# The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks

^{*}

## 1. Introduction

^{d}, with respect to periodic first order flexes is determined by a finite matrix, the associated periodic rigidity matrix. For essentially generic frameworks of this type in two dimensions, there is a deeper combinatorial characterisation, which is a counterpart of Laman's characterisation of the infinitesimal rigidity of generic placements of finite graphs in the plane. See Ross [1]. For related results and characterisations of other forms of periodic infinitesimal rigidity, see Borcea and Streinu [2,3], Connelly, Shen and Smith [4], Malestein and Theran [5], Owen and Power [6], Power [7,8] and Ross, Schulze and Whiteley [9].

^{d}, or as an equivalent subset of [0, 1)

^{d}, which arises from a choice of translation group . This set of multi-phases is determined by a matrix-valued function, Φ

_{ }(z), on

^{d}with the value at z = 1̂ = (1, 1, … , 1) providing the corresponding periodic rigidity matrix for .

_{ }(z).

^{3}. We show that for crystal frameworks whose RUM spectrum decomposes into a finite union of linear components, there is a corresponding vector space decomposition of the almost periodic flex space. The flexes in these subspaces are periodic in specific directions associated with certain symmetries of the crystallographic point group.

## 2. Crystal Frameworks and the RUM Spectrum

^{d}, is a pair consisting of a simple undirected graph G = (V, E) and an injective map p : V → ℝ

^{d}. A (real) infinitesimal flex of (G, p) is a field of velocities, or velocity vectors, u(v), assigned to the joints, p(v), such that, for every edge, vw ∈ E,

_{e}= (p(v), p(w)) with e = vw ∈ E.

#### Definition 1

_{v}⊆ p(V) and F

_{e}⊆ p(E), and a full rank translation group, , such that:

_{v}, F

_{e}) is called a motif for . The elements of F

_{v}are called motif vertices, and the elements of F

_{e}are called motif edges. The translation group, , is necessarily of the form:

_{1}, a

_{2},…, a

_{d}are linearly independent vectors in ℝ

^{d}. The translation $x\mapsto x+{\text{\u2211}}_{j=1}^{d}{k}_{j}{a}_{j}$ is denoted T

^{k}for each k = (k

_{1}, … , k

_{d}) ∈ ℤ

^{d}. For each motif vertex, p(v) ∈ F

_{v}, and each k ∈ ℤ

^{d}, we denote by (v, k) the unique vertex for which p(v, k) = T

^{k}(p(v)). For each motif edge, p

_{e}∈ F

_{e}, with p

_{e}= (p(v,l),p(w,m)), we let (e,k) denote the unique edge for which p

_{(}

_{e,k}

_{)}= (p(v, l+ k),p(w,m+k)). In Section 4, we provide a number of illustrative examples of crystal frameworks with natural choices of the motif and translation group.

_{v}× ℤ

^{d}→ ℂ

^{d}. Indeed, such vectors are infinitesimal flexes if and only if their real and imaginary parts are infinitesimal flexes. A velocity vector u : F

_{v}× ℤ

^{d}→ ℂ

^{d}for is said to be

- local if u(v, k) ≠ 0 for at least one and at most finitely many (v, k) ∈ F
_{v}× ℤ^{d}; - strictly periodic if u(v, k) = u(v, 0) for all (v, k) ∈ F
_{v}× ℤ^{d}; - supercell periodic if u(v, k) = u(v, 0) for each motif vertex v ∈ F
_{v}and for all k in a full rank subgroup of ℤ^{d}of the form m_{1}ℤ × ⋯ × m_{d}ℤ.

#### 2.1. The Symbol Function and Rigidity Matrix

_{ }(z), of a crystal framework and the rigidity matrix, R( ), from which it is derived. The d-dimensional torus is

^{d}= {(z

_{1}, …, z

_{d}) ∈ ℂ

^{d}: |z

_{1}| = ⋯ = |z

_{d}| = 1}. For k ∈ Z

^{d}, the associated monomial function, f :

^{d}→ ℂ, given by $f\left({z}_{1},\dots ,{z}_{d}\right)={z}_{1}^{{k}_{1}}\dots {z}_{d}^{{k}_{d}}$, is written simply as z

^{k}.

#### Definition 2

^{d}with motif (F

_{v}, F

_{e}), and for e = (v, k)(w, l) ∈ F

_{e}, let p(e) = p(v, k) − p(w, l). Then, Φ

_{ }(z) is a matrix-valued function on

^{d}, which assigns a finite |F

_{e}| × d|F

_{v}| matrix to each z ∈

^{d}. The rows of Φ

_{ }(z) are labelled by the edges of F

_{e}, and the columns are labelled by the vertex-coordinate pairs in F

_{v}× {1,…, d}. The row for an edge e = (v, k)(w, l) with v ≠ w takes the form,

#### Definition 3

^{d}with motif (F

_{v}, F

_{e}), and for e = vw ∈ F

_{e}, let p(e) = p(v) − p(w). Then, R( ) is the infinite matrix, whose rows are labelled by the edges, (e, k) in F

_{e}× ℤ

^{d}, and whose columns are labelled by the pairs, (v, k) in F

_{v}× ℤ

^{d}. The row for an edge (e, k) = (v, l + k)(w, m + k), with e = (v, l)(w, m) ∈ F

_{e}, takes the form:

_{v}× ℤ

^{d}→ ℝ

^{d}is an infinitesimal flex for if and only if R( ) u = 0.

#### 2.2. The RUM Spectrum

_{1},…, ω

_{d}) ∈

^{d}. A velocity vector u : F

_{v}× ℤ

^{d}→ ℂ

^{d}is said to be ω-phase-periodic if u(v, k) = ω

^{k}u(v, 0) for all (v, k) ∈ F

_{v}× ℤ

^{d}. Here, ω

^{k}is the product ${\omega}_{1}^{{k}_{1}}\dots {\omega}_{d}^{{k}_{d}}$. We also write u = b ⊗ e

_{ω}for this vector, where b is the vector (u(v, 0))

_{v}in ℂ

^{d|Fv|}and e

_{ω}is the multi-sequence (ω

^{k})

_{k∈ℤd}. We refer to ω as a multi-phase for u. The velocity vector u is said to be phase-periodic if it is ω-phase-periodic for some multi-phase ω ∈

^{d}.

^{iθ1}, … , e

^{iθd}). The k-th entry corresponds to the phase variation of the velocity vector, u, in the direction of the k-th lattice direction. The d-tuple of real numbers θ = (θ

_{1}, … , θ

_{d}) can be taken to have entries in the interval [0, 2π), in which case, θ can be considered as the reduced wave vector of the velocity vector, u (see [8] for further remarks on this dynamical connection).

#### Theorem 1

^{d}; let ω ∈

^{d}and let u = b ⊗ e

_{ω}, with b ∈ F

_{v}× ℂ

^{d}, be a ω-phase-periodic velocity vector. Then, the following conditions are equivalent:

- (i)
- R( )u = 0;
- (ii)
- Φ
_{ }(ω̅)b = 0.

_{ }(ω̅), is less than d|F

_{v}|.

^{d}. Evidently, the RUM spectrum must always contain the point, 1̂ ∈

^{d}, as every constant velocity vector, u, is a phase-periodic infinitesimal flex with multi-phase 1̂. Thus, the RUM spectrum of is a singleton (i.e., it contains only the single point, 1̂ if and only if every phase-periodic infinitesimal flex for is strictly periodic.

#### Corollary 1

^{d}with translation group . Then, the following statements are equivalent:

- (i)
- The -periodic real infinitesimal flexes of are trivial;
- (ii)
- The -periodic complex infinitesimal flexes of are trivial;
- (iii)
- The periodic rigidity matrix Φ
_{ }(1,…, 1) has rank equal to d|F_{v}| − d.

## 3. Almost Periodic Rigidity

^{d}, and this embraces the setting of velocity fields relevant to the almost periodic rigidity of crystal frameworks (see Definition 4). The constructive approximation theorem that we require is given in Theorem 2. This theorem together with Lemmas 1 and 2 leads to the almost periodic rigidity theorem.

#### 3.1. Almost Periodic Sequences

_{n}and their approximate identity property under convolution with a continuous periodic functions feature in a standard proof that a continuous 2π-periodic function, f(x), on the real line is uniformly approximable by the explicit trigonometric functions:

_{n}(x), has the form g

_{n}(x) = [f, R

_{x}K

_{n}], where [·, ·] is the mean inner product:

_{x}K

_{n}(s) = K

_{n}(s − x). It is classical that for a function, f(x) in AP(ℝ, ℂ), there is a sequence of Bochner-Fejér kernels ${K}_{n}^{\prime}$, which provide, by the same formula, a uniformly approximating sequence of trigonometric polynomials, g

_{n}(x). In this case, the frequencies, λ, in the nonzero terms, ae

^{iλx}, of these approximants appear in a countable set derived (by rational division) from the spectrum, Λ(f), of f defined by:

_{k∈ℤ}, that have a finite sum form:

_{n}(x), are given by the Fejér sequences:

^{iλ}. In particular:

#### Theorem 2

_{1},α

_{2}, … be a maximal subset of the Bohr spectrum Λ(h), which is independent over ℚ, and for n = 1, 2,… let:

_{1}, g

_{2}, … of trigonometric sequences in AP(ℤ, ℂ) given by:

^{d}, ℂ

^{r}). The approximating trigonometric sequences, g, now have a finite sum form:

_{ω}∈ ℂ

^{r}and e

_{ω}, for ω = (ω

_{1},…, ω

_{d}) in

^{d}, is the pure frequency sequence e

_{ω1}⊗ ⋯ ⊗ e

_{ωd}in AP(ℤ

^{d}, ℂ) with:

^{d}, which we freely identify with a countable subset of points, λ, in [0, 2π)

^{d}. For notational convenience, we state the general theorem only in the case d = 2.

_{∞}; for velocity fields h, h′, we have:

#### Theorem 3

^{2}, ℂ

^{r}), let (α

_{1}, β

_{1}), (α

_{2}, β

_{2}),… be a maximal subset of Λ (h) ⊂ [0,2π)

^{2}, which is independent over ℚ, and for n = 1,2,… let:

_{1}, g

_{2},… of trigonometric sequences in AP(ℤ

^{2},ℂ

^{r}) given by:

_{ℤ2}is the natural well-defined sesquilinear map from AP(ℤ

^{2}, ℂ

^{r}) × AP(ℤ

^{2}, ℂ) to ℂ

^{r}.

^{d}and on ℤ

^{d}is part of the abstract theory of almost periodic functions on locally compact abelian groups, due to Bochner and von Neumann [18]. For further details, see also Levitan and Zhikov [19], Loomis [20] and Shubin[21].

#### 3.2. Almost Periodic Rigidity

^{ℤd×|Fv|}⊗ ℂ

^{d}is the vector space of all velocity fields on p(V), that is, the space of functions h : ℤ

^{d}× |F

_{v}| → ℂ

^{d}. The codomain of R( ) is the vector space of complex-valued functions on the set of edges.

^{d}, are denoted by ${R}_{l}^{V}$ and ${R}_{l}^{E}$, respectively. Here, the right shift of a sequence, h(k, κ), by l is the sequence, h(k − l, κ). We note that:

#### Definition 4

^{d}× F

_{v}→ ℂ

^{d}be a velocity field.

- An integral vector, l in ℤ
^{d}, is an ε-translation vector for h if${\Vert {R}_{l}^{V}\left(h\right)-h\Vert}_{\infty}<\u220a$. - The velocity field, h, is Bohr almost periodic if for every ∊ > 0, the set of ∊-translation vectors, l, is relatively dense in ℝ
^{d}.

#### Lemma 1

_{ω}in ℂ

^{d|Fv|}. If g is a nonzero infinitesimal flex for , then each component sequence a

_{ω}⊗ e

_{ω}is a nonzero ω-phase-periodic infinitesimal flex.

#### Proof

_{1}, … , ω

_{d}) ∈

^{d}and N ∈ ℕ, let R

^{V}(ω, N) be the linear map on the normed space ℓ

^{∞}(ℤ

^{d}× F

_{v}, ℂ

^{d}) given in terms of the right shift operators, ${R}_{k}^{V}$, k ∈ ℤ

^{d}, by:

^{E}(ω, N) be the linear map on ℓ

^{∞}(ℤ

^{d}× F

_{e}, ℂ) given by,

_{ω}⊗ e

_{ω}) converges uniformly to a

_{ω}⊗ e

_{ω}, when ω′ = ω, and to the zero sequence otherwise, since for each (l, v

_{κ}) ∈ ℤ

^{d}× F

_{v},

^{∞}(ℤ

^{d}× F

_{v}, ℂ

^{d}) to ℓ

^{∞}(ℤ

^{d}× F

_{e}, ℂ), which commutes with the right shift operators. Thus, if ω is a multi-frequency for g, then:

#### Lemma 2

^{d}× F

_{v}, ℂ), and let h be an infinitesimal flex for in AP(ℤ

^{d}× F

_{v}, ℂ

^{d}). Then, the mean convolution multi-sequence g : ℤ

^{d}× F

_{v}→ ℂ

^{d}given by g(k, v

_{κ}) = [h, R

_{k}(K)]ℤ

_{d}is an infinitesimal flex for .

#### Proof

_{ω}, so that K(k) = ω

^{k}for k ∈ ℤ

^{d}. Then, g is the uniform limit of the sequence (g

_{N}), where:

_{N}, are contractive for the uniform norm, uniform convergence holds for a general almost periodic velocity sequence. Thus, since the vector space of infinitesimal flexes is invariant under translation, it follows that R( )g

_{N}= 0 for each N and, hence, that R( )g = 0.

#### Theorem 4

^{d}. The following statements are equivalent:

- (i)
- Every almost periodic infinitesimal flex of is trivial;
- (ii)
- Every strictly periodic infinitesimal flex of is trivial, and Ω( ) = {1̂}.

#### Proof

_{n}), of trigonometric sequences in AP(ℤ

^{d}× F

_{v}, ℂ

^{d}) given by:

_{n}, is an infinitesimal flex of , and so, by Lemma 1, each g

_{n}is a finite linear combination of phase-periodic infinitesimal flexes of . By hypothesis, the RUM spectrum, Ω( ), contains the single multi-phase, 1̂, and so, every phase-periodic infinitesimal flex is strictly periodic. In particular, g

_{n}is strictly periodic. It follows that u is strictly periodic and, hence, trivial, as desired.

#### Proposition 5

^{d}. Then, the space of supercell periodic infinitesimal flexes for m-fold periodicity with m = (m

_{1},…, m

_{d}) is equal to the linear span of:

_{m}( ) is the finite subset of the RUM spectrum given by the multi-phases ω, whose k-th component is an m

_{k–th}root of unity.

^{d}is the union of the Bohr spectra of all almost periodic infinitesimal flexes. Note that the spectra here depend on the translation group in the following manner. If ′ has the same underlying bar-joint framework as , but full rank translation group ′ ⊆ , then the infinitesimal flex, u, is represented anew as a sequence in $AP({\mathbb{Z}}^{d}\times {F}_{v}^{\prime},{\u2102}^{d})$ , where ${F}_{v}^{\prime}$ is a vertex motif for ′. The Bohr spectrum, Λ(u, ), is then the image of Λ(u, ) under the surjective map,

^{d}→

^{d}, induced by the inclusion, ′ ⊆ . This follows the same relationship as that for the RUM spectrum noted in [8]. It also follows from this that the dimension of Ω( ), as a topological space or as an algebraic variety, is independent of the translation group, and we refer to this integer, which takes values between zero and d, as the RUM dimension of .

^{d}, often decomposes as a union of linear components. This is the case, for example, in two dimensions if Φ

_{ }(z) is a square matrix function, whose determinant polynomial, det Φ

_{ }(z), either vanishes identically or factorises into simple factors of the form (z

^{n}− λw

^{m}) with |λ| = 1. It follows that each almost periodic flex, u, of admits a finite sum decomposition u

_{1}+ ⋯ + u

_{r}in which each component, u

_{i}, is an almost periodic flex, whose Bohr spectrum lies in the i-th linear component. Such component flexes are partially periodic, being periodic in certain directions of translational symmetry.

## 4. Gallery of Crystal Frameworks

^{d}.

#### Example 1

_{v}, F

_{e}) and translation group indicated in Table 1. Simplifying earlier notation, the motif vertex is labelled v and the motif edges are labelled e

_{0}= v(0, 0)v(1, 0), e

_{1}= v(1, 0)v(0, 1) and e

_{2}= v(0, 0)v(0, 1). The translation group is = {k

_{1}a

_{1}+ k

_{2}a

_{2}: k

_{1},k

_{2}∈ ℤ}, where a

_{1}= (1, 0) and${a}_{2}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. The symbol function for is,

_{ }(z, w) has rank two unless z = w = 1, and so, the RUM spectrum of is the singleton, (1, 1) ∈

^{2}. Furthermore, there are no non-trivial strictly periodic infinitesimal flexes of , and so, by Theorem 4, is almost periodically infinitesimally rigid.

_{1}⊂ G

_{2}⊂ ⋯ of G, such that every vertex of G is contained in some G

_{n}and each sub-framework (G

_{n}, p) is infinitesimally rigid. For example, for each n, take G

_{n}to be the vertex-induced subgraph on {v(k

_{1}, k

_{2}) : (k

_{1}, k

_{2}) ∈ ℤ

^{2}, |k

_{i}| ≤ n}. It follows that admits no nontrivial infinitesimal flexes and, so, is (absolutely) infinitesimally rigid as a bar-joint framework. In [22], we obtain a general characterisation of countable simple graphs, G, whose locally generic placements are infinitesimally rigid in this sense. The condition is that G should contain a vertex-complete chain of (2, 3)-tight subgraphs. The crystal framework, , may be viewed as a non-generic placement of such a graph, which remains infinitesimally rigid despite the crystallographic symmetry.

#### Example 2

_{0}= (0, 0) and${v}_{1}=\left(\frac{1}{2},\frac{1}{2}\right)$. The motif edges are e

_{0}= v

_{0}(0, 0)v

_{0}(1, 0), e

_{1}= v

_{0}(0, 0)v

_{0}(0, 1), e

_{2}= v

_{0}(0, 1)v

_{1}(0, 0) and e

_{3}= v

_{0}(1, 0)v

_{1}(0, 0). Note that the symbol function, Φ

_{ }(z,w), is a square matrix,

_{ }(z, w) vanishes identically, and so, the RUM spectrum of is

^{2}. A local infinitesimal flex of is evident by defining u(v

_{1}) = (1, 1) and u

_{v}= 0 for all v ≠ v

_{1}. A phase-periodic infinitesimal flex of for ω = (ω

_{1}, ω

_{2}) is obtained by taking u(v

_{0}, k) = 0 and$u\left({v}_{1},k\right)={\omega}_{1}^{{k}_{1}}{\omega}_{2}^{{k}_{2}}\left(1,1\right)$ for each k = (k

_{1}, k

_{2}) ∈ ℤ

^{2}. In particular, any finite linear combination of such phase-periodic flexes will be an almost periodic infinitesimal flex for .

_{κ}, 0) = u(v

_{κ}, k) for each motif vertex, v

_{κ}, and all k in a full rank subgroup of ℤ

^{d}.

#### Example 3

_{v}, F

_{e}) and translation group indicated in Table 3. Note that has symbol function,

_{0}, 0) and u(v

_{1}, 0) to the motif vertices and consider the multi-phase ω = (ω

_{1}, ω

_{2}) = (−1, 1) ∈ Ω( ). Define for each k = (k

_{1}, k

_{2}) ∈ ℤ

^{2},

_{0},0) = (1,−1) and u(v

_{1},0) = (−1,−1), then u is also an infinitesimal flex for . Note that u consists of alternating rotational motions. In the notation of Proposition 5, u has m-fold periodicity, where we have taken m = (2, 1); the multi-phase ω = (−1, 1) is contained in,

_{ω}, where b = (u(v

_{0},0), u(v

_{1}, 0)) ∈ ker Φ

_{ }(ω̅).

#### Example 4

^{2}, which is indicated in Table 4. The motif vertices are p(v

_{0}) = (0, 0) and$p\left({v}_{1}\right)=\left(\frac{1}{3},-\frac{2}{3}\right)$. The symbol function, Φ

_{ }(z, w) is,

^{2}. However, every strictly periodic velocity field, u, with (p(v

_{0})−p(v

_{1}))·(u(v

_{0})−u(v

_{1})) = 0 and u(v

_{0}) = u(v

_{1}) is a non-trivial strictly periodic infinitesimal flex of .

#### Example 5

_{0}= (0,0), ${v}_{1}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ and v

_{2}= (0, 1). The symbol function is a square matrix,

^{2}, whose representation in [0, 1)

^{2}consists of the points (s,t) in the line segments given by:

_{1}+ u

_{2}+ u

_{3}of three almost periodic flexes corresponding to this ordered decomposition. Furthermore, u

_{1}, with the Bohr spectrum in the line s = 0, is periodic in the direction of the period vector a

_{1}= (1, 0), while u

_{2}, with the Bohr spectrum in the line t = 0, is periodic in the direction of the period vector${a}_{2}=(1/2,(2+\sqrt{3})/2)$, and u

_{3}, with the Bohr spectrum, in the line s = t is periodic in the direction a

_{1}− a

_{2}.

#### Example 6

^{2}representation, it yields two horizontal and two vertical lines. From this, it follows that any almost periodic infinitesimal flex decomposes as a sum of two flexes, each of which is supercell periodic in one of the axial directions.

_{2v}. However, the former has RUM dimension one and is linearly indecomposable while the latter has RUM dimension zero. Note that four-regular crystal frameworks such as these have a square symbol function.

#### Example 7

_{e}| = d|F

_{v}| illustrated in Table 7. The symbol function, Φ

_{ }(z,w), is,

#### Example 8

_{e}| = d|F

_{v}|, and so, the symbol function, Φ

_{ }(z,w), is a square matrix,

#### Example 9

^{2}that satisfy ℜ(w) = aℜ(z) + (1 − a) for either$a=\sqrt{3}-\sqrt{2}$ or$a=\sqrt{3}+\sqrt{2}$. This set is illustrated in Figure 1 as a subset of the torus [0, 1)

^{2}, which consists of four closed curves with the common intersection point (0,0).

^{d}, one may construct a product framework = ×

_{ℤ}in ℝ

^{d+1}, whose intersection with the hyperplanes ℝ

^{d}× {n} are copies of and where these copies are connected by the edges ((p(v), n), (p(v), n + 1)). In the case that has square matrix symbol function Φ

_{ }(z

_{1},…,z

_{d}), it is straightforward to verify that:

^{3}of such frameworks. Further, three-dimensional examples not of this product form may be found in Power [8] and Wegner [12].

#### Example 10

^{3}.

#### Further Work

## Acknowledgments

## Conflicts of Interest

## References

- Ross, E. The rigidity of periodic body-bar frameworks on the fixed torus. Phil. Trans. R. Soc. A
**2014**, 372. [Google Scholar] [CrossRef] - Borcea, C.S.; Streinu, I. Periodic frameworks and flexibility. Proc. R. Soc. A
**2010**, 466, 2633–2649. [Google Scholar] - Borcea, C.S.; Streinu, I. Frameworks with crystallographic symmetry. Phil. Trans. R. Soc. A
**2014**, 372. [Google Scholar] [CrossRef] - Connelly, R.; Shen, J.D.; Smith, A.D. Ball packings with periodic constraints. Available online: http://arxiv.org/abs/1301.0664 (accessed on 18 April 2014).
- Malestein, J.; Theran, L. Generic combinatorial rigidity of periodic frameworks. Adv. Math.
**2013**, 233, 291–331. [Google Scholar] - Owen, J.C.; Power, S.C. Infinite bar-joint frameworks, crystals and operator theory. N. Y. J. Math.
**2011**, 17, 445–490. [Google Scholar] - Power, S.C. Crystal frameworks, symmetry and affinely periodic flexes. Available online: http://arxiv.org/pdf/1103.1914v3.pdf (accessed on 18 April 2014).
- Power, S.C. Polynomials for crystal frameworks and the rigid unit mode spectrum. Phil. Trans. R. Soc. A
**2014**, 372. [Google Scholar] [CrossRef] - Ross, E.; Schulze, B.; Whiteley, W. Finite motions from periodic frameworks with added symmetry. Int. J. Solids Struct.
**2011**, 48, 1711–1729. [Google Scholar] - Dove, M.T.; Pryde, A.K.A.; Heine, V.; Hammonds, K.D. Exotic distributions of rigid unit modes in the reciprocal spaces of framework aluminosilicates. J. Phys. Condens. Matter
**2007**, 19. [Google Scholar] [CrossRef] - Giddy, A.P.; Dove, M.T.; Pawley, G.S.; Heine, V. The determination of rigid unit modes as potential soft modes for displacive phase transitions in framework crystal structures. Acta Crystallogr.
**1993**, A49, 697–703. [Google Scholar] - Wegner, F. Rigid-unit modes in tetrahedral crystals. J. Phys. Condens. Matter
**2007**, 19, 406–218. [Google Scholar] - Kapko, V.; Dawson, C.; Rivin, I.; Treacy, M.M.J. Density of Mechanisms within the Flexibility Window of Zeolites. Phys. Rev. Lett.
**2011**, 107. [Google Scholar] [CrossRef] - Power, S.C. Crystal frameworks, matrix-valued functions and rigidity operators, Operator Theory: Advances and Applications. In Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation: 22nd International Workshop in Operator Theory and Its Application; Springer: Birkhäuser Basel, Switzerland, 2014; Volume 236. [Google Scholar]
- Partington, J.R. Linear operators and linear systems. An analytical approach to control theory. In London Mathematical Society Student Texts, 60; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Besicovitch, A.S.; Bohr, H. Almost periodicity and general trigonometric series. Acta Math.
**1931**, 57, 203–292. [Google Scholar] - Besicovitch, A.S. Almost Periodic Functions; Dover Publications, Inc.: New York, NY, USA, 1955. [Google Scholar]
- Bochner, S.; von Neumann, J. Almost periodic functions in groups. II. Trans. Amer. Math. Soc.
**1935**, 37, 21–50. [Google Scholar] - Levitan, B.M.; Zhikov, V.V. Almost Periodic Functions and Differential Equations; Cambridge University Press: Cambridge, UK, 1982; Longdon, L. W., Translator. [Google Scholar]
- Loomis, L.H. An Introduction to Abstract Harmonic Analysis; D. Van Nostrand Company, Inc.: New York, NY, USA, 1953. [Google Scholar]
- Šubin, M.A. Almost periodic functions and partial differential operators. Russ. Math. Surv.
**1978**, 33. [Google Scholar] [CrossRef] - Kitson, D.; Power, S.C. The rigidity of infinite graphs. Available online: http://arxiv.org/abs/1310.1860 (accessed on 18 April 2014).

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**MDPI and ACS Style**

Badri, G.; Kitson, D.; Power, S.C.
The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks. *Symmetry* **2014**, *6*, 308-328.
https://doi.org/10.3390/sym6020308

**AMA Style**

Badri G, Kitson D, Power SC.
The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks. *Symmetry*. 2014; 6(2):308-328.
https://doi.org/10.3390/sym6020308

**Chicago/Turabian Style**

Badri, Ghada, Derek Kitson, and Stephen C. Power.
2014. "The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks" *Symmetry* 6, no. 2: 308-328.
https://doi.org/10.3390/sym6020308