Solving the triangular Ising antiferromagnet by simple mean field

Few years ago, application of the mean field Bethe scheme on a given system was shown to produce a systematic change of the system intrinsic symmetry. For instance, once applied on a ferromagnet, individual spins are no more equivalent. Accordingly a new loopwise mean field theory was designed to both go beyond the one site Weiss approach and yet preserve the initial Hamitonian symmetry. This loopwise scheme is applied here to solve the triangular antiferromagnetic Ising model. It is found to yield Wannier's exact result of no ordering at non-zero temperature. No adjustable parameter is used. Simultaneously a non-zero critical temperature is obtained for the triangular Ising ferromagnet. This simple mean field scheme opens a new way to tackle random systems. Received 14 November 2001 / Received in final form 22 March 2002 Published online 19 July 2002     

Torsion divisors of plane curves with maximal flexes and Zariski pairs

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves that include a special smooth component, via the torsion properties induced by the divisors in the special curve associated to the remaining components, which is an arithmetic property. When this special curve has maximal flexes, there is a natural isomorphism between its Jacobian variety and the degree zero part of its Picard group. In this paper, we consider curve arrangements that contain a special smooth component with a maximal flex and exploit these properties to obtain Zariski tuples, which show the interplay between topology, geometry, and arithmetic.      

Partial symmetry, reflection monoids and Coxeter groups

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.     

Convergence spaces and diagonal conditions

Certain diagonal axioms due to Kowalsky, Cook and Fischer are studied and compared.     

对角线模n王后覆盖问题

所谓模ｎ王后是指在ｎ×ｎ国际象棋棋盘上不仅可以横走，竖走，而且可以沿对角线（没有折断的和折断了的）方向行走的棋子。我们记可以覆盖整个ｎ×ｎ棋盘且位于主对角线上的模ｎ王后的最小个数为，且对于所有．在本文中，我们证明了：当ｎ≡２（ｍｏｄ４）时，Ｄｉａｇ（ｎ）＝ｎ／２；当ｎ≡０（ｍｏｄ４）时，Ｄｉａｇ（ｎ）＝３ｎ／４－１；当ｎ为奇数时，Ｄｉａｇ（ｎ）＝ｎ－Ｒ（ｎ）．这里，ｎ＞２。     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

关于高斯消去法的主元估计

本文证明了对任意一个给定的6阶实阵 A=(a_(ij)),若其中|a_(ij)|≤1,则有 P_6<6.7883.对于一般 n 阶矩阵,本文给出了估计 P_n 的一个改进方法.     

Structural arrangements of the ternary metal boride carbide compounds MB2C4 (M=Mg, Ca, La and Ce) from first-principles theory

The structural arrangements of the ternary metal borocarbides MB₂C₄ (M=Mg, Ca; La and Ce) are investigated using density-functional theory (DFT) calculations within the generalized gradient approximation (GGA). Results indicate that these compounds adopt a layered structure consisting of graphite-like B₂C₄ layers alternating with metal sheets. Within the hexagonal layers, the coloring with the -C-C-C-B-C-B- sequence is energetically more stable than that with the -C-C-C-C-B-B- one. The electronic structures of these compounds, mainly determined by the B₂C₄ sheets, can be rationalized with the simple valence electron distribution M²⁺[B₂C₄]²⁻xe⁻, with the metals essentially acting as two-electron donors with respect to the boron-carbon network, the other x electrons remaining in the relatively narrow d and/or f bands of the metals. Accordingly, MB₂C₄ are narrow band-gap semiconductors (ΔE≈0.2-0.4 eV) with M=Mg and Ca. On the other hand, with M=La and Ce, the compounds are conducting with a relatively high density of states at the Fermi level predominantly metal in character with substantial B/Cπ* antibonding state admixture.     

On a class of diagonal equations over finite fields

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\cdots +x_n^{2^m}=0$ over a finite field of characteristic $p\equiv \pm 3(\mathrm{mod}8)$. All of the evaluations are effected in terms of parameters occurring in quadratic partitions of some powers of p.     

Convex combinations of continuous t-norms with the same diagonal function

The problem of whether a non-trivial convex combination of two continuous t-norms with the same diagonal function can be a t-norm is studied. It is shown that in both cases–of two nilpotent and of two strict t-norms–a non-trivial convex combination of t-norms with common diagonal function is associative only if the two t-norms involved coincide. For general continuous t-norms a similar result follows. An example of a convex class of non-continuous t-norms is also included.