Dual lattice

Dual lattice

A dual lattice is a lattice that is defined for both real and complex lattices, and is created by taking the points at the center of the elementary squares (plaquettes) of the original lattice. It is a separate lattice that is related to the original lattice, and is also a lattice in its own right.From: MIMO Processing for 4G and Beyond [2019], New construction approaches of uninorms on bounded lattices [2021], Statistical Mechanics [2021]

The Ising Model

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Published in Teunis C. Dorlas, Statistical Mechanics, 2021

Teunis C. Dorlas

We now present a proof, due essentially to Peierls (1936) but corrected by Griffiths (1964) and Dobrushin (1965), that the nearest-neighbour Ising model in dimensions greater than 1 has a phase transition. This is a very powerful argument and has been extended to a variety of situations, including the case of more general types of interactions which do not have a spin-up vs. spin-down symmetry. (This is called Pirogov-Sinai theory (Sinai (1982).) For simplicity we consider the two-dimensional case. The main idea is to separate regions of +-spins from regions of —spins by lines between points of the dual lattice. For the square lattice, the dual lattice is also a square lattice consisting of points at the centre of elementary squares (plaquettes) of the original lattice (figure 28-6).

New construction approaches of uninorms on bounded lattices

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Published in International Journal of General Systems, 2021

Gül Deniz Çaylı

A lattice (Birkhoff 1967) is a nonempty set L equipped with a partial order ≤ such that any two elements x and y have a greatest lower bound (called meet or infimum), denoted by , as well as a smallest upper bound (called join or supremum), denoted by . For , the symbol a<b means that and . The elements a and b in L are comparable if or b<a. Otherwise, a and b are incomparable, in this case we use the notation . For the fixed element a in L, the set of all with will be denoted by i.e. The transpose of a partial order ≤ of a lattice L, denoted by ≥, i.e. if and only if , is also a partial order on L and the meet and the join with respect to ≥ are ∨ and ∧, respectively. That is to say, is also a lattice, called the dual lattice of .

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The Ising Model

New construction approaches of uninorms on bounded lattices