Bi-Frobenius algebra structure on quantum complete intersections

This paper is to look for bi-Frobenius algebra structures on quantum complete intersections. We find a class of comultiplications, such that if $\sqrt{-1}\in k$, then a quantum complete intersection becomes a bi-Frobenius algebra with comultiplication of this form if and only if all the parameters $q_{ij} = \pm 1$. Also, it is proved that if $\sqrt{-1}\in k$ then a quantum exterior algebra in two variables admits a bi-Frobenius algebra structure if and only if the parameter $q = \pm 1$. While if $\sqrt{-1}\notin k$, then the exterior algebra with two variables admits no bi-Frobenius algebra structures. Since a quantum complete intersection over a field of characteristic zero admits no bialgebra structures, this gives a class of examples of bi-Frobenius algebras which are not bialgebras (and hence not Hopf algebras). On the other hand, a quantum exterior algebra admits a bialgebra structure if and only if ${\rm char} \ k = 2$. In commutative case, other two comultiplications on complete intersection rings are given, such that they admit non-isomorphic bi-Frobenius algebra structures

Bi-Frobenius quantum complete intersections with permutation antipodes

Quantum complete intersections $A= A({\bf q, a})$ are Frobenius algebras, but in the most cases they can not become Hopf algebras. This paper aims to find bi-Frobenius algebra structures on $A$. A key step is the construction of comultiplication, such that $A$ becomes a bi-Frobenius algebra. By introducing compatible permutation and permutation antipode, a necessary and sufficient condition is found, such that $A$ admits a bi-Frobenius algebra structure with permutation antipode; and if this is the case, then a concrete construction is explicitly given. Using this, intrinsic conditions only involving the structure coefficients $({\bf q, a})$ of $A$ are obtained, for $A$ admitting a bi-Frobenius algebra structure with permutation antipode. When $A$ is symmetric, $A$ admits a bi-Frobenius algebra structure with permutation antipode if and only if there exists a compatible permutation $\pi$ with $A$ such that $\pi^2 = {\rm Id}$

Mesopic visual quality after three kinds of aspheric acrylic monofocal intraocular lenses

AIM:To evaluate best corrected visual acuity(BCVA)and contrast sensitivity(CS)under mesopic condition with no glare in patients following implantation of three different kinds of aspheric acrylic monofocal intraocular lens(IOLs).<p>METHODS: Seventy-seven cases(90 eyes)of age-related cataract patients were selected, who were undergone phacoemulsification and intraocular lens(IOL)implantation in our hospital during December 2011 to November 2012. Preoperatively, the patients were randomly divided into three groups: 30 eyes(25 cases)were implanted with hydrophobic yellow-tinted acrylic(HOYA)IOLs in group 1; 30 eyes(28 cases)with hydrophilic acrylic IOLs(Rayner)in group 2; 30 eyes(24 cases)with hydrophilic acrylic surface heparin processing IOLs(XO)in group 3. All eyes were evaluated at 1 month and 3 months postoperatively. The BCVA and CS under mesopic condition without glare were measured and underwent statistical analysis.<p>RESULTS: There was neither statistically significant difference in the BCVA(<i>P</i>>0.05), nor statistically significant difference in CS results(after standardization of contrast sensitivity value: lgCS)(<i>P</i>>0.05)between groups under mesopic condition, but the lgCS of the HOYA group decreased slightly. <p>CONCLUSION: There were not marked differences of BCVA and CS between groups of patients at the follow-up intervals of 1 month and 3 months under mesopic condition. The blue-filter type artificial lens may protect the retina, thus, it is advisable to implant yellow-tinted filter blue artificial lens, especially for patients who are children or young persons

Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model

We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be $1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]$, which diverges around the critical point $J_z=(1/2)^+$.Comment: 7 pages, 6 figure

Spatial Heterogeneity in Light Supply Affects Intraspecific Competition of a Stoloniferous Clonal Plant

Spatial heterogeneity in light supply is common in nature. Many studies have examined the effects of heterogeneous light supply on growth, morphology, physiology and biomass allocation of clonal plants, but few have tested those effects on intraspecific competition. In a greenhouse experiment, we grew one (no competition) or nine ramets (with intraspecific competition) of a stoloniferous clonal plant, Duchesnea indica, in three homogeneous light conditions (high, medium and low light intensity) and two heterogeneous ones differing in patch size (large and small patch treatments). The total light in the two heterogeneous treatments was the same as that in the homogeneous medium light treatment. Both decreasing light intensity and intraspecific competition significantly decreased the growth (biomass, number of ramets and total stolon length) of D. indica. As compared with the homogeneous medium light treatment, the large patch treatment significantly increased the growth of D. indica without intraspecific competition. However, the growth of D. indica with competition did not differ among the homogeneous medium light, the large and the small patch treatments. Consequently, light heterogeneity significantly increased intraspecific competition intensity, as measured by the decreased log response ratio. These results suggest that spatial heterogeneity in light supply can alter intraspecific interactions of clonal plants

Coding Properties of Mouse Retinal Ganglion Cells with Dual-Peak Patterns with Respect to Stimulus Intervals

How visual information is encoded in spikes of retinal ganglion cells (RGCs) is essential in visual neuroscience. In the present study, we investigated the coding properties of mouse RGCs with dual-peak patterns with respect to visual stimulus intervals. We first analyzed the response properties, and observed that the latencies and spike counts of the two response peaks in the dual-peak pattern exhibited systematic changes with the preceding light-OFF interval. We then applied linear discriminant analysis (LDA) to assess the relative contributions of response characteristics of both peaks in information coding regarding the preceding stimulus interval. It was found that for each peak, the discrimination results were far better than chance level based on either latency or spike count, and were further improved by using the combination of the two parameters. Furthermore, the best discrimination results were obtained when latencies and spike counts of both peaks were considered in combination. In addition, the correct rate for stimulation discrimination was higher when RGC population activity was considered as compare to single neuron’s activity, and the correct rate was increased with the group size. These results suggest that rate coding, temporal coding, and population coding are all involved in encoding the different stimulus-interval patterns, and the two response peaks in the dual-peak pattern carry complementary information about stimulus interval