85,579 research outputs found

    Gorenstein triangular matrix rings and category algebras

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    We give conditions on when a triangular matrix ring is Gorenstein of a given selfinjective dimension. We apply the result to the category algebra of a finite EI category. In particular, we prove that for a finite EI category, its category algebra is 1-Gorenstein if and only if the given category is free and projective.Comment: 17 page

    The MCM-approximation of the trivial module over a category algebra

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    For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is Gorenstein-projective and is a maximal Cohen-Macaulay approximation of the trivial module. We give conditions on when the trivial module is Gorenstein-projective

    The spectrum of the singularity category of a category algebra

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    Let \C be a finite projective EI category and kk be a field. The singularity category of the category algebra k\C is a tensor triangulated category. We compute its spectrum in the sense of Balmer.Comment: 7 page

    The composite theory as the explanation of Haldane's rule should be abandoned

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    In 1922, JBS Haldane discovered an intriguing bias of postzygotic isolation during early speciation: the heterogametic sex of F1 hybrids between closely related species or subspecies is more susceptible to sterility or inviability than the homogametic sex. This phenomenon, now known as Haldane's rule, has been repeatedly confirmed across broad taxa in diecious animals and plants. Currently, the dominant view in the field of speciation genetics believes that Haldane's rule for sterility, inviability, male heterogamety and female heterogametic belongs to different entities; and Haldane's rule in these subdivisions has different causes, which operate coincidentally and/or collectively resulting in this striking bias against the heterogametic sex in hybridization. This view, known as the composite theory, was developed after many unsuccessful quests in searching for a unitary genetic mechanism. The composite theory has multiple sub-theories. The dominance theory and the faster male theory are the major ones. In this note, I challenge the composite theory and its scientific validity. By declaring Haldane's rule as a composite phenomenon caused by multiple mechanisms coincidentally/collectively, the composite theory becomes a self-fulfilling prophecy and untestable. I believe that the composite theory is an ad hoc hypothesis that lacks falsifiability, refutability and testability that a scientific theory requires. It is my belief that the composite theory does not provide meaningful insights for the study of speciation and should be abandoned.Comment: 14 pages, 25 references, no figure/tabl

    Magnetostatics of Magnetic Skyrmion Crystals

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    Magnetic skyrmion crystals are topological magnetic textures arising in the chiral ferromagnetic materials with Dzyaloshinskii-Moriya interaction. The magnetostatic fields generated by magnetic skyrmion crystals are first studied by micromagnetic simulations. For N\'eel-type skyrmion crystals, the fields will vanish on one side of the crystal plane, which depend on the helicity; while for Bloch-type skyrmion crystals, the fields will distribute over both sides, and are identical for the two helicities. These features and the symmetry relations of the magetostatic fields are understood from the magnetic scalar potential and magnetic vector potential of the hybridized triple-Q state. The possibility to construct magnetostatic field at nanoscale by stacking chiral ferromagnetic layers with magnetic skyrmion crystals is also discussed, which may have potential applications to trap and manipulate neutral atoms with magnetic moments.Comment: 5 pages, 2 figure

    Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains

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    In this paper, we generalize a recent work of Liu et al. from the open unit ball Bn\mathbb B^n to more general bounded strongly pseudoconvex domains with C2C^2 boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Carath\'eodory metric near the boundary of C2C^2 domains and the well-known Graham's estimate on the boundary behavior of the Carath\'eodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.Comment: Accepted by CAOT for publicatio

    Control of Ultracold Atoms with a Chiral Ferromagnetic Film

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    We show that the magnetic field produced by a chiral ferromagnetic film can be applied to control ultracold atoms. The film will act as a magnetic mirror or a reflection grating for ultracold atoms when it is in the helical phase or the skyrmion crystal phase respectively. By applying a bias magnetic field and a time-dependent magnetic field, one-dimensional or two-dimensional magnetic lattices including honeycomb, Kagome, triangular types can be created to trap the ultracold atoms. We have also discussed the trapping height, potential barrier, trapping frequency, and Majorana loss rate for each lattice. Our results suggest that the chiral ferromagnetic film can be a platform to develop artificial quantum systems with ultracold atoms based on modern spintronics technologies.Comment: 9 pages, 6 figure

    The Growth and Distortion Theorems for Slice Monogenic Functions

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    The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc DβŠ‚C\mathbb D\subset \mathbb C in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.Comment: 24 pages; Accepted by Pacific Journal of Mathematics for publicatio

    Extremal functions of boundary Schwarz lemma

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    In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Carath\'eodory theorem for univalent holomorphic self-mappings of the open unit disk DβŠ‚C\mathbb D\subset \mathbb C. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma

    On the curvature estimates for Hessian equations

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    The curvature estimates of kk curvature equations for general right hand side is a longstanding problem. In this paper, we totally solve the nβˆ’1n-1 case and we also discuss some applications for our estimate
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