On lattices embeddable into subsemigroup lattices. V. Trees

We prove that the class of finite lattices embeddable into the subsemilattice lattices of semilattices which are (n-ary) trees can be axiomatized by identities within the class of finite lattices, whence it forms a pseudovariety.     

Notes on Sublattice-Lattices

We prove that directly reducible lattices and selfdual subdirectly irreducible lattices of locally finite length are determined by their sublattice-lattices. As a corollary we obtain that splitting varieties are closed under the isomorphism of sublattice-lattices iff they are selfdual. A class of selfdual non-closed varieties is given too.     

Relations between site percolation thresholds

By decomposing certain lattices into two sublattices, and examining at percolation threshold the structure of their infinite clusters, an approximate relation between p _c ⁰ , of the original lattice and p _c ¹ , of the sublattice is established: p _c ⁰ (p _c ¹ )^1/2. It is conjectured that an inequality always holds: p _c ⁰ (p _c ¹ )^1/2, and heuristic arguments are given to substantiate it. By similar considerations good estimates forp _c of certain correlated percolation problems are also obtained.     

Neutron scattering by anharmonic crystals and the effect of sublattice displacements

A theory has been given for the scattering of neutrons by anharmonic crystals, for which terms of the typeV ⁽³⁾ (k _1j1; —k _1j1;o _j) which contribute to the sublattice displacements are not neglected. Using the standard perturbation theory in the interaction picture or Green’s function method, an expression has been derived for the differential scattering cross-section which brings in the shift and the width of the phonons in one-phonon energy exchange processes. It is shown that the sublattice displacements will modify the phase factor arising from the scattering by any atom in the unit cell, and the Debye-Waller factor also gets altered both by the sublattice displacements as well as by higher order terms arising from anharmonicity. It is shown that the differential scattering cross-section contains a term linearly depending on the third order anharmonicity coefficientV ⁽³⁾ (k _1j1;k _2j2;k _3j3) and neutron scattering by crystals should provide a useful method for evaluating the third order anharmonicity coefficients.     

Properties of Standard n-Ideals of a Lattice

An n-ideal of a lattice L is a convex sublattice containing a fixed element n L and it is called standard if it is a standard element of the lattice of n-ideals I_n(L). In this paper we have shown that, for a neutral element n of a lattice L, the principal n-ideal a_n of a lattice L is a standard n-ideal if and only if a n is standard and a n is dual standard. We have also shown that if n is a neutral element and (n] and [n) are relatively complemented, then every homomorphism n-kernels of L is a standard n-ideal and every standard n-ideal is the n-kernel of precisely one congruence relation. Finally, we have shown that, for a relatively complemented lattice L with 0 and 1, C(L) is a Boolean algebra if and only if every standard n-ideal of L is a principal n-ideal.AMS Subject Classification (2000) 06B10 06B99 06C15     

双循环半群上的同余关系

从双循环半群的同余关系出发,讨论了幂等元所在的同余类,证明了双循环半群上的一类同余ρd(d∈N)与其逆子半群之间的相互唯一确定关系,并对这种同余做成的集合进行了刻画,证明了这种同余做成的格与自然数集在某种偏序下做成的格同构,得到了一些有意义的结果.     

Ion beam mixing induced by electronic energy deposition in Fe-Si multilayers

Abstract

Spectroscopic properties of Ce³⁺ ions in GdAlO₃ crystal are presented. At least three Ce³⁺ nonequivalent centres (multisites) are present in this crystal. Energy transfer from the Ce³⁺ main in the UV emitting centres to the Ce³⁺ green emitting centres is observed. Ce³⁺ fluorescence decays are either fast (1.5–20 ns) or slower due to complicated processes of energy transfer and migration (Ce³⁺)_i → (Gd³⁺)_n-steps → (Ce³⁺)_j (energy transfer through Gd³⁺ sublattice).     

Anionic Sublattices in Halide Solid Electrolytes: A Case Study with the High-Pressure Phase of Li3ScCl6

The Li₃MX₆ compounds (M=Sc, Y, In; X=Cl, Br) are known as promising ionic conductors due to their compatibility with typical metal oxide cathode materials. In this study, we have successfully synthesized γ-Li₃ScCl₆ using high pressure for the first time in this family. Structural analysis revealed that the high-pressure polymorph crystallizes in the polar and chiral space group P6₃mc with hexagonal close-packing (hcp) of anions, unlike the ambient-pressure α-Li₃ScCl₆ and its spinel analog with cubic closed packing (ccp) of anions. Investigation of the known Li₃MX₆ family further revealed that the cation/anion radius ratio, r_M/r_X, is the factor that determines which anion sublattice is formed and that in γ-Li₃ScCl₆, the difference in compressibility between Sc and Cl exceeds the ccp r_M/r_X threshold under pressure, enabling the ccp-to-hcp conversion. Electrochemical tests of γ-Li₃ScCl₆ demonstrate improved electrochemical reduction stability. These findings open up new avenues and design principles for lithium solid electrolytes, enabling routes for materials exploration and tuning electrochemical stability without compositional changes or the use of coatings.     

纯正半群上的强同余及其格

给出了纯正半群S的强同余格上同余T的一些判别性质,证明了S上所有基础强同余所构成的集合FCP（S）是CP（S）的完备子格,最后讨论了由纯正半群的正规子半群决定的交完备子格的结构及由“求核”运算确定的（交完备格）同余K的若干性质,还顺带讨论了群同余格.     

Index of lattices and Hilbert polynomials

An upper bound for the index of a sublattice, which arises in relation to various versions of zero lemmas in the theory of linear forms in logarithms of algebraic numbers, in terms of the Hilbert polynomial is found. Simultaneously, a lower bound for the values of this polynomial is obtained.