fibre [25] communities, but has attracted only limited interest for PhC slabs [26, 27, 28]. The **kagome** **lattice** is a depleted version of the triangular **lattice**, consist- ing of a sparse super-**lattice** imposed on the standard triangular **lattice**. As shown in Fig. 1a, along every second diagonal line of a triangular **lattice** each sec- ond **lattice** site is removed to form the **kagome** **lattice**. Intuitively, the **kagome** **lattice** is better represented by a tight-binding model, compared to the triangu- lar **lattice** which is better approximated by a “free photon” (or free electron for electronic crystals) ap- proximation. This leads to larger photonic band gaps and flatter bands in the **kagome** **lattice** [27, 29]. If we consider a **lattice** of air holes in a 220 nm-thick silicon slab, the material system used for most slow light PhC work, the **kagome** **lattice** supports multiple optical bandgaps for TE polarized light, as shown in Fig. 1b. Furthermore, the sparse super-**lattice** results in a “compressed’ bandstructure, with each mode oc- cupying a narrower frequency range compared to the triangular **lattice**. For example, the first bandgap of the **kagome** **lattice** starts at a normalized frequency of 0.15, while that of a triangular **lattice** with the same **lattice** parameters starts at a normalized frequency of 0.21, see Fig. 1. The compressed bandstructure thus exhibits a reduced slope for all optical modes, result- ing in an increased group index, since the group index is defined as:

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In this work, we will present calculations of the specific heat at low T by the exact approach for small clus- ters. The exact approach is very powerful and extensively applied for study at zero T [15]-[17], although the size is severely limited. There are few researches on the exact approach to calculations of the specific heat at fi- nite T . Only some results on L = 18 or L = 24 clusters are found in [12] [13]. Here L is the total number of sites. By these previous works, it is understood that ordinary calculations on L = 27 or larger lattices are quite difficult because the system on the **kagome** **lattice** has the complicated structure at low T . We point out the Chebyshev polynomial expansion (CPE) as one method of the exact approaches at finite T [18] [19]. Another method is the finite temperature Lanczos method (FTLM) [20] [21].While the polymial expansion method is well mathematically understood, in the FTLM we should be careful to control numerical errors. Due to this weakness there are a fewer applications of the FTLM, compared with the CPE [18], although both methods give us the same calculation costs.

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Rb + . These compositions represent new extremes of both A/B′ cation size ratio and B′ absolute cation size within this family. By this size-directed crystal engineering, it was hoped that a perfect **kagome** **lattice** might be retained at low temperatures, thus prompting retention of a magnetically-frustrated ground state. However, it is found that Rb 2 TiCu 3 F 12 adopts a highly distorted, triclinic structural variant

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FIG. 4: (a) Reciprocal space of volborthite: the structural unit cell is indicated by the rectangle, and the extended Bril- louin zone of the **kagome** **lattice** by the dotted hexagon. Sym- bols represent Bragg peaks of the orders listed in the text. (b) The experimental S(Q, ω) measured at 0.05 K compared with the powder averaged S(Q, ω) derived from our empirical spin wave model. The dashed line in the left panel indicates the (Q, ω) window of the experiment.

This is however where the similarity with the triangular **lattice** ends. Because adjacent triangles on the **kagome** **lattice** only share a single spin, their local planes do not need to co- incide; rather, they may intersect on a line determined by the shared spin. Following a similar constructive approach for building up a ground state as for the triangular case, starting from a single triangle, each consecutive adjacent triangle that is added allows for the choice of one an- gle. However, as triangles start to close loops around the hexagonal plaquettes, their planes are not necessarily compatible. Hence, not all of the aforementioned angles can in fact be chosen freely, but are subject to intricate constraints themselves. The exact degeneracy of the ground state is not known. Still, nonplanar solutions do exist for which the local plane of the triangles is different and becomes independent for triangles at large distances. This is to be contrasted with the triangular **lattice**, where the constraint implies a coplanar order where all triangles share the same plane, even at (quasi-)long range, thus breaking the O ( 3 ) spin-rotational sym- metry.

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which can be accounted for by series expansion for the breathing **kagome** model. The determination of the ra- tio of the interactions within the breathing **kagome** lat- tice, J O /J M = 0.55(4), will allow for further theoreti- cal investigations into the effects of the breathing of the S = 1/2 KAFM **lattice** that are specific to DQVOF. It is hoped that this in turn will shed light on our experimen- tal observation of a gapless spin liquid ground state. The breathing of the **kagome** **lattice** appears to be an inter- esting tuning parameter that may be used to control the spin liquid physics and one that warrants further explo- ration. Investigating new anisotropic **kagome** compounds with different layered structures is also a promising ex- perimental route to furthering our understanding of the rich physics of the S = 1/2 KAFM [35].

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– the former corresponds to the doubly degenerate ○ and ⬠s and the latter the singular degenerate □s and △s. Example orderings are shown in Fig. 5. It should be noted that the spins are not constrained to the same plane as the **kagome** **lattice** because only the relative angle between spins is important to achieve an energy minimum. Stability of the nuclear moment orderings on **kagome**. To test the stability of the mean field order- ings found above against fluctuations, we investigate how spin-waves modify or destroy these orderings. To do this we select a basis for the directions in spin-space such that the spin on the first site of the Fourier-space unit cell is parallel to the spin z-axis, I Q z 1 ( ) n . Within this basis, because the regular periodic orderings are co-planar, we can effectively rotate the remaining spins in the unit cell about the y-axis to be parallel with the first spin by defining I Q ′ ( ) = R ( ) ( ) θ I Q

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Geometrically frustrated magnets are crystalline solids in which the arrangement of spin moments prevents long range magnetic order that typically prevails in magnetic materials upon cooling. Instead, the frustration that can be imposed on the magnetic interactions within a material as a result of its structure leads to a wealth of unusual magnetic behaviours [1]. An archetypal geometrically frustrated structure is the **kagome** **lattice**, formed from a two-dimensional corner-sharing triangular network of magnetic ions. When the **kagome** **lattice** is dressed with a set of antiferromagnetically interacting S = ½ species an exciting possibility presents itself – the realization of a quantum spin liquid (QSL) [2]. The QSL is a long-sought after magnetic phase of matter that was originally proposed as an alternative to the ordered Néel ground state of an antiferromagnet [3]. It has many intriguing properties, for instance, despite being a system of highly entangled, correlated spins it can evade a conventional symmetry breaking magnetic phase transition even at T = 0 K and instead may exhibit a topological order that could have important consequences for quantum information technology [4]. Furthermore, it can support the existence of unusual magnetic excitations known as spinons, which are relatively well understood in one-dimensional magnets but far less so for magnetic systems of higher dimensionality [5]. In two- dimensions the S = ½ **kagome** antiferromagnet is the prime candidate to host a QSL state [6].

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The reason that there are two distinct transitions in the regular lattices is because sets of braces can form redundancies fairly easily. The coupling of a large number of floppy modes together is not sufficient to completely rigidify the system. In the regular square **lattice**, this coupling does not even create a single large rigid component, though it does for the regular **kagome** **lattice**. Despite the fact that most of the floppy modes become coupled together, many floppy modes remain “isolated”—that is, decoupled from all other modes—and added braces tend to create redundancies rather than remove degrees of freedom. In the bipartite graph representation of the regular square **lattice**, these floppy modes correspond indeed to isolated vertices, and adding enough braces to the system to couple them all to the giant floppy mode yields a coupon-collector problem [42]. The rigidity transition in the regular **kagome** **lattice** proceeds through a similar, but more complicated, process. Thus, in both cases, there is a separate transition to rigidity of the system which occurs much later: when p ∼ ln L/L.

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Abstract: We present a systematic experimental study on the optical properties of plasmonic crystals (PlC) with hexagonal symmetry. We compare the dispersion and avoided crossings of surface plasmon modes around the Γ-point of Au-metal hole arrays with a hexagonal, honeycomb and **kagome** **lattice**. Symmetry arguments and group theory are used to label the six modes and understand their radiative and dispersive properties. Plasmon-plasmon interaction are accurately described by a coupled mode model, that contains e ff ective scattering amplitudes of surface plasmons on a **lattice** of air holes under 60 ◦ , 120 ◦ , and 180 ◦ . We determine these rates in the experiment and find that they are dominated by the hole-density and not on the complexity of the unit-cell. Our analysis shows that the observed angle-dependent scattering can be explained by a single-hole model based on electric and magnetic dipoles.

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COMPASS was parameterized using ab initio calculations and empirical data, which entailed adding new molecular classes to PCFF. In addition, non-bond parameters were re-parameterized whereby the electrostatic and van der Waals terms combine quantum mechanical calculations and fitting to experimental condensed phase properties of liquids and crystals. COMPASS II extends this to include parameters specific to polymers and drug-like molecules, hence its wide adoption in the pharmaceutical industry. However, given the current availability of DFD-DISP options, is it time to re-think this default position of using classical methods to calculate **lattice** energies? Attempting to answer this question generated the research we present in this paper, in which we assess whether state-of-the-art DFT-DISP methods are comparably predictive to, or better than classical methods We use two different simulation methods to calculate the **lattice** energies of the organic molecular crystals, namely classical molecular mechanics using a force field, and ab initio density functional theory (DFT) plus dispersion (DFT-DISP). The molecular mechanics was performed using the Forcite module of Materials Studio 62 , with the

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a **lattice** include a contribution from the self-energy of the static source which is unphysical. Thus only energy differ- ences have a physical significance and hence we concentrate especially on the binding energies—the difference of BB en- ergy from twice the B energy. In the special case of R ⫽ 0, we show the actual **lattice** energy values in Table II to allow us to discuss the extrapolation to large t needed to extract the ground state. Other results are given in Table III for the case of R ⫽ 3 and in Figs. 6–9. We show the results from both 12 3 and 16 3 spatial lattices with the same parameters in order to explore finite size effects. Within errors, we do not see sig- nificant differences in the results between spatial sizes of L ⫽ 12 and 16, which is not unexpected since a study of the B meson using L ⫽ 8 and 12 found 关 16 兴 agreement for the en- ergies of the ground state mesons and a relatively localized Bethe-Saltpeter wave function.

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In spite of the heavy work duty as president of the INFN, he continued to work on **lattice** QCD. A very remarkable paper of 2007 is QCD with light Wilson quarks on fine lattices: first experiences and physics results [60]. In this paper the universality of the continuum limit and the applicability of renormalized perturbation theory are tested in the SU(2) **lattice** gauge theory by computing two differ- ent non-perturbatively defined running couplings over a large range of energies [61]. The **lattice** data (which were generated on the powerful APE computers at Rome II and DESY) are extrapolated to the continuum limit by simulating sequences of lattices with decreasing spacings. The results confirmed the expected universality at all energies to a precision of a few percent. The author found, however, that perturbation theory must be used with care when matching different renormalized couplings at high energies.

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Theorem 2.1 gives us a direct correspondence between a class of plane partitions of an integer ν and a class of partitions with n copies of n of the integer ν. Also, since partitions with n copies of n have a **lattice** path representation. So, we obtain a **lattice** path representation for a class of restricted plane partitions. But there is another class of plane partitions for which **lattice** path representation is still to be found.

Being the extremal case, totally local classes have a series of specific properties. In particular we note that for every non-negative integer n the lattices of all n-multiply local formations, of all n-multiply local hered- itary formations, of all n-multiply local normally hereditary formations etc. are modular but all of them are not distributive even in the class of all soluble groups S (see [13, Chapter 2] and [14, Chapter 4]). Moreover as it was mentioned in [15] (see also [13, 14]) for every two non-negative integers n and m the systems of all laws of the lattices of all n-multiply local and all m-multiply local formations coincide. On the other hand the **lattice** of all soluble totally local formations is distributive [13] but we know nothing about it in the general case (see [14, Question 4.2.14] and [16, Question 14.80]).

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a polynomial time when d is less than N 0.25 for N = pq and q < p < 2q. Using **lattice** reduction approach based on the Coppersmith techniques [7] for finding small solutions of modular bivariate integer polynomial equations, D. Boneh and G. Durfee [4] improved the wiener result from N 0.25 to N 0.292 in 2000 and J. Bl¨ omer and A. May [5] has given an RSA attack for d less than N 0.29 in 2001, that requires lattices of dimension smaller than the approach by Boneh and Durfee. In 2006, E. Jochemsz and A. May [10], described a strategy for finding small modular and integer roots of multivariate polynomial using **lattice**-based Coppersmith tech- niques and by implementing this strategy they gave a new attack on an RSA variant called common prime RSA. In our paper [2], first we described an attack on RSA when ϕ(N ) has small multiplicative inverse k of modulo e, the public encryption exponent by using **lattice** and sublattice based techniques. Let N = pq, q < p < 2q, p − q = N β and e = N α > p + q. As (e, ϕ(N )) = 1, there exist unique r, s such that (p − 1)r ≡

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a i ). A partially ordered set (X , ≤) is called a meet semi **lattice** if a ∧ b (=glb{a, b}) exists for all a and b ∈ X . (X , ≤) is called a join semi **lattice** if a ∨ b (=lub{a, b}) exists for all a and b ∈ X . A poset (X , ≤) is called a **lattice** if it is both a meet and join semi **lattice**. Equivalently, **lattice** can also be defined as an algebraic system (X , ∧, ∨), where ∧ and ∨ are binary operations which are associative, commutative and idempotent and satisfying the absorption laws, namely a ∧ (a ∨ b) = a = a ∨ (a ∧ b) for all a, b ∈ X ; in this case the partial order ≤ on X is such that a ∧ b and a ∨ b are respectively the glb and lub of {a, b}. The algebraic operations ∧ and ∨ and the partial order ≤ are related by

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Abstract. **Lattice** theory play an important role in mathematics as well as in other disciplines such as computer science, engineering, cryptography, etc. In this paper, we introduce the concept of generalized soft **lattice** (gs **lattice**) and investigate some of its fundamental properties. Further we define soft valuation on a generalized soft **lattice** (gs **lattice**) and study its major properties. In the last section we discuss the notion of soft distance function and express it in terms of soft valuation. Here we discuss the notions of soft pseudo metric **lattice** and soft metric **lattice**.

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Concept **lattice** is an efficient tool for knowledge representation and knowledge discovery and is applied to many fields successfully. However, in many real life applications, the problem under investigation cannot be described by formal concepts. Such concepts are called the non-definable concepts. The hierarchical structure of formal concept (called concept **lattice**) represents a structural information which obtained automatically from the input data table. We deal with the problem in which how further additional information be supplied to utilize the basic object attribute data table. In this paper , we provide rough concept **lattice** to incorporate the rough set into the concept **lattice** by using equivalence relation. Some results are established to illustrate the paper.

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