Hindawi Publishing Corporation Journal of Mathematics Volume 2015, Article ID 967417, 5 pages http://dx.doi.org/10.1155/2015/967417

Research Article Symplectic Toric Geometry and the Regular

Elisa Prato

Dipartimento di Matematica e Informatica “U. Dini”, Piazza Ghiberti 27, 50122 Firenze, Italy

Correspondence should be addressed to Elisa Prato; [email protected]

Received 30 July 2015; Accepted 26 October 2015

Academic Editor: Fred´ eric´ Mynard

Copyright © 2015 Elisa Prato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The regular dodecahedron is the only simple among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold.

1. Introduction Definition 1 (simple polytope). A dimension 𝑛 convex poly- 𝑛 ∗ tope Δ⊂(R ) is said to be simple if each of its vertices is According to a well-known theorem by Delzant [1], there contained in exactly 𝑛 facets. exists a one-to-one correspondence between symplectic toric 𝑛 manifolds and simple rational convex satisfying a Definition 2 (quasilattice). A quasilattice in R is the Z-span 𝑛 special integrality condition. One of the important features of a set of R-spanning vectors, 𝑌1,...,𝑌𝑑,ofR . of Delzant’s theorem is that it provides an explicit procedure Notice that SpanZ{𝑌1,...,𝑌𝑑} is an ordinary lattice if and for computing the symplectic manifold that corresponds to 𝑛 only if it admits a set of generators which is a basis of R . eachgivenpolytope.Asitturnsout,therearemanyimport- 𝑛 ∗ Consider now a dimension 𝑛 Δ⊂(R ) ant examples of simple convex polytopes that do not fall into 𝑛 this class, either because they do not satisfy Delzant’sintegral- having 𝑑 facets. Then, there exist elements 𝑋1,...,𝑋𝑑 in R ity condition, or, even worse, because they are not rational. and 𝜆1,...,𝜆𝑑 in R such that The regular dodecahedron is one of the most remarkable 𝑑 𝑛 ∗ examples of a simple polytope that is not rational. In this Δ=⋂ {𝜇 ∈R ( ) |⟨𝜇,𝑋𝑗⟩≥𝜆𝑗}. (1) paper, we apply a generalization of Delzant’s construction [2] 𝑗=1 to simple nonrational convex polytopes and we associate to Definition 3 (quasirational polytope). Let 𝑄 be a quasilattice 𝑛 𝑛 ∗ the regular dodecahedron a symplectic toric quasifold.Quasi- in R . A convex polytope Δ⊂(R ) is said to be quasirational folds are a natural generalization of manifolds and orbifolds with respect to 𝑄 if the vectors 𝑋1,...,𝑋𝑑 in (1) can be chosen introduced by the author in [2]; they are not necessarily in 𝑄. Hausdorff spaces and they are locally modeled by quotients 𝑛 ∗ of manifolds by the action of discrete groups. We remark that each polytope in (R ) is quasirational This paper is structured as follows. In Section 2, we withrespecttosomequasilattice𝑄:justtakethequasilattice recall the generalized Delzant construction. In Section 3, we that is generated by the elements 𝑋1,...,𝑋𝑑 in (1). Notice that apply this construction to the regular dodecahedron and we if 𝑋1,...,𝑋𝑑 canbechoseninsideanordinarylattice,then describe the corresponding symplectic toric quasifold. thepolytopeisrationalintheusualsense. 𝑛 2. The Generalized Delzant Construction Definition 4 (quasitorus). Let 𝑄⊂R be a quasilattice. We call quasitorus of dimension 𝑛 thegroupandquasifold 𝑛 𝑛 We begin by recalling a few useful definitions. 𝐷 = R /𝑄. 2 Journal of Mathematics

For the definition and main properties of symplectic quasifolds and of Hamiltonian actions of quasitori on sym- plectic quasifolds, we refer the reader to [2, 3]. On the other hand, the basic facts on polytopes that are needed can be found in Ziegler’s book [4]. We are now ready to recall from [2] the generalized Delzant construction. For the purposes of this paper, we will restrict our attention to the special case 𝑛=3.

3 3 ∗ Theorem 5. Let 𝑄 be a quasilattice in R and let Δ⊂(R ) be a simple convex polytope that is quasirational with respect to 𝑄. Then, there exists a 6-dimensional compact connected symplectic quasifold 𝑀 and an effective Hamiltonian action of 3 3 the quasitorus 𝐷 = R /𝑄 on 𝑀 such that the image of the corresponding moment mapping is Δ.

𝑑 Proof. Let us consider the space C endowed with the 𝑑 standard symplectic form 𝜔0 =1/(2𝜋𝑖)∑𝑗=1 𝑑𝑧𝑗 ∧𝑑𝑧𝑗 and 𝑑 𝑑 𝑑 the action of the torus 𝑇 = R /Z given by

𝑑 𝑑 𝑑 𝜏: 𝑇 × C 󳨀→ C (2) Figure 1: The regular dodecahedron. 2𝜋𝑖𝜃1 2𝜋𝑖𝜃𝑑 2𝜋𝑖𝜃1 2𝜋𝑖𝜃𝑑 ((𝑒 ,...,𝑒 ),𝑧) 󳨃󳨀→ (𝑒 𝑧1,...,𝑒 𝑧𝑑).

This is an effective Hamiltonian action with moment map- Remark 6. We will say that 𝑀 is a symplectic toric quasifold ping given by associated to the polytope Δ.Thequasifold𝑀 depends on our ∗ choice of quasilattice 𝑄 with respect to which the polytope is 𝐽 C𝑑 󳨀→ ( R𝑑) : quasirational and on our choice of vectors 𝑋1,...,𝑋𝑑.Note 𝑑 󵄨 󵄨2 ∗ (3) that the case where the polytope is simple and rational, but 𝑧 󳨃󳨀→ ∑ 󵄨𝑧 󵄨 𝑒∗ +𝜆, 𝜆∈(R𝑑) . 󵄨 𝑗󵄨 𝑗 constant does not necessarily satisfy Delzant’s integrality condition, 𝑗=1 was treated by Lerman and Tolman in [5]. They allowed 𝐽 orbifold singularities and introduced the notion of symplectic The mapping is proper and its image is given by the cone toric orbifold. C𝜆 =𝜆+C,whereC denotes the positive orthant of 𝑑 ∗ (R ) .Takenowvectors𝑋1,...,𝑋𝑑 ∈𝑄and real numbers 𝜆 ,...,𝜆 1 𝑑 as in (1). Consider the surjective linear mapping 3. The Regular Dodecahedron from 𝑑 3 𝜋: R 󳨀→ R , a Symplectic Viewpoint (4) Δ 𝑒𝑗 󳨃󳨀→ 𝑋 𝑗. Let be the regular dodecahedron centered at the origin and having vertices 3 𝐷3 = R3/𝑄 Consider the dimension quasitorus .Then, (±1, ±1, ±1) the linear mapping 𝜋 induces a quasitorus epimorphism 𝑑 3 Π: 𝑇 →𝐷. Define now 𝑁 to be the kernel of the mapping 1 𝑑 ∗ (0, ±𝜙, ± ) Π and choose 𝜆=∑𝑗=1 𝜆𝑗𝑒𝑗 .Denoteby𝑖 the Lie algebra 𝜙 𝑑 ∗ inclusion Lie(𝑁) → R and notice that Ψ=𝑖 ∘𝐽 is a 1 (5) 𝑑 (± ,0,±𝜙) moment mapping for the induced action of 𝑁 on C .Then, 𝜙 3.1 𝑀=Ψ−1(0)/𝑁 according to [2, Theorem ], the quotient 1 isasymplecticquasifoldwhichisendowedwiththeHamil- (±𝜙, ± ,0), 𝑑 ∗ 𝜙 tonian action of the quasitorus 𝑇 /𝑁.Since𝜋 is injective and 𝐽 is proper, the quasifold 𝑀 is compact. If we identify 𝜙 = (1 + √5)/2 3 𝑑 where is the golden ratio and satisfies the quasitori 𝐷 and 𝑇 /𝑁 via the epimorphism Π,weget 𝜙=1+1/𝜙 3 (see Figure 1). It is a well-known fact that the a Hamiltonian action of the quasitorus 𝐷 whose moment polytope Δ is simple but not rational. However, consider the ∗ −1 ∗ 3 mapping Φ has image equal to (𝜋 ) (C𝜆 ∩ ker 𝑖 )= quasilattice 𝑃 that is generated by the following vectors in R : ∗ −1 ∗ ∗ −1 ∗ (𝜋 ) (C𝜆 ∩ im 𝜋 )=(𝜋 ) (𝜋 (Δ)) which is exactly Δ.This −1 1 action is effective since the level set Ψ (0) contains points of 𝑌1 =( ,1,0) 𝑑 𝑑 𝜙 the form 𝑧 ∈ C , 𝑧𝑗 =0̸ , 𝑗 = 1,...,𝑑,wherethe𝑇 -action is 𝑀=2𝑑−2 𝑁=2𝑑−2(𝑑− 1 free. Notice finally that dim dim 𝑌 =(0, ,1) 3) = 6. 2 𝜙 Journal of Mathematics 3

1 𝑌 = (1, 0, ) 3 𝜙 1 𝑌 =(− ,1,0) 4 𝜙 1 𝑌 =(0,− ,1) 5 𝜙 1 𝑌 = (1, 0, − ). 6 𝜙 (6) We remark that these six vectors and their opposites point to thetwelveverticesofaregularicosahedronthatisinscribed in the sphere of radius √3−𝜙 (see Figures 2 and 3). The quasilattice 𝑃 is known in physics as the simple icosahedral lattice [6]. Now, an easy computation shows that

12 3 ∗ Δ=⋂ {𝜇∈(R ) | ⟨𝜇,𝑗 𝑋 ⟩ ≥−𝜙} , (7) 𝑗=1 where 𝑋𝑖 =𝑌𝑖 and 𝑋6+𝑖 =−𝑌𝑖,for𝑖 = 1,...,6. Therefore, Δ 𝑃 is quasirational with respect to .Letusperformthe ±𝑌 ,...,±𝑌 generalized Delzant construction with respect to 𝑃 and the Figure 2: The vectors 1 6. vectors 𝑋1,...,𝑋12. Following the proof of Theorem 5, we consider the surjective linear mapping Moreover, the moment mapping for the induced 𝑁-action 12 3 12 ∗ 𝜋: R 󳨀→ R Ψ: C →(n) has the 9 components below: (8) 𝑒𝑖 󳨃󳨀→ 𝑋 𝑖. 󵄨 󵄨2 󵄨 󵄨2 󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 −2𝜙 It is easy to see that the following relations 󵄨 1󵄨 󵄨 7󵄨 󵄨 󵄨2 󵄨 󵄨2 1 1 󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 −2𝜙 −1 󵄨 2󵄨 󵄨 8󵄨 𝑌 𝜙 𝜙 𝑌 4 1 󵄨 󵄨2 󵄨 󵄨2 1 1 󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 −2𝜙 (𝑌 )=(−1 )(𝑌 ) 󵄨 3󵄨 󵄨 9󵄨 5 𝜙 𝜙 2 (9) 󵄨 󵄨2 󵄨 󵄨2 𝑌 1 1 𝑌 󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 −2𝜙 6 −1 3 󵄨 4󵄨 󵄨 10󵄨 𝜙 𝜙 󵄨 󵄨2 󵄨 󵄨2 󵄨𝑧5󵄨 + 󵄨𝑧11󵄨 −2𝜙 (11) imply that the kernel of 𝜋, n,isthe9-dimensional subspace 12 󵄨 󵄨2 󵄨 󵄨2 of R that is spanned by the vectors 󵄨𝑧6󵄨 + 󵄨𝑧12󵄨 −2𝜙 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 𝑒1 +𝑒7 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑧1󵄨 + 󵄨𝑧2󵄨 −𝜙(󵄨𝑧3󵄨 + 󵄨𝑧4󵄨 )+2 𝑒 +𝑒 2 8 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨𝑧2󵄨 + 󵄨𝑧3󵄨 −𝜙(󵄨𝑧1󵄨 + 󵄨𝑧5󵄨 )+2 𝑒3 +𝑒9 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨𝑧1󵄨 + 󵄨𝑧3󵄨 −𝜙(󵄨𝑧2󵄨 + 󵄨𝑧6󵄨 )+2. 𝑒4 +𝑒10 −1 The level set Ψ (0) is described by the 9 equations that 𝑒5 +𝑒11 (10) are obtained by setting these components to 0.Finally,the 𝑀 6 𝑒6 +𝑒12 symplectic toric quasifold is given by the compact - −1 3 3 dimensional quasifold Ψ (0)/𝑁.Thequasitorus𝐷 = R /𝑃 𝑒 +𝑒 −𝜙(𝑒 +𝑒 ) 1 2 3 4 acts on 𝑀 in a Hamiltonian fashion, with image of the corresponding moment mapping given exactly by the dodec- 𝑒 +𝑒 −𝜙(𝑒 +𝑒 ) 3 2 3 1 5 ahedron Δ.Theactionof𝐷 has 20 fixed points, and the

𝑒1 +𝑒3 −𝜙(𝑒2 +𝑒6). moment mapping sends each of them to a different of the dodecahedron. The quasifold 𝑀 has an atlas made of Since the vectors 𝑋𝑖, 𝑖 = 1,...,12, generate the quasilattice 20 charts, each of which is centered around a different fixed 𝑄,thegroup𝑁 is connected and given by the group exp(n). point. To give an idea of the local behavior of 𝑀,wewill 4 Journal of Mathematics

√ 󵄨 󵄨2 𝜏7 (𝑧)= 2𝜙 − 󵄨𝑧1󵄨

√ 󵄨 󵄨2 𝜏8 (𝑧)= 2𝜙 − 󵄨𝑧2󵄨

√ 󵄨 󵄨2 𝜏9 (𝑧)= 2𝜙 − 󵄨𝑧3󵄨

󵄨 󵄨2 1 󵄨 󵄨2 󵄨 󵄨2 𝜏 (𝑧)=√󵄨𝑧 󵄨 − (󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 )+2 10 󵄨 3󵄨 𝜙 󵄨 1󵄨 󵄨 2󵄨

󵄨 󵄨2 1 󵄨 󵄨2 󵄨 󵄨2 𝜏 (𝑧)=√󵄨𝑧 󵄨 − (󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 )+2 11 󵄨 1󵄨 𝜙 󵄨 2󵄨 󵄨 3󵄨

󵄨 󵄨2 1 󵄨 󵄨2 󵄨 󵄨2 𝜏 (𝑧)=√󵄨𝑧 󵄨 − (󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 )+2. 12 󵄨 2󵄨 𝜙 󵄨 1󵄨 󵄨 3󵄨 (14)

Figure 3: The regular icosahedron. The mapping 𝜏̃ induces a homeomorphism

𝑈̃ 𝜏 󳨀→ 𝑈 describe the chart around the fixed point that maps to the Γ (15) vertex (−1, −1, −1). Consider the open neighborhood 𝑈̃ of 0 3 [(𝑧 ,𝑧 ,𝑧 )] 󳨃󳨀→ [ 𝜏(𝑧̃ ,𝑧 ,𝑧 )] , in C defined by the inequalities below: 1 2 3 1 2 3 󵄨 󵄨2 where the open subset 𝑈 of 𝑀 is the quotient 󵄨𝑧1󵄨 <2𝜙 −1 󵄨 󵄨2 {𝑤 ∈Ψ (0) |𝑤𝑖 =0,̸ 𝑖=4,...,12} 󵄨𝑧2󵄨 <2𝜙 (16) 𝑁 󵄨 󵄨2 󵄨𝑧 󵄨 <2𝜙 󵄨 3󵄨 Γ (12) and the discrete group is given by 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 −2 < 󵄨𝑧1󵄨 + 󵄨𝑧2󵄨 −𝜙󵄨𝑧3󵄨 <2𝜙 2𝜋𝑖𝜙(ℎ+𝑙) 2𝜋𝑖𝜙(ℎ+𝑘) 2𝜋𝑖𝜙(𝑘+𝑙) 3 {(𝑒 ,𝑒 ,𝑒 )∈𝑇 |ℎ,𝑘,𝑙∈Z}. (17) 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 −2 < 󵄨𝑧2󵄨 + 󵄨𝑧3󵄨 −𝜙󵄨𝑧1󵄨 <2𝜙 The triple (𝑈, 𝜏, 𝑈/Γ)̃ defines a chart around the fixed point 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 corresponding to the vertex (−1, −1, −1).Theotherchartscan −2 < 󵄨𝑧1󵄨 + 󵄨𝑧3󵄨 −𝜙󵄨𝑧2󵄨 <2𝜙 be described in a similar way. Ψ−1(0) and consider the following slice of that is transversal Remark 7. According to joint work with Battaglia [7], to 𝑁 to the -orbits any nonrational simple polytope one can also associate a complex toric quasifold. If we apply the explicit construction ̃ 𝜏̃ −1 𝑈 󳨀→ { 𝑤 ∈Ψ (0) |𝑤𝑖 =0,𝑖=4,...,12}̸ given in [7] to the regular dodecahedron, similarly to the symplectic case, we get a complex 3-quasifold 𝑀C endowed (𝑧 ,𝑧 ,𝑧 ) 󳨃󳨀→ ( 𝑧 ,𝑧 ,𝑧 ,𝜏 (𝑧),𝜏 (𝑧),𝜏 (𝑧),𝜏 (𝑧), (13) 3 3 1 2 3 1 2 3 4 5 6 7 with an action of the complex quasitorus 𝐷C = C /𝑃.This action is holomorphic and has a dense open orbit. Moreover, 𝜏8 (𝑧),𝜏9 (𝑧),𝜏10 (𝑧),𝜏11 (𝑧),𝜏12 (𝑧)) , 3 by [7, Theorem 3.2], the quasifold 𝑀C is 𝐷 -equivariantly 𝑀 𝑀 3 diffeomorphic to and the induced symplectic form on C where 𝑧 =(𝑧1,𝑧2,𝑧3)∈C , 𝑤 =(𝑤1,𝑤2,𝑤3,𝑤4,𝑤5,𝑤6, 12 is Kahler.¨ 𝑤7,𝑤8,𝑤9,𝑤10,𝑤11,𝑤12)∈C ,and Remark 8. The only other platonic solids that are simple are 1 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 the and the regular . They both satisfy the 𝜏 (𝑧)=√ (󵄨𝑧 󵄨 + 󵄨𝑧 󵄨 −𝜙󵄨𝑧 󵄨 +2) 4 𝜙 󵄨 1󵄨 󵄨 2󵄨 󵄨 3󵄨 hypotheses of the Delzant theorem. By applying the Delzant procedure to the cube, with respect to the standard lattice 3 2 2 2 Z ,wegetthesymplectictoricmanifold𝑆 ×𝑆 ×𝑆;by 1 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 𝜏5 (𝑧)=√ (󵄨𝑧2󵄨 + 󵄨𝑧3󵄨 −𝜙󵄨𝑧1󵄨 +2) applying it to the tetrahedron, with respect to the sublattice 𝜙 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 3 of Z that is generated by the 8 vectors (±1, ±1, ±1),we 3 get the symplectic toric manifold CP . On the other hand, 1 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 the remaining platonic solids, the regular octahedron and 𝜏6 (𝑧)=√ (󵄨𝑧1󵄨 + 󵄨𝑧3󵄨 −𝜙󵄨𝑧2󵄨 +2) 𝜙 the regular icosahedron, are not simple. In these cases, the Journal of Mathematics 5

Delzant procedure and our generalization will not work. The octahedron is rational, thus, the standard toric geometry applies: the toric variety associated to the octahedron is described, for example, in [8, Section 1.5]. The icosahedron, on the other hand, is not rational. However, by work of Battaglia on arbitrary convex polytopes [9, 10], one can asso- ciate to the icosahedron, both in the symplectic and complex category, a space that is stratified by quasifolds. Incidentally, Battaglia’s approach can also be applied to the octahedron, yielding, in this case, a space that is stratified by manifolds.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This research was partially supported by Grant PRIN 2010NNBZ78 012 (MIUR, Italy) and by GNSAGA (INDAM, Italy). All pictures were drawn using ZomeCAD.

References

[1] T. Delzant, “Hamiltoniens periodiques´ et images convexes de l’application moment,” Bulletin de la SMF,vol.116,pp.315–339, 1988. [2] E. Prato, “Simple non-rational convex polytopes via symplectic geometry,” Topology,vol.40,no.5,pp.961–975,2001. [3] F.BattagliaandE.Prato,“ThesymplecticPenrosekite,”Commu- nications in Mathematical Physics,vol.299,no.3,pp.577–601, 2010. [4]G.M.Ziegler,Lectures on Polytopes,vol.152ofGraduate Texts in Mathematics, Springer, New York, NY, USA, 1995. [5]E.LermanandS.Tolman,“Hamiltoniantorusactionsonsym- plectic orbifolds and toric varieties,” Transactions of the Ameri- can Mathematical Society,vol.349,no.10,pp.4201–4230,1997. [6] D. S. Rokhsar, N. D. Mermin, and D. C. Wright, “Rudimentary quasicrystallography: the icosahedral and decagonal reciprocal lattices,” Physical Review. B. Condensed Matter, vol. 35, no. 11, pp. 5487–5495, 1987. [7] F. Battaglia and E. Prato, “Generalized toric varieties for sim- ple nonrational convex polytopes,” International Mathematics Research Notices, vol. 24, pp. 1315–1337, 2001. [8] W. Fulton, Introduction to Toric Varieties,vol.131ofAnnals of Mathematics Studies, Princeton University Press, 1993. [9] F. Battaglia, “Convex polytopes and quasilattices from the sym- plectic viewpoint,” Communications in Mathematical Physics, vol. 269, no. 2, pp. 283–310, 2007. [10] F. Battaglia, “Geometric spaces from arbitrary convex poly- topes,” International Journal of Mathematics,vol.23,no.1, ArticleID1250013,39pages,2012. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

The Scientific International Journal of World Journal Differential Equations Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Submit your manuscripts at http://www.hindawi.com

International Journal of Advances in Combinatorics Mathematical Physics Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Journal of Mathematical Problems Abstract and Discrete Dynamics in Complex Analysis Mathematics in Engineering Applied Analysis Nature and Society Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

International Journal of Journal of Mathematics and Mathematical Discrete Mathematics Sciences

Journal of International Journal of Journal of Function Spaces Stochastic Analysis Optimization Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014