Gorenstein triangular matrix rings and category algebras

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

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

Let \C be a finite projective EI category and $k$ 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

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

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

In this paper, we generalize a recent work of Liu et al. from the open unit ball $\mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ 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 $C^2$ 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

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

The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc $\mathbb D\subset \mathbb C$ 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

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 $\mathbb D\subset \mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma

Julia theory for slice regular functions

Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternionic versions of the Julia lemma, the Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B$ and of the right half-space $\mathbb H^+$. Our quaternionic boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity of quaternions. Together with some explicit examples, it shows that the slice derivative of a slice regular self-mapping of $\mathbb B$ at a boundary fixed point is not necessarily a positive real number, in contrast to that in the complex case, meaning that its commonly believed version turns out to be totally wrong.Comment: To appear in Transactions of the American Mathematical Society. arXiv admin note: substantial text overlap with arXiv:1412.420