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Pavel Etingof - One of the best experts on this subject based on the ideXlab platform.
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symmetric tensor categories in characteristic 2
Advances in Mathematics, 2019Co-Authors: Dave Benson, Pavel EtingofAbstract:Abstract We construct and study a Nested Sequence of finite symmetric tensor categories Vec = C 0 ⊂ C 1 ⊂ ⋯ ⊂ C n ⊂ ⋯ over a field of characteristic 2 such that C 2 n are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius–Perron dimension. This generalizes the category C 1 described by Venkatesh [28] and the category C 2 defined by Ostrik. The Grothendieck rings of the categories C 2 n and C 2 n + 1 are both isomorphic to the ring of real cyclotomic integers defined by a primitive 2 n + 2 -th root of unity, O n = Z [ 2 cos ( π / 2 n + 1 ) ] .
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symmetric tensor categories in characteristic 2
arXiv: Representation Theory, 2018Co-Authors: Dave Benson, Pavel EtingofAbstract:We construct and study a Nested Sequence of finite symmetric tensor categories ${\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset \mathcal{C}_n\subset\cdots$ over a field of characteristic $2$ such that $\mathcal{C}_{2n}$ are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius--Perron dimension. This generalizes the category $\mathcal{C}_1$ described by Venkatesh and the category $\mathcal{C}_2$ defined by Ostrik. The Grothendieck rings of the categories $\mathcal{C}_{2n}$ and $\mathcal{C}_{2n+1}$ are both isomorphic to the ring of real cyclotomic integers defined by a primitive $2^{n+2}$-th root of unity, $\mathcal{O}_n=\mathbb Z[2\cos(\pi/2^{n+1})]$.
Andrew Willis - One of the best experts on this subject based on the ideXlab platform.
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Tectonic control of Nested Sequence architecture in the Sego Sandstone, Neslen Formation and Upper Castlegate Sandstone (Upper Cretaceous), Sevier Foreland Basin, Utah, USA
Sedimentary Geology, 2000Co-Authors: Andrew WillisAbstract:Abstract The Sego Sandstone, Neslen Formation and Upper Castlegate Sandstone are laterally equivalent formations of the Upper Cretaceous Mesaverde Group, which crop out along the Book Cliffs of eastern Utah and represent sediment shed into a foreland basin from the Sevier orogenic belt to the west. The studied interval contains three regional unconformities which bound stratigraphic Sequences around 100 m thick and of one-to-three million year duration, here termed high-order Sequences I–III. The high-order Sequence boundaries display increasing depth of erosion northwestward toward the Sevier orogenic belt and mark changes in sediment dispersal patterns and provenance, none of which can be explained by eustatic sea-level falls. Their formation is attributed to erosion and isostatic rebound of proximal parts of the foreland basin following thrust events in the adjacent Sevier orogenic belt. In the southeastern part of the study area, high-order Sequence II (∼Sego Sandstone) contains several (>4) Nested stratigraphic Sequences around 20 m thick and of hundred thousand year duration, here termed low-order Sequences. The low-order Sequence boundaries are disconformities rather than angular unconformities and are not marked by changes in sediment dispersal patterns or provenance. Each of these low-order Sequences consists of: (a) LST of incised valleys (up to 14 m thick) filled with estuarine sandstone, and (b) TST of marine shale and sandstone. No HSTs are preserved. The origin of these Sequences (eustatic/tectonic) cannot be determined. When traced to the northwest, the low-order Sequence boundaries become conformable and unrecognizable, and no low-order stratigraphic Sequences corresponding to those in the Sego Sandstone can be recognized within the alluvial Upper Castlegate Sandstone. The observed Nested Sequence architecture of the Upper Mesaverde Group is considered to be controlled by variation in subsidence rate related to the emplacement and erosion of thrust sheets in the adjacent Sevier orogenic belt. During emplacement of thrust sheets, subsidence rates in proximal areas of the basin were sufficiently high to offset any low-order base-level falls, resulting in the deposition of a conformable fluvial succession. In distal parts of the basin where long-term subsidence rates were lower, low-order base-level falls produced relative falls in sea level and generated spatially restricted low-order Sequences.
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Tectonic control of Nested Sequence architecture in the Castlegate Sandstone (Upper Cretaceous), Book Cliffs, Utah
Journal of Sedimentary Research, 1996Co-Authors: Shuji Yoshida, Andrew Willis, Andrew D. MiallAbstract:ABSTRACT The Castlegate Sandstone at its type section, Price Canyon, near Price, Utah, encompasses a single stratigraphic Sequence spanning approximately 5 m.y. It includes a sandstone member corresponding to a lowstand systems tract, consisting of braided-fluvial sheet sandstones, and a mudstone member, in which shales are more abundant and some evidence of tidal influence is present. This is the transgressive to highstand systems tract. From near Trail Canyon eastward the mudstone member passes laterally into the Sego Sandstone and Neslen Formation, a succession of at least six higher-frequency Sequences of fluvial-estuarine origin. The Buck Tongue, a marine shale unit separating the Castlegate Sandstone and the Sego Sandstone east of Green River, is erosionally truncated below the Sego Sand tone northwest of Trail Canyon. We suggest that the origin of the Sequences is related to flexural loading and intraplate stress on two time scales. Eustasy cannot be ruled out, but there is no independent evidence for this process. The main 5 m.y. Sequence reflects regional tectonism, with the sandstone member developing at a time of slow subsidence, and the mudstone member reflecting a higher long-term subsidence rate. The higher-order Sequences Nested within the third-order Sequence east of Trail Canyon are interpreted as a basinal response to episodes of crustal shortening on a 105 yr time scale. This study amplifies the model of Posamentier and Allen (1993a), in which ramp-type foreland basins are divided into areas of rapid and slow subsidence (Zones A and B). We postulate that these zones migrated asinward and landward in response to variations in long-term subsidence rate (an effect not predicted in the original model), and can be mapped by reference to the distribution of Type 1 Sequence boundaries in the higher-order Sequences. Differences in Sequence architecture east and west of Trail Canyon may have been amplified by differences in crustal rheology. The Sequence architecture changes at the boundary of the underlying Paleozoic Paradox Basin, a zone of NW-SE-oriented folds, faults, and salt diapirs, which we suspect were reactivated by Cretaceous tectonism. The high-frequency Sequences are within the area of the Paradox Basin, an area that may have been more prone to vertical structural movements in response to intraplate stresses. Incipient uplift of Laramide structures may also have modified fluvial patterns and controlled the orientation of incised valleys on several of the Sequence boundaries.
Dave Benson - One of the best experts on this subject based on the ideXlab platform.
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symmetric tensor categories in characteristic 2
Advances in Mathematics, 2019Co-Authors: Dave Benson, Pavel EtingofAbstract:Abstract We construct and study a Nested Sequence of finite symmetric tensor categories Vec = C 0 ⊂ C 1 ⊂ ⋯ ⊂ C n ⊂ ⋯ over a field of characteristic 2 such that C 2 n are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius–Perron dimension. This generalizes the category C 1 described by Venkatesh [28] and the category C 2 defined by Ostrik. The Grothendieck rings of the categories C 2 n and C 2 n + 1 are both isomorphic to the ring of real cyclotomic integers defined by a primitive 2 n + 2 -th root of unity, O n = Z [ 2 cos ( π / 2 n + 1 ) ] .
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symmetric tensor categories in characteristic 2
arXiv: Representation Theory, 2018Co-Authors: Dave Benson, Pavel EtingofAbstract:We construct and study a Nested Sequence of finite symmetric tensor categories ${\rm Vec}=\mathcal{C}_0\subset \mathcal{C}_1\subset\cdots\subset \mathcal{C}_n\subset\cdots$ over a field of characteristic $2$ such that $\mathcal{C}_{2n}$ are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius--Perron dimension. This generalizes the category $\mathcal{C}_1$ described by Venkatesh and the category $\mathcal{C}_2$ defined by Ostrik. The Grothendieck rings of the categories $\mathcal{C}_{2n}$ and $\mathcal{C}_{2n+1}$ are both isomorphic to the ring of real cyclotomic integers defined by a primitive $2^{n+2}$-th root of unity, $\mathcal{O}_n=\mathbb Z[2\cos(\pi/2^{n+1})]$.
A. Chakrabarti - One of the best experts on this subject based on the ideXlab platform.
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a Nested Sequence of projectors 2 multiparameter multistate statistical models hamiltonians s matrices
arXiv: Quantum Algebra, 2006Co-Authors: B. Abdesselam, A. ChakrabartiAbstract:Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a Nested Sequence of projectors. Statistical models associated to such $N^2\times N^2$ matrices for odd $N$ are studied here. Presence of $\frac 12(N+3)(N-1)$ free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are $N$ possible states at each site. The trace of the transfer matrix is shown to depend on $\frac 12(N-1)$ parameters. For order $r$, $N$ eigenvalues consitute the trace and the remaining $(N^r-N)$ eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the $r$-th roots of unity, or lower dimensional multiplets corresponding to factors of the order $r$ when $r$ is not a prime number. The modulus of any eigenvalue is of the form $e^{\mu\theta}$, where $\mu$ is a linear combination of the free parameters, $\theta$ being the spectral parameter. For $r$ a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable $S$-matrices are constructed starting from our braid matrices. Perspectives are discussed.
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Nested Sequence of projectors. II. Multiparameter multistate statistical models, Hamiltonians, S-matrices
Journal of Mathematical Physics, 2006Co-Authors: B. Abdesselam, A. ChakrabartiAbstract:Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a Nested Sequence of projectors. Statistical models associated to such N2×N2 matrices for odd N are studied here. Presence of 12(N+3)(N−1) free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are N possible states at each site. The trace of the transfer matrix is shown to depend on 12(N−1) parameters. For order r, N eigenvalues constitute the trace and the remaining (Nr−N) eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the rth roots of unity, or lower dimensional multiplets corresponding to factors of the order r when r is not a prime number. The modulus of any eigenvalue is of the form eμθ, where μ is a linear combination of the free parameters, θ being the spectral parameter. For r a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians an...
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a Nested Sequence of projectors and corresponding braid matrices r θ 1 odd dimensions
Journal of Mathematical Physics, 2005Co-Authors: A. ChakrabartiAbstract:A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R(θ), θ being the spectral parameter. Only odd values of N are considered here. Our ansatz for the projectors Pα appearing in the spectral decomposition of R(θ) leads to exponentials exp(mαθ) as the coefficient of Pα. The sums and differences of such exponentials on the diagonal and the antidiagonal, respectively, provide the (2N2−1) nonzero elements of R(θ). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 12(N+3)(N−1) free parameters mα. The diagonalizer of R(θ) is presented for all N. Transfer matrices t(θ) and L(θ) operators corresponding to our R(θ) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2 constant N×N matrices. The θ dependence factors out for such combinations. R(θ) is developed in a power series in θ. Th...
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A Nested Sequence of projectors and corresponding braid matrices $\hat R(\theta)$: (1) Odd dimensions
Journal of Mathematical Physics, 2005Co-Authors: A. ChakrabartiAbstract:A basis of $N^2$ projectors, each an ${N^2}\times{N^2}$ matrix with constant elements, is implemented to construct a class of braid matrices $\hat{R}(\theta)$, $\theta$ being the spectral parameter. Only odd values of $N$ are considered here. Our ansatz for the projectors $P_{\alpha}$ appearing in the spectral decomposition of $\hat{R}(\theta)$ leads to exponentials $exp(m_{\alpha}\theta)$ as the coefficient of $P_{\alpha}$. The sums and differences of such exponentials on the diagonal and the antidiagonal respectively provide the $(2N^2 -1)$ nonzero elements of $\hat{R}(\theta)$. One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves ${1/2}(N+3)(N-1)$ free parameters $m_{\alpha}$. The diagonalizer of $\hat{R}(\theta)$ is presented for all $N$. Transfer matrices $t(\theta)$ and $L(\theta)$ operators corresponding to our $\hat{R}(\theta)$ are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of $N^2$ constant $N\times N$ matrices. The $\theta$-dependence factors out for such combinations. $\hat R(\theta)$ is developed in a power series in $\theta$. The basic difference arising for even dimensions is made explicit. Some special features of our $\hat{R}(\theta)$ are discussed in a concluding section.
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a Nested Sequence of projectors and corresponding braid matrices hat r theta 1 odd dimensions
arXiv: Quantum Algebra, 2004Co-Authors: A. ChakrabartiAbstract:A basis of $N^2$ projectors, each an ${N^2}\times{N^2}$ matrix with constant elements, is implemented to construct a class of braid matrices $\hat{R}(\theta)$, $\theta$ being the spectral parameter. Only odd values of $N$ are considered here. Our ansatz for the projectors $P_{\alpha}$ appearing in the spectral decomposition of $\hat{R}(\theta)$ leads to exponentials $exp(m_{\alpha}\theta)$ as the coefficient of $P_{\alpha}$. The sums and differences of such exponentials on the diagonal and the antidiagonal respectively provide the $(2N^2 -1)$ nonzero elements of $\hat{R}(\theta)$. One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves ${1/2}(N+3)(N-1)$ free parameters $m_{\alpha}$. The diagonalizer of $\hat{R}(\theta)$ is presented for all $N$. Transfer matrices $t(\theta)$ and $L(\theta)$ operators corresponding to our $\hat{R}(\theta)$ are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of $N^2$ constant $N\times N$ matrices. The $\theta$-dependence factors out for such combinations. $\hat R(\theta)$ is developed in a power series in $\theta$. The basic difference arising for even dimensions is made explicit. Some special features of our $\hat{R}(\theta)$ are discussed in a concluding section.
Jianmin Wang - One of the best experts on this subject based on the ideXlab platform.
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Combined use of real-time PCR and Nested Sequence-based typing in survey of human Legionella infection.
Epidemiology and infection, 2016Co-Authors: T. Qin, Haijian Zhou, Hongyu Ren, W. Shi, Huiming Jin, X. Jiang, M. Zhou, Jianmin WangAbstract:Legionnaires' disease (LD) is a globally distributed systemic infectious disease. The burden of LD in many regions is still unclear, especially in Asian countries including China. A survey of Legionella infection using real-time PCR and Nested Sequence-based typing (SBT) was performed in two hospitals in Shanghai, China. A total of 265 bronchoalveolar lavage fluid (BALF) specimens were collected from hospital A between January 2012 and December 2013, and 359 sputum specimens were collected from hospital B throughout 2012. A total of 71 specimens were positive for Legionella according to real-time PCR focusing on the 5S rRNA gene. Seventy of these specimens were identified as Legionella pneumophila as a result of real-time PCR amplification of the dotA gene. Results of Nested SBT revealed high genetic polymorphism in these L. pneumophila and ST1 was the predominant Sequence type. These data revealed that the burden of LD in China is much greater than that recognized previously, and real-time PCR may be a suitable monitoring technology for LD in large sample surveys in regions lacking the economic and technical resources to perform other methods, such as urinary antigen tests and culture methods.
