the thermal stabilization process. Moreover, increasing temperature resulted in reduction of tensile strength and tensile modulus, which was mainly resulted by the high degree of cyclization instead of structural defects. These results guide the design and optimization of temperature program and adjusting the processing parameters for all commercially PAN fibers without any additional conditions.

Exploring pNIPAM lyogels: Experimental study on swelling equilibria in various organic solvents and mixtures, supported by COSMO-RS analysis

Stimuli-responsive lyogels are considered to be smart materials due to their capability of undergoing significant macroscopic changes in response to external triggers. Due to their versatile and unique properties, smart lyogels exhibit great potential in various applications such as drug delivery or actuation processes. While Poly-N-isopropylacrylamide (pNIPAM) is widely known as a thermo-responsive material, it also shows significant solvent-responsive swelling behavior. For the systems tested, polar solvents induce strong swelling due to hydrogen bonding with the amide group, nonpolar solvents lead to significant shrinkage of the lyogels. The aim of this study is to investigate and to model this behavior for future application in chemical or biochemical reactors. As a current area of research, incorporating smart lyogel technology into (bio-)chemical reactors facilitates the development of smart reactor systems. Applying thermodynamic modelling with the gE model COSMO-RS on the monomer or oligomers of pNIPAM, the correlation between solvent-polymer interactions and the degree of swelling can be observed. pNIPAM derivatives exhibit low infinite dilution activity coefficients (IDACs) in polar solvents with large degrees of swelling, while displaying an increase of IDACs in nonpolar solvents. Hydrogen bonds dominate the swelling behavior of lyogels not only in pure solvents but also in mixtures of solvents with varying polarity. Even in mixtures containing high amounts of nonpolar solvents, large degrees of swelling were observed due to the uptake of the polar solvent in the lyogel matrix. This effect can be observed in binary solvent mixtures but also in representative mixtures along an esterification reaction with varying carboxylic chain length of alcohol and carboxylic acids.

Carrier diffusion in long wavelength InGaN quantum well LEDs after injection through V-defects

The efficiency of operation of GaN-based light emitting diodes (LEDs) to a large degree relies on realization of a uniform hole distribution between multiple quantum wells (QWs) of the active region. Since the direct thermionic transport between the QWs is inefficient, the hole injection through semipolar 101¯1 QWs that form on the facets of V-defects has been suggested as an alternative approach. However, for an efficient LED operation, the carrier distribution should be uniform not only vertically, between the QWs but also laterally, within individual QWs. In this work, the lateral carrier distribution in long wavelength InGaN/GaN QW LEDs is studied by the scanning near-field optical microscopy. The measurements have shown that emission is concentrated around the V-defect injectors. At high currents, the diffusion length of holes in polar QWs was found to be ∼0.6–1 μm and the hole diffusion coefficient ∼0.6 cm2/s. The obtained data should aid design of the V-defect injectors for a laterally uniform carrier distribution in the active region QWs.

Effects of temperature gradient on functional fruit traits: an elevation-for-temperature approach

Fruit traits mediate animal-plant interactions and have to a large degree evolved to match the sensory capacities and morphology of their respective dispersers. At the same time, fruit traits are affected by local environmental factors, which may affect frugivore-plant trait match. Temperature has been identified as a major factor with a strong effect on the development of fruits, which is of serious concern because of the rising threat of global warming. Nonetheless, this primarily originates from studies on domesticated cultivars in often controlled environments. Little is known on the effect of rising temperatures on fruit traits of wild species and the implications this could have to seed dispersal networks, including downstream consequences to biodiversity and ecosystem functioning. In a case study of five plant species from eastern Madagascar, we addressed this using the elevation-for-temperature approach and examined whether a temperature gradient is systematically associated with variation in fruit traits relevant for animal foraging and fruit selection. We sampled across a gradient representing a temperature gradient of 1.5–2.6 °C, corresponding to IPCC projections. The results showed that in most cases there was no significant effect of temperature on the traits evaluated, although some species showed different effects, particularly fruit chemical profiles. This suggests that in these species warming within this range alone is not likely to drive substantial changes in dispersal networks. While no systemic effects were found, the results also indicate that the effect of temperature on fruit traits differs across species and may lead to mismatches in specific animal-plant interactions.

Heffter spaces

The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of r mutually orthogonal Heffter systems for any r. Such a set is equivalent to a resolvable partial linear space of degree r whose parallel classes are Heffter systems: this is a new combinatorial design that we call a Heffter space. We present a series of direct constructions of Heffter spaces with odd block size and arbitrarily large degree r obtained with the crucial use of finite fields. Among the applications we establish, in particular, that if q=2kw+1 is a prime power with kw odd and k≥3, then there are at least ⌈w4k4⌉ mutually orthogonal k-cycle systems of order q.

Targeted immunization thresholds for the contact process on power-law trees

Scale-free configuration models are intimately connected to power law Galton–Watson trees. It is known that contact process epidemics can propagate on these trees and therefore these networks with arbitrarily small infection rate, and this continues to be true after uniformly immunizing a small positive proportion of vertices. So, we instead immunize those with largest degree: above a threshold for the maximum permitted degree, we discover the epidemic with immunization has survival probability similar to without, by duality corresponding to comparable metastable density. With maximal degree below a threshold on the same order, this survival probability is severely reduced or zero.

Autologous bone grafting in combination with autologous chondrocyte implantation yields favourable outcomes in the treatment of osteochondral defects of the knee: A systematic literature review.

This study addresses the gap in the current literature by evaluating the combined treatment of autologous bone grafting and autologous chondrocyte implantation (ABCI) for osteochondral defects of the knee. It aims to evaluate clinical outcomes against methodological quality and to summarize histological results and surgical techniques. A thorough search was conducted across Pubmed, Cochrane and Embase databases. Studies reporting clinical outcomes of ABCI for osteochondral defects of the knee were included. Patient-reported outcome measures (PROMs), failure rates, methodological quality and potential conflicts of interest were evaluated. Histological results and surgical techniques were summarized. Eighteen studies with 344 analyzed patients met the eligibility criteria for inclusion. All studies showed a significant improvement (p < 0.05) across different PROMs (subjective International Knee Documentation Committee score, Cincinnati Knee Rating System, Visual Analogue Scale, Lysholm Score, Tegner Activity Scale, Knee injury and Osteoarthritis Outcome Score and Knee Society Score) compared to the preoperative status. Failure rates ranged from 0% to 17.6%, with a mean follow-up of 73.2 months (range: 9.0-143.6 months). Methodological quality was low to medium, including only one comparative study. Six studies reviewed reported a potential conflict of interest. The histological assessment showed effective bonding between autologous chondrocytesand bone graft. A large degree of variability was observed in the operative technique used. The current literature suggests that ABCI yields good clinical outcomes at mid- to long-term follow-up with favourable histological results for osteochondral defects of the knee. However, future research should focus on high-quality comparative studies to better guide treatment choices. Introducing ABCI as the standard abbreviation may enhance clarity in future research. Level IV.

Energy transfer and a novel SBR thermometry of SrY2O4:Sm3+/Eu3+ phosphor based on redshift of charge transfer band edge

In this study, a solid-state reaction was employed to synthesize SrY2O4: Sm3+, SrY2O4: Eu3+ and SrY2O4: Sm3+/Eu3+ phosphors. An intriguing redshift phenomenon from charge transfer band (CTB) edge was investigated for single-band ratiometric (SBR) thermometry applications. The phosphors synthesized exhibited an orthogonal CaFe2O4 structure with Pnam (62) space group. The Y3+ sites from host lattice were most likely to be replaced by Sm3+/Eu3+ ions. The sample of SrY2O4: 0.03 Sm3+/0.3Eu3+ showed a large degree of agglomeration with elongated particles, having an average size of approximately 4 μm. The energy bandgap decreased due to increased surface imperfections, resulting in enhanced defect level concentration. The dipole-dipole interaction could be used to explain energy transfer (ET) of Sm3+-Sm3+ and Sm3+-Eu3+. Furthermore, the energy transfer (ET) efficiency of Sm3+→Eu3+ in Sr2YO4 reached 77.7 %. The sample exhibited a good thermal stability (90.956 %@423 K) with Ea of 0.31 eV, which was an important parameter for broadening thermometry range. A thermometry strategy utilized this redshift phenomenon from CTB edge with anti-thermal quenching behavior and other peaks or bands with thermal quenching was therefore proposed. The high Sr value of 1.533 % K−1@298 K provides a great potential for optical thermometry application, contributing significantly to the advancement of single band ratiometric thermometry technologies.

Improving the Representation of Moisture and Convective Instability in Baroclinic-Wave Channel Simulations

Abstract We present an improved approach to generating moist baroclinically unstable background states for f-plane-channel simulations via potential vorticity (PV) inversion. Previous studies specified PV distributions with constant values in the troposphere and the stratosphere, but this produces unrealistic static-stability profiles that decrease sharply with height in the troposphere. Adding moisture to such environments can yield unrealistically large values of convective available potential energy (CAPE) even for reasonable relative humidity (RH) distributions. In our modified approach, we specify a PV distribution that increases with height in the troposphere and the stratosphere, yielding background states with more realistic values of static stability and CAPE. This modification produces environments that are better suited for representing moist processes, namely, deep convection, in idealized extratropical-cyclone simulations. Also, we present a method for introducing moisture that preserves a specified RH distribution while maintaining hydrostatic balance. Our approach allows for a large degree of control over the initial conditions, as background states with different jet strengths and shapes, average temperatures, moisture contents, or horizontal shears can easily be obtained without changing the underlying PV formula and inadvertently producing unreasonable values of static stability or CAPE. We demonstrate the characteristics of idealized extratropical cyclones developing in our background states by adding localized perturbations that represent an upper-level trough passing over a low-level frontal zone. In particular, we illustrate the impacts of horizontal shear, moisture, and grid spacing on baroclinic-wave development.

Terminal Hydride Complex of High-Spin Mn.

The iron-molybdenum cofactor of nitrogenase (FeMoco) catalyzes fixation of N2 via Fe hydride intermediates. Our understanding of these species has relied heavily on the characterization of well-defined 3d metal hydride complexes, which serve as putative spectroscopic models. Although the Fe ions in FeMoco, a weak-field cluster, are expected to adopt locally high-spin Fe2+/3+ configurations, synthetically accessible hydride complexes featuring d5 or d6 electron counts are almost exclusively low-spin. We report herein the isolation of a terminal hydride complex of four-coordinate, high-spin (d5; S = 5/2) Mn2+. Electron paramagnetic resonance and electron-nuclear double resonance studies reveal an unusually large degree of spin density on the hydrido ligand. In light of the isoelectronic relationship between Mn2+ and Fe3+, our results are expected to inform our understanding of the valence electronic structures of reactive hydride intermediates derived from FeMoco.

The Ramsey Numbers for Trees of Large Maximum Degree Versus the Wheel Graph W8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_8$$\\end{document}

The Ramsey numbers R(Tn,W8)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(T_n,W_8)$$\\end{document} are determined for each tree graph Tn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_n$$\\end{document} of order n≥7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 7$$\\end{document} and maximum degree Δ(Tn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (T_n)$$\\end{document} equal to either n-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-4$$\\end{document} or n-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-5$$\\end{document}. These numbers indicate strong support for the conjecture, due to Chen, Zhang and Zhang and to Hafidh and Baskoro, that R(Tn,Wm)=2n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(T_n,W_m) = 2n-1$$\\end{document} for each tree graph Tn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_n$$\\end{document} of order n≥m-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge m-1$$\\end{document} with Δ(Tn)≤n-m+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (T_n)\\le n-m+2$$\\end{document} when m≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\\ge 4$$\\end{document} is even.