Generalised Kummer constructions and Weil restrictions

We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in Calabi-Yau threefolds defined over Q. The Calabi-Yau manifolds are smooth models of quotients of the Weil restrictions of elliptic curves with CM of class number three.     

Skew-symmetric identities of octonions

Quadratic alternative superalgebras are introduced and their super-identities and central functions on one odd generator are described. As a corollary, all multilinear skew-symmetric identities and central polynomials of octonions are classified.     

Hopf algebras constructed by the FRT-construction

Given any bialgebra A and a braiding product 〈|〉 on A, a bialgebra U_〈|〉 was constructed in [R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum Lie algebra”, Comm. Algebra 19 (1991) 3295-3345], contained in the finite dual of A. This construction generalizes a (not very well known) construction of Fadeev, Reshetikhin and Takhtajan [L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97-110]. In the present paper it is proved that when U_〈|〉 is finite-dimensional (even if A is not), then it is a quasitriangular Hopf algebra.     

Normal toric ideals of low codimension

Every normal toric ideal of codimension two is minimally generated by a Gröbner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal.     

Size of random Galois lattices and number of closed frequent itemsets

Given a sample of binary random vectors with i.i.d. Bernoulli(p) components, that is equal to 1 (resp. 0) with probability p (resp. 1−p), we first establish a formula for the mean of the size of the random Galois lattice built from this sample, and a more complex one for its variance. Then, noticing that closed α-frequent itemsets are in bijection with closed α-winning coalitions, we establish similar formulas for the mean and the variance of the number of closed α-frequent itemsets. This can be interesting for the study of the complexity of some data mining problems such as association rule mining, sequential pattern mining and classification.     

Large deviations for the Boussinesq equations under random influences

A Boussinesq model for the Bénard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier–Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion.     

Homotopic properties of confluent maps in the proper category

We establish some properties of homotopical nature for confluent maps in the proper category. We analyze in this setting the characterization of tree-like continua by J.H. Case and R.E. Chamberlin as well as the theorem by T.B. McLean on the preservation of tree-likeness under confluent maps. We give counterexamples for the corresponding proper analogues and we extend results of several authors in classical continuum theory to non-compact spaces. Finally, we describe the behavior of these maps with respect to the fundamental pro-group, generalizing results of J. Grispolakis and other authors. Two questions of interest are still open (Open Question 15 and Conjecture 24).     

Constrained variational refinement

A non-uniform, variational refinement scheme is presented for computing piecewise linear curves that minimize a certain discrete energy functional subject to convex constraints on the error from interpolation. Optimality conditions are derived for both the fixed and free-knot problems. These conditions are expressed in terms of jumps in certain (discrete) derivatives. A computational algorithm is given that applies to constraints whose boundaries are either piecewise linear or spherical. The results are applied to closed periodic curves, open curves with various boundary conditions, and (approximate) Hermite interpolation.     

基于单体的Q和e值绘制二元共聚双曲线*

采用Matlab中的Plot函数,首次以单体的Q、e值为中间变量,绘制出二元共聚双曲线。将二元共聚物组成曲线F1-f1和F2-f2同时呈现在一个图上,使隐含的第二单体的共聚行为可视化,有助于加深学习者对二元共聚的理解。     

A Supported Bismuth Halide Perovskite Photocatalyst for Selective Aliphatic and Aromatic C–H Bond Activation

Direct selective oxidation of hydrocarbons to oxygenates by O₂ is challenging. Catalysts are limited by the low activity and narrow application scope, and the main focus is on active C−H bonds at benzylic positions. In this work, stable, lead-free, Cs₃Bi₂Br₉ halide perovskites are integrated within the pore channels of mesoporous SBA-15 silica and demonstrate their photocatalytic potentials for C−H bond activation. The composite photocatalysts can effectively oxidize hydrocarbons (C₅ to C₁₆ including aromatic and aliphatic alkanes) with a conversion rate up to 32900 μmol g_cat⁻¹ h⁻¹ and excellent selectivity (>99 %) towards aldehydes and ketones under visible-light irradiation. Isotopic labeling, in situ spectroscopic studies, and DFT calculations reveal that well-dispersed small perovskite nanoparticles (2–5 nm) possess enhanced electron–hole separation and a close contact with hydrocarbons that facilitates C(sp³)−H bond activation by photoinduced charges.