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To determine the frequency, magnitude, and direction of toric intraocular lens (IOL) planning discrepancies between the Johnson & Johnson (J&J) official calculator and European Society of Cataract and Refractive Surgeons (ESCRS)-hosted toric engines, and to identify preoperative predictors of clinically relevant disagreements. This retrospective, single-center study included 182 eyes of 127 patients undergoing toric IOL implantation (TECNIS). For each eye, planning results were generated using identical biometric data (IOLMaster 700, CASIA2 keratometry) across all calculators. The primary comparison was J&J with posterior corneal astigmatism correction enabled (PCA-ON) versus ESCRS-Barrett - secondary comparisons included Kane, EVO 2.0, and Hoffer QST. Clinically relevant discrepancy (ClinRel) was defined as a toric cylinder step difference ≥ 1 and/or axis difference ≥ 10°. Logistic regression, including Firth-penalized estimation, identified predictors of ClinRel. The primary analysis set comprised one eye per patient (n=127). ClinRel between J&J PCA-ON and Barrett occurred in 59.8% of eyes (76/127; 95% CI: 51.1-68.0%), driven almost only by step differences. J&J PCA-ON recommended a higher step in 56.7% of eyes, with only 2.4% showing the opposite. Discrepancies were concentrated in with-the-rule eyes (72.3%) and nearly absent in against-the-rule eyes (5.6%). Rates were higher for secondary engines (Kane 83.8%, EVO 74.4%, Hoffer QST 76.7%). An Alpins-style paired-planning analysis yielded a Correction Index of 1.20, with the Difference Vector concentrated at the vertical meridian (84°). Posterior corneal astigmatism grade was the strongest independent predictor (Firth-adjusted OR = 203.5; p=0.014). An empirical PCA threshold of 0.36 D (Youden index; sensitivity 91.6%, specificity 56.0%) identified eyes at elevated risk of calculator disagreement. Calculators' variability in toric IOL planning is substantial (~60% of eyes) and concentrated in with-the-rule eyes with elevated posterior corneal astigmatism. In this subgroup, consulting multiple calculators is recommended; in against-the-rule eyes, calculator choice rarely alters the recommendation.

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

Michal Klimek,1,2 Maximilian Tamberger,1,2 Stephanie Schauer,1,2 Dejan Botic,1,2 Kata Mihaltz,1,2 Pia Veronika Vécsei-Marlovits1,2 1Department of Ophthalmology, Klinik Hietzing, Wiener Gesundheitsverbund, Vienna, Austria; 2Karl Landsteiner Institut für Prozessoptimierung und Qualitätsmanagement in der Katarakt-Chirurgie, Vienna, AustriaCorrespondence: Michal Klimek, Department of Ophthalmology, Klinik Hietzing, Wiener Gesundheitsverbund, Wolkersbergenstraße 1, Vienna, 1130, Austria, Tel +43 664 8132466, Fax +43 1 801 10 2266, Email michal.klimek@gesundheitsverbund.atPurpose: To determine the frequency, magnitude, and direction of toric intraocular lens (IOL) planning discrepancies between the Johnson & Johnson (J&J) official calculator and European Society of Cataract and Refractive Surgeons (ESCRS)-hosted toric engines, and to identify preoperative predictors of clinically relevant disagreements.Methods: This retrospective, single-center study included 182 eyes of 127 patients undergoing toric IOL implantation (TECNIS). For each eye, planning results were generated using identical biometric data (IOLMaster 700, CASIA2 keratometry) across all calculators. The primary comparison was J&J with posterior corneal astigmatism correction enabled (PCA-ON) versus ESCRS-Barrett – secondary comparisons included Kane, EVO 2.0, and Hoffer QST. Clinically relevant discrepancy (ClinRel) was defined as a toric cylinder step difference ≥ 1 and/or axis difference ≥ 10°. Logistic regression, including Firth-penalized estimation, identified predictors of ClinRel. The primary analysis set comprised one eye per patient (n=127).Results: ClinRel between J&J PCA-ON and Barrett occurred in 59.8% of eyes (76/127; 95% CI: 51.1– 68.0%), driven almost only by step differences. J&J PCA-ON recommended a higher step in 56.7% of eyes, with only 2.4% showing the opposite. Discrepancies were concentrated in with-the-rule eyes (72.3%) and nearly absent in against-the-rule eyes (5.6%). Rates were higher for secondary engines (Kane 83.8%, EVO 74.4%, Hoffer QST 76.7%). An Alpins-style paired-planning analysis yielded a Correction Index of 1.20, with the Difference Vector concentrated at the vertical meridian (84°). Posterior corneal astigmatism grade was the strongest independent predictor (Firth-adjusted OR = 203.5; p=0.014). An empirical PCA threshold of 0.36 D (Youden index; sensitivity 91.6%, specificity 56.0%) identified eyes at elevated risk of calculator disagreement.Conclusion: Calculators’ variability in toric IOL planning is substantial ( 60% of eyes) and concentrated in with-the-rule eyes with elevated posterior corneal astigmatism. In this subgroup, consulting multiple calculators is recommended; in against-the-rule eyes, calculator choice rarely alters the recommendation.Keywords: toric IOL, calculator comparison, posterior corneal astigmatism, toric planning variability, ESCRS toric calculator, Johnson & Johnson toric calculator

To compare toric intraocular lens (IOL) alignment assisted by image-guided surgery or manual marking methods and its impact on visual quality. This prospective comparative study enrolled 80 eyes with cataract and astigmatism ≥1.5 D to undergo phacoemulsification with toric IOL alignment by manual marking method using bubble marker (group I, n=40) or Callisto eye and Z align (group II, n=40). Postoperatively, accuracy of alignment and visual quality was assessed with a ray tracing aberrometer. Primary outcome measure was deviation from the target axis of implantation. Secondary outcome measures were visual quality and acuity. Follow-up was performed on postoperative days (PODs) 1 and 30. Deviation from the target axis of implantation was significantly less in group II on PODs 1 and 30 (group I: 5.5°±3.3°, group II: 3.6°±2.6°; =0.005). Postoperative refractive cylinder was -0.89±0.35 D in group I and -0.64±0.36 D in group II ( =0.003). Visual acuity was comparable between both the groups. Visual quality measured in terms of Strehl ratio ( <0.05) and modulation transfer function (MTF) ( <0.05) was significantly better in the image-guided surgery group. Significant negative correlation was observed between deviation from target axis and visual quality parameters (Strehl ratio and MTF) ( <0.05). Image-guided surgery allows precise alignment of toric IOL without need for reference marking. It is associated with superior visual quality which correlates with the precision of IOL alignment.

Nous étudions la fonction zêta des hauteurs anticanonique d’une variété torique (non nécessairement déployée) définie sur un corps global de caractéristique positive. Nous nous inspirons pour cela de la méthode utilisée par Batyrev et Tschinkel pour traiter la situation analogue en caractéristique zéro, méthode que nous rappelons d’ailleurs en détail. We investigate the anticanonical height zeta function of a (not necessarily split) toric variety defined over a global field of positive characteristic, drawing our inspiration from the method used by Batyrev and Tschinkel to deal with the analogous problem over a number field. By the way, we give a detailed account of their method.

The Eyhance Toric intraocular lens (IOL) builds upon the Tecnis Toric platform, initially associated with considerable post-operative rotational instability. Version 2, the Eyhance Toric IOL has been modified to enhance rotational stability. This study evaluates the post-operative rotational stability of the Eyhance Toric IOL compared to the Clareon Toric IOL, recognized for its stable performance. Patients undergoing cataract surgery received either the Eyhance or Clareon Toric IOLs. Placement was guided by the Barrett Toric Calculator at baseline (P0). IOL stability, uncorrected distance visual acuity (UDVA), corrected distance visual acuity (CDVA), and refractive astigmatism were assessed at 6-24 hours (P1) and 3 weeks to 6 months (P2) post-operatively. IOL rotational measurements were recorded at each interval. The study included 187 patients (median age: 74 for Clareon, 79 for Eyhance, p = 0.004). No significant differences were found in UDVA, CDVA, or refractive astigmatism at P2. Median rotation from P0 to P1 (3.0 vs 4.0 degrees, p = 0.091) and P0 to P2 (1.0 vs -0.5 degrees, p = 0.482) were not statistically different. However, the Clareon IOL showed less rotation between P1 and P2 (0.0 vs 1.0 degrees, p = 0.049). Absolute rotation from P0 to P1 (4.0 degrees), P1 to P2 (1.0 vs 2.0 degrees, p = 0.064), and P0 to P2 (4.0 vs 3.5 degrees, p = 0.095) were comparable. The Eyhance Toric IOL demonstrated comparable rotational stability and visual outcomes to the Clareon Toric IOL. Modifications in the Eyhance design have successfully improved its rotational stability, positioning it as a viable alternative to the Clareon Toric IOL in clinical practice.

When geometric quantization is applied to a manifold using a real polarization which is “nice enough”, a result of Śniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld leaves. Subsequently, several authors have taken this as motivation for counting Bohr-Sommerfeld leaves when studying the quantization of manifolds which are less “nice”. In this paper, we examine the quantization of compact symplectic manifolds that can locally be modelled by a toric manifold, using a real polarization modelled on fibres of the moment map. We compute the results directly, and obtain a theorem similar to Śniatycki’s, which gives the quantization in terms of counting Bohr-Sommerfeld leaves. However, the count does not include the Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is different from the quantization obtained using a Kähler polarization.

Since the introduction of the first toric intraocular lens (IOLs) in the early 1990s, these lenses have become the preferred choice for surgeons across the globe to correct corneal astigmatism during cataract surgery. These lenses allow patients to enjoy distortion-free distance vision with excellent outcomes. They also have their own set of challenges. Inappropriate keratometry measurement, underestimating the posterior corneal astigmatism, intraoperative IOL misalignment, postoperative rotation of these lenses, and IOL decentration after YAG-laser capsulotomy may result in residual cylindrical errors and poor uncorrected visual acuity resulting in patient dissatisfaction. This review provides a broad overview of a few important considerations, which include appropriate patient selection, precise biometry, understanding the design and science behind these lenses, knowledge of intraoperative surgical technique with emphasis on how to achieve proper alignment manually and with image-recognition devices, and successful management of postoperative complications.

In this article we use techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras. The Calabi-Yau property appears throughout geometry and string theory and is increasingly being studied in algebra. We further show that the algebras constructed are examples of non-commutative crepant resolutions (NCCRs), in the sense of Van den Bergh, of Gorenstein affine toric threefolds. Dimer models, first studied in theoretical physics, give a way of writing down a class of non-commutative algebras, as the path algebra of a quiver with relations obtained from a ‘superpotential’. Some examples are Calabi-Yau and some are not. We consider two types of ‘consistency’ conditions on dimer models, and show that a ‘geometrically consistent’ dimer model is ‘algebraically consistent’. We prove that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows us to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entov-Polterovich theory of spectral symplectic quasi-states and quasi-morphisms by incorporating The most novel part of this paper is to use open-closed Gromov-Witten-Floer theory (operator We use this open-closed Gromov-Witten-Floer theory to produce new examples. Especially using the calculation of Lagrangian Floer cohomology with bulk deformation in Fukaya, et al. (2010, 2011, 2016), we produce examples of compact symplectic manifolds Many of these applications were announced in Fukaya, et al. (2010, 2011, 2012).