Parallel Closure Research Articles

The Novel Parallel Closure Technique Compared to Single-Layer Closure of the Uterus After Primary Cesarean Section Decreases the Incidence of Isthmocele Formation and Increases Residual Myometrial Thickness.

Background Isthmocele or a scar defect is a relatively common consequence of cesarean section resulting in menstrual disturbances and infertility and may compromise the myometrial integrity of the uterus in women contemplating subsequent vaginal birth. Several preventive measures have been suggested, including the modification of surgical techniques used for the closure of the uterine incision. The current study aimed to compare the incidence of isthmocele and assess residual myometrial thickness in women who underwent single versus parallel layered closure to approximate the endo-myometrial layer during cesarean section. Methodology This retrospective study evaluated data of women undergoing their first cesarean section under elective conditions (n = 497) where the uterine incision was closed using a single (n = 295) or a parallel layer (n = 202) technique. Patients were evaluated twice, at 3-6 months and 18 months postpartum, with a transvaginal ultrasound noting the presence or absence of an isthmocele and measurement of the residual myometrial thickness. Results Regardless of the closure technique, 64 (12.9%) women had an ultrasound-diagnosed isthmocele. Significantly fewer patients in the parallel-layer closure group presented with an isthmocele both at 3-6 (13.6 vs. 6.9%; p = 0.019) and 18 months (16.3 vs. 7.8%; p = 0.009) postpartum. Residual myometrium was significantly thicker in the parallel-layer closure group (8.0 vs. 13.2 mm at 3-6 months postpartum; p = 0.000 and 7.2 vs. 12.3 mm at 18 months postpartum; p = 0.004). For all patients, a retroverted position of the uterus at 3-6 months follow-up examination significantly increased the frequency of isthmocele (36/395 (9.1%) with an anteverted uterus and 18/102 (17.6%) with a retroverted uterus; p = 0.002). In patients with a single-layer closure, a retroverted uterus at the 3-6-month follow-up was associated with an isthmocele in 29.5% (18/61) of patients, while no isthmocele was recorded when the uterus was retroverted in the parallel-layer closure group (0/41) (p = 0.001). At 18 months postpartum, of the 64 patients with an isthmocele, 26 (40.6%) presented with abnormal uterine bleeding mainly in the form of postmenstrual spotting. Of the 26 patients with abnormal bleeding, 23 were in the single-layer and three were in the parallel-layer closure group. Conclusions The parallel-layer closure when compared to a single-layer closure of the uterine incision in patients undergoing primary cesarean section decreased the incidence of isthmocele formation and increased residual myometrial thickness. More patients in the single-layer closure group had menstrual cycle disturbances at 18 months postpartum.

Non-overlapping High-accuracy Parallel Closure for Compact Schemes: Application in Multiphysics and Complex Geometry

Compact schemes are often preferred in performing scientific computing for their superior spectral resolution. Error-free parallelization of a compact scheme is a challenging task due to the requirement of additional closures at the inter-processor boundaries. Here, sources of the error due to sub-domain boundary closures for the compact schemes are analyzed with global spectral analysis. A high-accuracy parallel computing strategy devised in “ A high-accuracy preserving parallel algorithm for compact schemes for DNS. ACM Trans. Parallel Comput. 7, 4, 1-32 (2020)” systematically eliminates error due to parallelization and does not require overlapping points at the sub-domain boundaries. This closure is applicable for any compact scheme and is termed here as non-overlapping high-accuracy parallel (NOHAP) sub-domain boundary closure. In the present work, the advantages of the NOHAP closure are shown with the model convection equation and by solving the compressible Navier–Stokes equation for three-dimensional Rayleigh–Taylor instability simulations involving multiphysics dynamics and high Reynolds number flow past a natural laminar flow airfoil using a body-conforming curvilinear coordinate system. Linear scalability of the NOHAP closure is shown for the large-scale simulations using up to 19,200 processors.

Alkanal transfer hydrogenation catalyzed by solid Brønsted acid sites

Catalytic pathway and requirements for transfer hydrogenation of n-alkanals (CnH2nO, n=3–6) on Brønsted acid sites (H+) immobilized in microporous MFI and FAU crystalline structures or dispersed on H4SiW12O40 polyoxometalate clusters are established by isolating its rates from those of the various concomitant catalytic cycles. Transfer hydrogenation of alkanals involves a kinetically-relevant, inter-molecular hydride transfer step from substituted tetralins or cyclohexadienes produced from the parallel alkanal coupling and ring closure reactions as the hydride donor (R′H2) to protonated alkanals (RCH2CHOH+) as the hydride acceptor, via a bi-molecular transition state with a shared hydride ion, (RCH2CHOH+--H−–R′H+)‡. The rate constants for the inter-molecular hydride transfer step correlate directly to the hydride ion affinity difference between the carbenium ions of the H-donors (R′H+) and the protonated alkanals (RCH2CHOH+). As a result, smaller alkanals with higher hydride ion affinities are more effective in abstracting hydride ions and in transfer hydrogenation (C4>C5>C6). Propanal is an exception, as it is less effective in transfer hydrogenation than butanal. The deviation of propanal from the reactivity trend is apparently caused by its smaller transition state for hydride transfer, which is solvated to a lesser extent in FAU cages. The transfer hydrogenation occurs much more effectively on partially confined H+ sites in FAU structures than in smaller pore MFI or unconfined H4SiW12O40 polyoxometalate clusters, an indication that FAU solvates and stabilizes the bulky transition state of hydride transfer via van der Waals interactions. These effects of local site structures and the thermochemical properties of reactant determine the reactivity of alkanal transfer hydrogenation and thus selectivity ratio of alkenes, dienes, aromatics, and larger oxygenates during deoxygenation catalysis.

The impact of collisionality, FLR, and parallel closure effects on instabilities in the tokamak pedestal: Numerical studies with the NIMROD code

The extended-MHD NIMROD code [C. R. Sovinec and J. R. King, J. Comput. Phys. 229, 5803 (2010)] is verified against the ideal-MHD ELITE code [H. R. Wilson et al., Phys. Plasmas 9, 1277 (2002)] on a diverted tokamak discharge. When the NIMROD model complexity is increased incrementally, resistive and first-order finite-Larmour radius effects are destabilizing and stabilizing, respectively. The full result is compared to local analytic calculations which are found to overpredict both the resistive destabilization and drift stabilization in comparison to the NIMROD computations.

Experimental Validation of a Filament Transport Model in Turbulent Magnetized Plasmas.

In a wide variety of natural and laboratory magnetized plasmas, filaments appear as a result of interchange instability. These convective structures substantially enhance transport in the direction perpendicular to the magnetic field. According to filament models, their propagation may follow different regimes depending on the parallel closure of charge conservation. This is of paramount importance in magnetic fusion plasmas, as high collisionality in the scrape-off layer may trigger a regime transition leading to strongly enhanced perpendicular particle fluxes. This work reports for the first time on an experimental verification of this process, linking enhanced transport with a regime transition as predicted by models. Based on these results, a novel scaling for global perpendicular particle transport in reactor relevant tokamaks such as ASDEX-Upgrade and JET is found, leading to important implications for next generation fusion devices.

Linearly exact parallel closures for slab geometry

Parallel closures are obtained by solving a linearized kinetic equation with a model collision operator using the Fourier transform method. The closures expressed in wave number space are exact for time-dependent linear problems to within the limits of the model collision operator. In the adiabatic, collisionless limit, an inverse Fourier transform is performed to obtain integral (nonlocal) parallel closures in real space; parallel heat flow and viscosity closures for density, temperature, and flow velocity equations replace Braginskii's parallel closure relations, and parallel flow velocity and heat flow closures for density and temperature equations replace Spitzer's parallel transport relations. It is verified that the closures reproduce the exact linear response function of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] for Landau damping given a temperature gradient. In contrast to their approximate closures where the vanishing viscosity coefficient numerically gives an exact response, our closures relate the heat flow and nonvanishing viscosity to temperature and flow velocity (gradients).

The dynamisation of locking plate osteosynthesis by means of dynamic locking screws (DLS)—An experimental study in sheep

In this in vivo study a new generation of locking screws was tested. The design of the dynamic locking screw (DLS) enables the dynamisation of the cortex underneath the plate (cis-cortex) and, therefore, allows almost parallel interfragmentary closure of the fracture gap. A 45° angle osteotomy was performed unilaterally on the tibia of 37 sheep. Groups of 12 sheep were formed and in each group a different osteotomy gap (0, 1 and 3mm) was fixed using a locking compression plate (LCP) in combination with the DLS. The healing process was monitored radiographically every 3 weeks for 6, respectively 12 weeks. After this time the sheep were sacrificed, the bones harvested and the implants removed. The isolated bones were evaluated in the micro-computed tomography unit, tested biomechanically and evaluated histologically.The best results of interfragmentary movement (IFM) were shown in the 0mm configuration. The bones of this group demonstrated histomorphometrically the most distinct callus formation on the cis-cortex and the highest torsional stiffness relative to the untreated limb at 12 weeks after surgery.This animal study showed that IFM stimulated the synthesis of new bone matrix, especially underneath the plate and thus, could solve a current limitation in normal human bone healing. The DLS will be a valuable addition to the locking screw technology and improve fracture healing.

Parallel transitive closure algorithm

Abstract Using the BSP/CGM model, with $$p$$ processors, where $$p \ll n$$ , we present a parallel algorithm to compute the transitive closure of a digraph $$D$$ with $$n$$ vertices and $$m$$ edges. Our algorithm uses $$\log p + 1$$ communication rounds if the input is an acyclic directed graph labeled using a linear extension. For general digraphs, the algorithm requires $$p$$ communication rounds in the worst case. In both cases, $$O(M/p)$$ local computation is performed per round, where $$M$$ is the amount of computation needed to compute a sequential transitive closure of a digraph. The presented algorithm can be implemented using any sequential algorithm that computes the transitive closure of a digraph $$D$$ . We have implemented the algorithm using the Warshall transitive closure algorithm on two Beowulf clusters using MPI. The implementation results show its efficiency and scalability. It also compares favorably with other parallel implementations. The worst case (communication rounds) for a digraph was derived through an artificially elaborated example. However, in all our test cases, the algorithm never requires more than $$\log p + 1$$ communication rounds and presents very promising implementation results. The algorithm also can be applied to any $$n \times n$$ matrix that represents a binary relation.

Parallel transport of long mean-free-path plasma along open magnetic field lines: Parallel heat flux

In a long mean-free-path plasma where temperature anisotropy can be sustained, the parallel heat flux has two components with one associated with the parallel thermal energy and the other the perpendicular thermal energy. Due to the large deviation of the distribution function from local Maxwellian in an open field line plasma with low collisionality, the conventional perturbative calculation of the parallel heat flux closure in its local or non-local form is no longer applicable. Here, a non-perturbative calculation is presented for a collisionless plasma in a two-dimensional flux expander bounded by absorbing walls. Specifically, closures of previously unfamiliar form are obtained for ions and electrons, which relate two distinct components of the species parallel heat flux to the lower order fluid moments such as density, parallel flow, parallel and perpendicular temperatures, and the field quantities such as the magnetic field strength and the electrostatic potential. The plasma source and boundary condition at the absorbing wall enter explicitly in the closure calculation. Although the closure calculation does not take into account wave-particle interactions, the results based on passing orbits from steady-state collisionless drift-kinetic equation show remarkable agreement with fully kinetic-Maxwell simulations. As an example of the physical implications of the theory, the parallel heat flux closures are found to predict a surprising observation in the kinetic-Maxwell simulation of the 2D magnetic flux expander problem, where the parallel heat flux of the parallel thermal energy flows from low to high parallel temperature region.

Nonlocal closures for plasma fluid simulations

The application of fluid models in studies of transport and macroscopic stability of magnetized, nearly collisionless plasmas requires closure relations that are inherently nonlocal. Such closures address the fact that particles are capable of carrying information over macroscopic parallel scale lengths. In this work, generalized closures that embody Landau, collisional and particle-trapping physics are derived and discussed. A gyro/bounce-averaged drift kinetic equation is solved via an expansion in eigenfunctions of the pitch-angle scattering operator and the resulting system of algebraic equations is solved by integrating along characteristics. The desired closure moments take the form of integral equations involving perturbations in the flow and temperature along magnetic field lines. Implementation of the closures in massively parallel plasma fluid simulation codes is also discussed. This implementation includes the use of a semi-implicit time advance of the fluid equations to stabilize the dominant closure terms which are introduced explicitly. Application of the nonlocal, parallel heat flow closure, q∥, in studies of temperature flattening across helical magnetic islands in toroidal geometry reveal a scaling of temperature versus critical island width for flattening of T∼wd−1.5. This result predicts more robust flattening at small island widths when compared to the diffusive scaling, T∼wd−1.7, which assumes a Braginskii-type parallel heat conductivity. Preliminary application of q∥ to tokamak disruption simulations shows qualitative agreement of wall heat loads with experimental observations, smooth distribution in toroidal angle, and striation in the poloidal direction along the wall.

Unified form for parallel ion viscous stress in magnetized plasmas

In this work a unified form for the parallel ion viscous stress in a magnetized plasma is presented. Approximately valid for arbitrary collisionality, the integral nature of this generalized closure results from assuming the maximal ordering between collisional pitch-angle scattering and free streaming effects and from taking a Chapman–Enskog-type approach which includes the parallel ion viscous stress itself as a drive. The ion drift kinetic equation is solved in a sheared slab using an expansion in eigenfunctions of the Lorentz scattering operator. Integrating the coefficient equations in space and taking the proper velocity space moments couples the parallel viscous stress closure to an integral momentum restoring term, thus generalizing the concept of momentum conservation for simplified Coulomb collision operators. The integral closure involves following ions along magnetic field lines which are the ideal, time-independent characteristics of the perturbed distribution function. The fact that the viscous stress and momentum restoring term appear in the kernels of these field line integrals means that in general the closure has the form of coupled Volterra equations with an inhomogeneous term supplied by the traditional flow gradient drive. It is shown that the unified closure agrees both qualitatively and quantitatively with previous results and hence represents a generalized physical form for the parallel ion viscous stress in magnetized plasmas.

Comparison between kinetic and fluid simulations of slab ion temperature gradient driven turbulence

A detailed comparison between kinetic and fluid simulations of collisionless slab ion temperature gradient driven turbulence is made. The nondissipative closure model (NCM) for linearly unstable modes, which is presented by Sugama, Watanabe, and Horton [Phys. Plasmas 8, 2617 (2001)], and the dissipative closure model by Hammett and Perkins (HP) [Phys. Rev. Lett. 64, 3019 (1990)] are used in separate fluid simulations. The validity of these closure models for quantitative prediction of the turbulent thermal transport is examined by comparing nonlinear results of the fluid simulations with those of the collisionless kinetic simulation of high accuracy. Simulation results show that, in the saturated turbulent state, the turbulent thermal diffusivity χ obtained from the HP model is significantly larger than the χ given by the NCM which is closer to χ measured in the kinetic simulation. Contrary to the dissipative form of the parallel heat flux closure relation assumed in the HP model, the NCM describes well the exact kinetic simulation, in which for some unstable wave numbers k, the imaginary part of the ratio of the parallel heat flux qk to the temperature fluctuation Tk is a oscillatory function of time and sometimes takes positive values. The positive values of Im(qk/Tk), imply the negative parallel heat diffusivity, correlate with the occasional inward heat flux occurring for the wave numbers k, and reduce the total χ.

Sketches from "My Voice Locked in: The Lives of Basil Bunting"

Despite the thirty-year rule, MI6 and the Foreign Office files on Basil Bunting remain closed. His neglect by the English literary establishment would suggest a parallel closure. All his life Basil Bunting tried to ensure that friends and colleagues destroyed the letters he wrote to them, and burnt all those he received, once read. That's why you won't find anything but footnotes to Bunting in the collected letters of Yeats, Ford Maddox Ford, T.S. Eliot, W.C. Williams, Ezra Pound amongst others. He wanted only his work to speak for him and his life. He also destroyed, or sought to have destroyed, manuscripts and early versions including those by his friends that came into his possession. There is no doubt that he was amused at the frustration this would cause to literary historians. He wrote to me: is no excuse for literary criticism.(1) But such activities could also arouse genuine anger in him. When R.B. Woodings, Editorial Director of Faber and Faber, wrote a polite letter to him in 1977 asking for a quotable comment on Ezra Pound's Collected Early Poems, he replied: Dear Mr Wooding - Thank you for sending me the book - It makes me shudder. No doubt it is fitting that maggots consume us in the end, or at least the rubbish we scatter as we go; but I'd rather leave the lid on my dustbin and the earth on my friends' graves. - Piety takes curious forms: the toenail clippings of Saint What's His Name are revered. I don't think religion is much advanced by that. It would be more profitable, more to his glory, to throw away some of the poems Pound printed than to print those he threw away himself. - I apologise for my lack of sympathy for the industrious compilers.(2) His own letters were always entertaining or informative so very few of his friends, including myself, complied with his request to destroy them. I met Bunting in 1963 when I was seventeen and he was sixty-three. We became and remained close friends. In 1982 when I expressed a desire to write his biography he sent this typed note: TAKE HEED My biography appears in Who's Who, complete except for the date of my death which will not be delayed very long. Whatever anybody else writes must be looked on as fiction. Tom Pickard has a vigorous imagination and I hope whatever he writes will amuse you. I haven't read it; and once it's in print I shall not trouble to deny it unless it seems to libel someone else. There is another note worth quoting. This is from the MS notebook of Briggflatts, given to me in 1965, and from which I was meant to learn how to write a long poem. In the note, crossed out in the MS and unpublished in the final poem, he may only have been trying to sketch a character, perhaps (as it is an autobiography) that of Bunting himself: The scholar ought to be like the poet, an Ishmael, scouted and feared; a magician, impious, to be consulted in secrecy and shame. Only the neglected by man can keep either truth or beauty in view. The moment advantage has a part in his studies or his craft, his work perishes.(3) School After a year and a half at Newcastle Grammar School Bunting was sent to a Quaker boarding school at Ackworth in North Yorkshire where they instilled the general Quaker attitude and every morning you had to get a lump of Bible by heart before breakfast. At breakfast the Bible was read to you. At tea time, after tea, the Bible was read to you again. And on Sundays there were very large lumps of the Bible, beside Scripture lessons in between. Consoling himself in the library he found a copy of Wait Whitman's Leaves of Grass hidden behind the oak bookshelves. He was swept up by the open expansive verse and developed a lifelong love of the American poet. So enamored was he of Whitman that he secretly wrote an essay on the book and entered it for a national competition. It won first prize and the displeasure of the headmaster who disapproved of the exceedingly wicked book. …

Specific Deactivation of the Mouse Mammary Tumor Virus Long Terminal Repeat Promoter upon Continuous Hormone Treatment

We have studied the transcriptional behavior of the mouse mammary tumor virus long repeat (MMTV-LTR) promoter during a prolonged exposure to glucocorticoids. When integrated into XC-derived cells, MMTV-LTR expression reached its maximum during the first day of dexamethasone treatment, but longer exposure to the hormone resulted in the deactivation of the promoter. In contrast, glucocorticoid-responsive resident genes or MMTV-based transiently transfected plasmids maintained or even increased their mRNA levels during the same period of hormone treatment. An integrated chimeric construct containing the hormone-responsive elements from MMTV-LTR but in different sequence context became also deactivated after a prolonged hormone treatment but with a deactivation kinetics significantly slower than constructs containing the entire, chromatin-positioning MMTV-LTR sequence. The decrease on MMTV-LTR-driven transcription was concomitant with a parallel closure of the MMTV-LTR chromatin and with a decrease in glucocorticoid receptor (GR) concentration in the cell. We concluded that the chromatin-organized MMTV-LTR promoter is particularly sensitive to any decrease on GR levels. We propose that chromatin structure may contribute decisively to the differential expression of MMTV-LTR by two mechanisms: limiting MMTV-LTR accessibility to activating transcription factors and accelerating its shutting down upon a decrease on GR levels.

Parallel transitive closure computation in relational databases

The transitive closure operation is an important extension to relational algebra. Because of its high computation cost, it is of great interest to design efficient parallel algorithms for computing the transitive closure in relational database systems. In this paper, we present a new algorithm to compute transitive closures on SIMD meshes based on relational algebra operations. Double-hash distribution is used to avoid rehashing new tuples for the next join phase. There presently exists no extra step for the redistribution of these tuples. Possible redundant computation between different join phases has been prevented without using global operations. As only regular linear communication occurs on the mesh, and the workload is fully distributed, a speedup of O( n × n) has been achieved, where n × n is the size of mesh. Therefore, this algorithm is an optimal parallel version of the transitive closure algorithms based on relational algebra operations on SIMD meshes.

A new fragmentation scheme for recursive query processing

Data fragmentation is a well-known technique used in distributed database design to support efficient query processing. In deductive databases and knowledge bases, one distinct operation is the computation of transitive closures. In this paper, we employ a fragmentation approach to deductive databases. Fragmentation techniques are investigated in distributed database environments for parallel recursive query processing. Based on the novel fragmentation scheme, a new parallel transitive closure algorithm is proposed. Compared with previous algorithms, our algorithm can explore a much higher level of parallelism with very little communication overhead increase.

Blade-deviated fenestrated clips for anterior communicating artery aneurysms

Blade-deviated fenestrated clips developed by one of the authors were applied very successfully in 15 cases of A Com A aneurysms which corresponded to 32% of the aneurysms upon which were operated. Of the 15, 13 cases were projected superiorly, which is the most difficult and hazardous type if ruptured at the neck. The characteristic feature of the clip is the lateral deviation of occluding blades. The superiorly projecting type A Com A aneurysm is located posterolateral to the A Com A in the operative field, and the clips were extremely useful in the aneurysms in providing safer and more exact occlusion via parallel closure of the A Com A or the aneurysm neck.

Triplet–singlet intersystem crossing as the second step of the cycloaddition of triplet penta-1,4-diene. An ab initio MO study

The mechanism of the second step in the photosensitized cycloaddition reaction of penta-1,4-diene has been investigated. This step, which starts from an aliphatic cyclic intermediate on the first excited triplet (T1) surface, consists of an intersystem crossing from the T1 state to the ground singlet state (So) and ring closure on the So surface. For two possible reaction paths, cross addition and parallel closure, we have calculated by an ab initio MO method, the lowest point on the crossing seam between the So and T1 states, which can be regarded as the transition state for this second step. The triplet cyclic intermediate for cross addition, 2, requires only ca. 1 kcal mol–1 to reach the lowest crossing point accompanied by 90° internal rotation of the terminal methylene group with relatively small changes in the skeletal structure. On the other hand, the triplet cyclic intermediate for parallel closure, 4, requires ca. 5 kcal mol–1, accompanied by a drastic change in the skeletal molecular structure. The calculation confirms that this second step is not the rate-determining step of the entire cycloaddition reaction.

ChemInform Abstract: Parallel and Cross Cycloaddition of Triplet Penta‐1,4‐diene. An ab initio MO Study

ChemInformVolume 21, Issue 36 Physical Organic Chemistry ChemInform Abstract: Parallel and Cross Cycloaddition of Triplet Penta-1,4-diene. An ab initio MO Study M. OHSAKU, M. OHSAKU Inst. Mol. Sci., Okazaki Natl. Res. Inst., Myodaiji, Okazaki 444, JapanSearch for more papers by this authorK. MOROKUMA, K. MOROKUMA Inst. Mol. Sci., Okazaki Natl. Res. Inst., Myodaiji, Okazaki 444, JapanSearch for more papers by this author M. OHSAKU, M. OHSAKU Inst. Mol. Sci., Okazaki Natl. Res. Inst., Myodaiji, Okazaki 444, JapanSearch for more papers by this authorK. MOROKUMA, K. MOROKUMA Inst. Mol. Sci., Okazaki Natl. Res. Inst., Myodaiji, Okazaki 444, JapanSearch for more papers by this author First published: September 4, 1990 https://doi.org/10.1002/chin.199036041Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume21, Issue36September 4, 1990 RelatedInformation

Parallel and cross cycloaddition of triplet penta-1,4-diene. An ab initio MO study

ab initio UHF molecular orbital calculations using the 3-21G through 6-31G* basis sets, and including spin projection and electron correlation (APUMP2), have been carried out to investigate the photosensitized cycloaddition reactions of penta-1,4-diene. Two forms of triplet penta-1,4-diene, cis-trans and envelope, with nearly equal energy but with excitation localized on different bonds, are the starting points for the transition states TS1 of cross closure and TS2 of parallel closure, respectively. TS2 has a barrier of 14 kcal mol–1, about 22 kcal mol–1 lower than TS1, supporting the experimental dominance of parallel closure. The transition states have structures where a significant ring strain has already developed, and the TS1 – TS2 energy difference is mainly dictated by the strain energy difference of the cyclic triplet product intermediates.