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Flexural properties of visually graded southern pine structural lumber were evaluated. Several grade controlling characteristics were considered relative to bending properties and compared with current design values. A total of 751 southern pine lumber specimens were obtained from a broad spectrum of regions in the southeastern United States. Visually graded No. 2, nominally two inch thick specimens, in four and six inch widths, were obtained from commercial sawmills. All specimens were evaluated by a certified grader in the laboratory. Actual dimensions, weight, and moisture content (MC) were measured. Growth and manufacturing related characteristics were identified and classified into two categories: strength reducing characteristics (SRC) and grade reducing characteristics (GRC). Specific gravity (SG), bending modulus of elasticity (MOE), and modulus of rupture (MOR), were determined for each specimen. The presence of knots was identified as the most significant SRC; their presence had the most significant impact on SG, MOE and MOR. For GRC, specimens with knots, warp and specimens that fell into the category none, were significantly lower in SG, MOE and MOR. MOE and the allowable design bending strength values yielded in this study met the current design value criteria for both widths tested. [ABSTRACT FROM AUTHOR]

Crossarms are heavy timbers used to support electrical lines on utility poles. The requirements for crossarm material are fairly strict especially with respect to limitations on the presence of knots, which represent a gap or hole in the timber and also result in grain deviations that can affect properties. The strict grading rules can sometimes result in shortages of acceptable material. The potential for accepting arms with minor, but currently unacceptable defects was explored on a population of 200 rejected Douglas-fir crossarms and 50 arms that met current standards. Knot diameter and location on the arms were mapped and the arms were tested to failure in a specially designed apparatus that simulated an actual line load. The resulting data were used to calculate Modulus of Rupture (MOR) and Modulus of Elasticity (MOE). Nearly all arms failed above the minimum American National Standards assumed MOR value of 53.78 MPa and many of the rejected arms had higher MOR values than acceptable arms. While there was more variability in properties for the rejected arm population, correlations between knot diameter or knot location and MOR were all exceeding poor (r2 ≤ 0.25). The results suggest the need for further evaluation of the emphasis on knots in grading rules. [ABSTRACT FROM AUTHOR]

We prove that a minimal disc in a CAT.0/space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fáry-Milnor's theorem in the CAT.0/setting. [ABSTRACT FROM AUTHOR]

We consider -difference equations for colored Jones polynomials. These sequences of polynomials are invariants for the knots and their asymptotics plays an important role in the famous volume conjecture for the complement of the knot to the d sphere. We give an introduction to the theory of hyperbolic volume of the knots complements and study the asymptotics of the solutions of -recurrence relations of high order. [ABSTRACT FROM AUTHOR]

A group of matrices with entries in a number field K is integral if it has a finite index subgroup of matrices whose entries are algebraic integers. In this paper, we show that the fundamental groups, G (p / q , (m , n)) , of the orbifolds (p / q , (m , n) are integral as subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , for all the rational knots and links p / q and all the isotropies with m > 2 , n > 2. We obtain the same result for the fundamental group G (m , m , m) of the orbifold B (m , m , m) , m > 2 , where B are the Borromean rings. The only groups G (m , n , p) with m , n , p ∈ 3 , 4 , 5 , 6 , ∞ , which are integral subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , are for the following (m , n , p) : { (3 , 3 , 3) , (3 , 3 , ∞) , (3 , 4 , 4) , (3 , 4 , ∞) , (3 , 6 , 6) , (3 , ∞ , ∞) , (4 , 4 , 4) , (4 , 4 , ∞) , (4 , ∞ , ∞) , (6 , 6 , 6) , (6 , 6 , ∞) , (∞ , ∞ , ∞) , (3 , 3 , 5) , (3 , 5 , 5) , (3 , 5 , ∞) , (4 , 4 , 5) , (4 , 5 , ∞) , (5 , 5 , 5) , (5 , 5 , ∞) , (5 , 6 , 6) , (5 , ∞ , ∞) }. [ABSTRACT FROM AUTHOR]

The variability of longitudinal bending strength of timber beams due to the presence of knots and other defects is analyzed. The weak zones model of timber beam bending strength used in the analysis consists of short weak zones (knots or group of them) and strong sections of clear wood. The load bearing capacity of timber beams is defined as an extreme (minimum) value problem or as a first downcrossing problem. Assuming a marked Poisson random field with zero correlation between all random variables, cumulative probability distribution functions of the load carrying capacity of timber beams were determined by analytical methods for typical load cases: pure bending, midspan point load and one-third point load. These results, as well as the marked Poisson random field as a model of longitudinal variability of the bending strength of timber beams, can be applied in the reliability analysis of timber structures. Furthermore, the analytical formulae for the cumulative distribution functions of load carrying capacity of timber beams can provide a good reference for numerical analysis conducted with Monte Carlo simulation to determined statistics for specific timber members. [ABSTRACT FROM AUTHOR]

Mechanically Interlocked Molecules (MIMs), such as catenanes, knots, and rotaxanes, are a class of molecules that possess mechanical bonds. These bonds constitute an entanglement in space between components of the MIMs that cannot be separated without breaking a participating covalent bond. Although it is possible to synthesize chiral MIMs with covalent stereogenic centers, this new bond opens up the possibilities of axial, planar and helical chirality, as well as topologically chiral compounds. A growing number of chiral MIMs have been investigated and employed in numerous applications, ranging from sensing to catalysis. While the investigation of the properties of MIMs is highly developed, it is also essential to study their chiroptical properties. In this review, we focus on the chiroptical properties of chiral MIMs. We discuss their electronic circular dichroism (ECD) since these properties have been analyzed in some detail in relation to their structures and stereochemistry. Although only a few examples have been described to date, we discuss the encouraging circularly polarized luminescence (CPL) properties of MIMs. The review also includes a recent study of vibrational circular dichroism (VCD) in mechanically planar chiral rotaxanes suggesting that this technique is a promising tool for analyzing the structures of MIMs. The study of the chiroptical properties of MIMs can have a large impact on their use in materials science or as catalysts. One of the key advantage of chiral MIMs is that they provide a means of generating efficient chiroptical switches that have a bright future in the context of multiple applications. [ABSTRACT FROM AUTHOR]

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of S 3 branched along a knot α ⊂ S 3 . Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. [ABSTRACT FROM AUTHOR]

This article describes the simplest members of an infinite family of knots and links that have achiral piecewise‐linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner‐ and stick‐2‐transitive – the smallest possible for achiral knots. [ABSTRACT FROM AUTHOR]

The conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of N ≫ 1 edges has only logarithmic, and not power law corrections to the known leading exponential law: Punknot(N) ~ exp[-N/N0 + o(ln N)] with N0 being referred to as the random knotting length. This conjecture is consistent with the numerical result of 2010 by Baiesi, Orlandini, and Stella. [ABSTRACT FROM AUTHOR]

Copyright of Culture & Folk Literature / Farhang va Adabiyyāt-i ̒āmah is the property of Tarbiat Modares University Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

John Conway was one of the most versatile mathematicians in modern times, who made important contributions to several areas of mathematics. In this article, we highlight his contributions to two areas—knot theory and group theory. [ABSTRACT FROM AUTHOR]

Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or self-intersections. This article develops efficient algorithms for (self-)repulsion of plane and space curves that are well-suited to problems in computational design. Our starting point is the so-called tangent-point energy, which provides an infinite barrier to self-intersection. In contrast to local collision detection strategies used in, e.g., physical simulation, this energy considers interactions between all pairs of points, and is hence useful for global shape optimization: local minima tend to be aesthetically pleasing, physically valid, and nicely distributed in space. A reformulation of gradient descent based on a Sobolev-Slobodeckij inner product enables us to make rapid progress toward local minima—independent of curve resolution. We also develop a hierarchical multigrid scheme that significantly reduces the per-step cost of optimization. The energy is easily integrated with a variety of constraints and penalties (e.g., inextensibility, or obstacle avoidance), which we use for applications including curve packing, knot untangling, graph embedding, non-crossing spline interpolation, flow visualization, and robotic path planning. [ABSTRACT FROM AUTHOR]

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices. [ABSTRACT FROM AUTHOR]

This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram D , it is known that s (D) , the number of Seifert circles in D , equals the braid index b (D) of D if D contains no lone crossings, where a crossing in D is called a lone crossing if it is the only crossing between two Seifert circles in D. If D contains lone crossings, then one can reduce the number of Seifert circles in D using link-type preserving moves such as the Murasugi–Przytycki operation. Let r (D) ≥ 1 be the maximum number of Seifert circles in D that can be reduced, then we have b (D) ≤ s (D) − r (D). On the other hand, if the a -span of the HOMFLY polynomial of D satisfies the equality a -span / 2 + 1 = s (D) − r (D) , then the Morton–Frank–Williams (MFW) inequality a -span / 2 + 1 ≤ b (D) leads us to the simple braid index formula b (D) = s (D) − r (D). In this paper, we derive explicit formulas for many alternating links based on minimum projections of these links by establishing the equality a -span / 2 + 1 = s (D) − r (D). Our methods depend on the structures of the link diagrams under consideration and our results lead to explicit braid index formulas that are applicable to a very large class of links, a proper subset of which contains all two bridge links, all alternating pretzel links, and more generally all alternating Montesinos links. The derived braid index formula for an alternating Montesinos link is a function whose inputs are the signs of the crossings in the rational tangles of the Montesinos link. Finally, by applying the now proven Jones Conjecture on the writhe of minimum braids of a link, our results also allow us to obtain explicit formulas of the writhe of minimum braids for the links discussed in this paper from the minimum projections of these links. [ABSTRACT FROM AUTHOR]

Networks of flexible filaments often involve regions of tight contact. Predictively understanding the equilibrium configurations of these systems is challenging due to intricate couplings between topology, geometry, large nonlinear deformations, and friction. Here, we perform an in-depth study of a simple, yet canonical, problem that captures the essence of contact between filaments. In the orthogonal clasp, two filaments are brought into contact, with each centerline lying in one of a pair of orthogonal planes. Our data from X-ray tomography (CT) and mechanical testing experiments are in excellent agreement with finite element method (FEM) simulations. Despite the apparent simplicity of the physical system, the data exhibit strikingly unintuitive behavior, even when the contact is frictionless. Specifically, we observe a curvilinear diamond-shaped ridge in the contactpressure field between the two filaments, sometimes with an inner gap. When a relative displacement is imposed between the filaments, friction is activated, and a highly asymmetric pressure field develops. These findings contrast to the classic capstan analysis of a single filament wrapped around a rigid body. Both the μCT and FEM data indicate that the cross-sections of the filaments can deform significantly. Nonetheless, an idealized geometrical theory assuming undeformable tube cross-sections and neglecting elasticity rationalizes our observations qualitatively and highlights the central role of the small, but nonzero, tube radius of the filaments. We believe that our orthogonal clasp analysis provides a building block for future modeling efforts in frictional contact mechanics of more complex filamentary structures. [ABSTRACT FROM AUTHOR]

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants c 1 , c 2 > 0 such that the ribbonlength is bounded above by c 1 Cr (K) 2 , and also by c 2 Cr (K) 3 / 2 . We use a different method for each bound. The constant c 1 is quite small in comparison to c 2 , and the first bound is lower than the second for knots and links with Cr (K) ≤ 12,748. [ABSTRACT FROM AUTHOR]

Effect of sawn zone on the quality of lumber in the evaluation of selected pine wood defects. In the course of research on the sawn pine raw material with defined distribution of the defects, a variable level of change in the presence of knots was assessed. Initially, the experimental material was classified in terms of the general-purpose timber, and then the strength classes of wood for structural applications were assigned. The proportion of sound knots increased in case of wood obtained from the middle and top zones. In the case of butt-end logs, an increase in the share of the rotten knots having an average diameter of 2-4 cm was observed. The intensity of the defect's occurrence corresponded with the zone of origin along the large-sized roundwood length. [ABSTRACT FROM AUTHOR]

We study and characterize the topology of connectivity circuits observed in natively folded protein structures whose coordinates are deposited in the Protein Data Bank (PDB). Polypeptide chains of some proteins naturally fold into unique knotted configurations. Another kind of nontrivial topology of polypeptide chains is observed when, in addition to covalent bonds connecting consecutive amino acids in polypeptide chains, one also considers disulfide and ionic bonds between non-consecutive amino acids. Bonds between non-consecutive amino acids introduce bifurcation points into connectivity circuits defined by bonds between consecutive and nonconsecutive amino acids in analyzed proteins. Circuits with bifurcation points can form θ-curves with various topologies. We catalog here the observed topologies of θ-curves passing through bridges between consecutive and non-consecutive amino acids in studied proteins., (© 2024 The Protein Society.)

The article comprises brief appreciations about the importance and contribution of the Miniature Art to the development of the culture, art and civilization of the Middle Ages. The small Celtic island – Ireland has influenced in a great manner the European creations and the miniature manuscripts from the 6th – 9th century. As surprisingly as it may seem, the majority of the manuscripts are essentially dominated by the abstract approach, which is close to the non-figurative creations. The testimony of this approach is represented by the Irish ornaments of Celtic origin, that are unique, complicated, non-repeatable, and full of unforeseen, which prove an unending fantasy and a great technical virtuosity.

A spectral sequence is established whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology in Khovanov's classification. [ABSTRACT FROM AUTHOR]

In studies on Polish pine wood, the parameters determining its usefulness for processing are highly variable. In contrast, international studies on the optimum utilization of coniferous sawn wood describes its visual features as those indicating the distribution of defects and their impact on the selected mechanical and physical properties of sawn wood. Knottiness is one of the qualitative properties of sawn wood that is essential for wood grading. The objective of this study was to determine a correlation between variable knots and their soundness on the one hand and their selected strength parameters on the other. One of the basic issues of this work includes structural timber optimization. For Polish sawn wood, these indicators are still relatively new. The study material involved edged butt-end log lumber from a 120-year-old forest stand. The results confirmed a correlation between knots frequency and modulus of elasticity. The correlation was strong, indicating that sound knots play a major role in the process of the strength assessments. The study did not demonstrate a significant share of rotten knots and their effect on strength. [ABSTRACT FROM AUTHOR]

Lehmer’s question, an open question about the Mahler measure of monic integral polynomials, is shown to be equivalent to a question about the complexity growth rate of signed 1-periodic graphs. If G is a d-periodic graph (i.e. G has a co-finite free ℤd-action by automorphisms), then a d-variable polynomial ∆G can be defined with Mahler measure equal to the logarithmic growth rate γG of a complexity defined for the finite quotients of G. A plane 1-periodic graph determines a link via projection and the medial graph construction. The polynomial ∆G can be determined from the Alexander polynomial of the link. The complexity growth rate γG of any d-periodic graph is at least log 2. An investigation of plane 1- and 2-periodic graphs yields more connections with knot theory including work of A. Champanerkar and I. Kofman. [ABSTRACT FROM AUTHOR]

Copyright of Annals of Warsaw University of Life Sciences - SGGW, Forestry & Wood Technology is the property of Wydawnictwo SGGW and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The knots 8 1 , 8 2 , 8 3 , 8 5 , 8 6 , 8 7 , 8 8 , 8 1 0 , 8 1 1 , 8 1 2 , 8 1 3 , 8 1 4 , 8 1 5 , 9 7 , 9 1 6 , 9 2 0 , 9 2 6 , 9 2 8 , 9 3 2 , and 9 3 3 all have superbridge index equal to 4. This follows from new upper bounds on superbridge index not coming from the stick number and increases the number of knots from the Rolfsen table for which superbridge index is known from 29 to 49. Appendix A gives the current state of knowledge of superbridge index for prime knots through 10 crossings. [ABSTRACT FROM AUTHOR]

Mathematical knots and links are described as piecewise linear – straight, non‐intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ('vertex'‐transitive). Corner‐ and stick‐transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest‐stick) embeddings. Stick 2‐transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted. [ABSTRACT FROM AUTHOR]

Study Objective: To investigate why security of identical knot sequences is variable and how to avoid occasionally insecure knots.Design: A factorial design was used to assess factors affecting the security of half knot (H) and half-hitch (S) knot combinations. The effect of tying forces and the risk factors to transform H knots into S knots were investigated. The risk factors evaluated were as follows: starting with an H1 or H2 instead of an H3 knot, inexperience, short sutures, and monomanual knot tying. Security of transformed knots, S2S1 and S2S2 knots, and their recuperation with 2 additional half hitches, SSb or SbSb, were evaluated.Setting: Training center for laparoscopic suturing.Patients: Not applicable.Interventions: Security of knots was evaluated in vitro.Measurements and Main Results: The forces that caused knot combinations to open before breaking of the suture were used to calculate the risk of opening with low forces. Tying more strongly increased the security of half knots (H2H1sH1s) (p <.02) and half hitches (p <.001). The forces needed to transform an H3 into an S3 are higher than those for an H2 (p <.001), and the risk increases when the surgeon is inexperienced (p <.001), when sutures are short (p <.001), and when monomanual knot tying (p <.001) is used. Inadvertently made S2S1 and S2S2 knots are dangerous, with the exception of the symmetric S2S2, which is stable. Unstable knots such as S2S1a and S2S2a knot combinations improve with 2 additional blocking half hitches (SbSb), but S2S2aSbSb remains occasionally insecure.Conclusion: To reduce the risk of accidentally transforming a first H into an S knot, it is recommended to start with an H3, tie with force, avoid short sutures, and use bimanual suturing. This permits the recommendation to use preferentially H3H2 knots or 5 half hitches (SSSbSbSb). When in doubt, half knot combinations should be secured with at least 2 blocking half hitches. [ABSTRACT FROM AUTHOR]

For any classical knot k 1 , we can construct a ribbon 2 -knot spun (k 1) by spinning an arc removed a small segment from k 1 about R 2 in R 4 . A ribbon 2 -knot is an embedded 2 -sphere in R 4 . If k 1 has an n -crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun (k 1). Thus, we can define naturally a notion on ribbon 2 -knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer n there exists an amphicheiral ribbon 2 -knot with the ribbon crossing number n. [ABSTRACT FROM AUTHOR]

A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a nonalternating form, and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millett, but these rely on a special continued fraction expansion of the rational number in which all partial denominators are even (called the all-even form). In this paper we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced alternating form, or, equivalently, the nonalternating form of the corresponding rational number. [ABSTRACT FROM AUTHOR]

We prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading. [ABSTRACT FROM AUTHOR]

Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus. [ABSTRACT FROM AUTHOR]

The production of poles in Chile is done by choosing trees specially selected for that purpose, however, the forestry that is massively applied is oriented to the simultaneous production of pruned, sawn and pulpable logs. Silviculture for the specific production of poles in Chile is a technically complex issue, not addressed to date, whose description and analysis is what this review attempts to outline. The technical requirements of poles in terms of size, stem shape, knots, nodal thickening, straightness, spiral grain and resistance were analysed. These requirements were related to the variables site, stand density, genotype, pruning, thinning and harvest age. From the analysis of the variables it was concluded that a high-density forestry is required with multinodal plants In sites of regular too bad quality, but that generates high density wood with a low pruning and a selection thinning. [ABSTRACT FROM AUTHOR]

For an unoriented link 𝒦 , let L (𝒦) be the ropelength of 𝒦. It is known that in general L (𝒦) is at least of the order O ((Cr (𝒦)) 3 / 4) , and at most of the order O (Cr (𝒦) ln 5 (Cr (𝒦)) where Cr (𝒦) is the minimum crossing number of 𝒦. Furthermore, it is known that there exist families of (infinitely many) links with the property L (𝒦) = O (Cr (𝒦)). A long standing open conjecture states that if 𝒦 is alternating, then L (𝒦) is at least of the order O (Cr (𝒦)). In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant a > 0 such that L (𝒦) ≥ a B (𝒦) for any 𝒦 , where B (𝒦) is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of 𝒦 (called the maximum braid index of 𝒦). Consequently, L (𝒦) ≥ O (Cr (𝒦)) for any link 𝒦 whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links. [ABSTRACT FROM AUTHOR]

The valuable luminescence, magnetic, and catalytic properties of lanthanide cations are beginning to be exploited in conjunction with structurally exotic mechanically interlocked molecules (MIMs) such as rotaxanes, catenanes and knots. This Minireview provides an account of this rapidly developing research area commencing with the use of lanthanides in extended MIM-containing frameworks. Then, attention turns to discrete lanthanide-containing pseudorotaxanes, followed by fully interlocked rotaxanes, catenanes and knots – where lanthanides have not only been incorporated into MIM architectures but have also been used to template formation of the interlocked structure. Particular focus is paid to examples where the lanthanide MIMs have been put to useful applications, in what is still a relatively youthful avenue of research in both lanthanide coordination chemistry and the chemistry of mechanically interlocked molecules. [ABSTRACT FROM AUTHOR]

Simultaneous applications of thinning and pruning are common silvicultural practices in radiata pine (Pinus radiata) forest plantations. Their separate effects on tree growth and wood quality have been well studied, but their combined effect is not clear yet. The aim of this study is to identify how thinning and pruning together affect the stem structure and properties of the pruned but also the unpruned section of the trees (the portions formed immediately after the application of these management efforts). The effects of pruning and thinning on the number of growth units per year, internode length, number of branches, and branch diameters was analyzed in managed and unmanaged stands of radiata pine grown in Chile. When used jointly, these practices generated larger individual tree volumes (135% more) and clear wood in the pruned logs; however, they also reduced the sawn wood quality of the unpruned stem section for some years after the silvicultural interventions. The managed trees showed more growth units per annual shoot and shorter internodes, thus generating more knotty wood. Moreover, managed trees showed more taper. As trees of the managed stand restore the foliar biomass lost due to pruning, managed and unmanaged stands approach the same level of canopy closure, and differences minimize.