Topological Field Theory with Haagerup Symmetry

We construct a (1+1)$d$ topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup $\mathcal{H}_3$ fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to the this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup $\mathcal{H}_1$ and $\mathcal{H}_2$ fusion categories can be obtained by gauging algebra objects. This note makes a modest offering in our pursuit of exotica and the quest for their eventual conformity.Comment: 41+11 pages, 1 figure, 3 tables; v2: corrected statements about the literature, revised Appendix

The F-Symbols for Transparent Haagerup-Izumi Categories with G = Z_(2n+1)

The notion of a transparent fusion category is defined. For the Haagerup-Izumi fusion rings with G=Z_(2n+1) (the Z_3 case is the Haagerup H_3 fusion ring), the transparent property reduces the number of independent F-symbols from order O(n6) to O(n^2), rendering the pentagon identity practically solvable. Transparent fusion categories are constructed up to Z_(15), and the explicit F-symbols are compactly presented. The potential construction of categories for new families of fusion rings is discussed

The F-Symbols for Transparent Haagerup-Izumi Categories with G = Z_(2n+1)

The notion of a transparent fusion category is defined. For the Haagerup-Izumi fusion rings with G=Z_(2n+1) (the Z_3 case is the Haagerup H_3 fusion ring), the transparent property reduces the number of independent F-symbols from order O(n6) to O(n^2), rendering the pentagon identity practically solvable. Transparent fusion categories are constructed up to Z_(15), and the explicit F-symbols are compactly presented. The potential construction of categories for new families of fusion rings is discussed

Adult morphology and redescription of Lestidiops indopacificus (Ege, 1953), with comments on the features of related species (Teleostei, Aulopiformes, Paralepididae)

Two specimens representing the first known adults of Lestidiops indopacificus (Ege, 1953) are reported and described from Taiwan, and the validity and generic assignment of this species are confirmed. The origin of the pelvic fin directly below the dorsal-fin base shows that L. indopacificus belongs to the L. mirabilis species complex. It can be separated from its congeners by the position of the nostrils above the posterior end of the maxilla, the light body color with unevenly distributed melanophores in adults, and a distinct combination of meristic values and other morphological characteristics. New geographic records are reported for the two other current members of this species complex, L. mirabilis (Ege, 1933) and L. extremus (Ege, 1953). The diagnostic features that separate these three very similar species are discussed

Integrating a Hive Triangle Pattern with Subpixel Analysis for Noncontact Measurement of Structural Dynamic Response by Using a Novel Image Processing Scheme

This work presents a digital image processing approach with a unique hive triangle pattern by integrating subpixel analysis for noncontact measurement of structural dynamic response data. Feasibility of proposed approach is demonstrated based on numerical simulation of a photography experiment. According to those results, the measured time-history displacement of simulated image correlates well with the numerical solution. A small three-story frame is then mounted on a small shaker table, and a linear variation differential transformation (LVDT) is set on the second floor. Experimental results indicate that the relative error between data from LVDT and analyzed data from digital image correlation is below 0.007%, 0.0205 in terms of frequency and displacement, respectively. Additionally, the appropriate image block affects the estimation accuracy of the measurement system. Importantly, the proposed approach for evaluating pattern center and size is highly promising for use in assigning the adaptive block for a digital image correlation method

Simultaneous Cranioplasty and Subdural-Peritoneal Shunting for Contralateral Symptomatic Subdural Hygroma following Decompressive Craniectomy

Background. Contralateral subdural hygroma caused by decompressive craniectomy tends to combine with external cerebral herniation, causing neurological deficits. Material and Methods. Nine patients who underwent one-stage, simultaneous cranioplasty and contralateral subdural-peritoneal shunting were included in this study. Clinical outcome was assessed by Glasgow Outcome Scale as well as Glasgow Coma Scale, muscle power scoring system, and complications. Results. Postoperative computed tomography scans demonstrated completely resolved subdural hygroma and reversed midline shifts, indicating excellent outcome. Among these 9 patients, 4 patients (44%) had improved GOS following the proposed surgery. Four out of 4 patients with lethargy became alert and orientated following surgical intervention. Muscle strength improved significantly 5 months after surgery in 7 out of 7 patients with weakness. Two out of 9 patients presented with drowsiness due to hydrocephalus at an average time of 65 days after surgery. Double gradient shunting is useful to eliminate the respective hydrocephalus and contralateral subdural hygroma. Conclusion. The described surgical technique is effective in treating symptomatic contralateral subdural hygroma following decompressive craniectomy and is associated with an excellent structural and functional outcome. However, subdural-peritoneal shunting plus cranioplasty thoroughly resolves the subdural hygroma collection, which might deteriorate the cerebrospinal fluid circulation, leading to hydrocephalus