Loading [MathJax]/jax/output/SVG/jax.js
Research article

Enhanced flowering of the F1 long-day strawberry cultivars ‘Tarpan’ and ‘Gasana’ with nitrogen and daylength management

  • Received: 01 December 2016 Accepted: 26 December 2016 Published: 28 December 2016
  • Interest in sustainable self-sufficiency, particularly when considering produce has increased. Strawberries, especially particularly the seed propagated, long-day, F1 hybrids such as ‘Tarpan’ and ‘Gasana’, fit self-sufficient, sustainable production models well. They provide aesthetic as well as culinary benefits producing colorful, attractive blossoms and flavorful fruit. To determine if ‘Tarpan’ and ‘Gasana’ flowering would respond to photoperiod and nitrogen, seedlings were fertilized with 100 or 800 ppm nitrogen for 4 weeks in September beginning one week after exposure to either short days, the natural photoperiod, or long days, the natural photoperiod supplemented with 24 hours of incandescent radiation. Plants were then greenhouse forced under both photoperiods and floral phenology evaluated. Photoperiod and N fertility during floral initiation in September affected subsequent flowering of both cultivars. The response was rapid with significant differences observed 4 weeks after the commencement of treatment. Both cultivars responded with increased rate (enhanced precocity) and intensity (enhanced inflorescence/flower number) of flowering with elevated N. In ‘Gasana’ elevated N accelerated flowering by 2–3 weeks. There was a slight acceleration of flowering in lower N plants with long-day forcing, however, elevated N was much more effective in accelerating flowering than long-day forcing. In ‘Tarpan’, long-day forcing and elevated N were equally effective in accelerating flowering. Inflorescence production per plant or crown and the number of flowers per plant were enhanced with elevated N in both cultivars. Long-day forcing stimulated the number of inflorescences produced per crown in both cultivars. The number of flowers per plant in ‘Tarpan’ was also enhanced by long-day initiation or forcing. Elevated N, short-day initiation or long-day forcing enhanced the number of flowers per inflorescence in ‘Gasana’. Elevated N under long-day initiation or under short-day initiation when followed by long-day forcing enhanced the number of flowers per inflorescence in ‘Tarpan’.

    Citation: Edward F. Durner. Enhanced flowering of the F1 long-day strawberry cultivars ‘Tarpan’ and ‘Gasana’ with nitrogen and daylength management[J]. AIMS Agriculture and Food, 2017, 2(1): 1-15. doi: 10.3934/agrfood.2017.1.1

    Related Papers:

    [1] Kengo Matsumoto . C-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, 2021, 29(4): 2645-2656. doi: 10.3934/era.2021006
    [2] Peigen Cao, Fang Li, Siyang Liu, Jie Pan . A conjecture on cluster automorphisms of cluster algebras. Electronic Research Archive, 2019, 27(0): 1-6. doi: 10.3934/era.2019006
    [3] Hongyan Guo . Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29(4): 2673-2685. doi: 10.3934/era.2021008
    [4] Doston Jumaniyozov, Ivan Kaygorodov, Abror Khudoyberdiyev . The algebraic classification of nilpotent commutative algebras. Electronic Research Archive, 2021, 29(6): 3909-3993. doi: 10.3934/era.2021068
    [5] Jin-Yun Guo, Cong Xiao, Xiaojian Lu . On n-slice algebras and related algebras. Electronic Research Archive, 2021, 29(4): 2687-2718. doi: 10.3934/era.2021009
    [6] Bing Sun, Liangyun Chen, Yan Cao . On the universal α-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, 2021, 29(4): 2619-2636. doi: 10.3934/era.2021004
    [7] Shishuo Fu, Zhicong Lin, Yaling Wang . Refined Wilf-equivalences by Comtet statistics. Electronic Research Archive, 2021, 29(5): 2877-2913. doi: 10.3934/era.2021018
    [8] Dušan D. Repovš, Mikhail V. Zaicev . On existence of PI-exponents of unital algebras. Electronic Research Archive, 2020, 28(2): 853-859. doi: 10.3934/era.2020044
    [9] Ling-Xiong Han, Wen-Hui Li, Feng Qi . Approximation by multivariate Baskakov–Kantorovich operators in Orlicz spaces. Electronic Research Archive, 2020, 28(2): 721-738. doi: 10.3934/era.2020037
    [10] José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar . Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29(1): 1783-1801. doi: 10.3934/era.2020091
  • Interest in sustainable self-sufficiency, particularly when considering produce has increased. Strawberries, especially particularly the seed propagated, long-day, F1 hybrids such as ‘Tarpan’ and ‘Gasana’, fit self-sufficient, sustainable production models well. They provide aesthetic as well as culinary benefits producing colorful, attractive blossoms and flavorful fruit. To determine if ‘Tarpan’ and ‘Gasana’ flowering would respond to photoperiod and nitrogen, seedlings were fertilized with 100 or 800 ppm nitrogen for 4 weeks in September beginning one week after exposure to either short days, the natural photoperiod, or long days, the natural photoperiod supplemented with 24 hours of incandescent radiation. Plants were then greenhouse forced under both photoperiods and floral phenology evaluated. Photoperiod and N fertility during floral initiation in September affected subsequent flowering of both cultivars. The response was rapid with significant differences observed 4 weeks after the commencement of treatment. Both cultivars responded with increased rate (enhanced precocity) and intensity (enhanced inflorescence/flower number) of flowering with elevated N. In ‘Gasana’ elevated N accelerated flowering by 2–3 weeks. There was a slight acceleration of flowering in lower N plants with long-day forcing, however, elevated N was much more effective in accelerating flowering than long-day forcing. In ‘Tarpan’, long-day forcing and elevated N were equally effective in accelerating flowering. Inflorescence production per plant or crown and the number of flowers per plant were enhanced with elevated N in both cultivars. Long-day forcing stimulated the number of inflorescences produced per crown in both cultivars. The number of flowers per plant in ‘Tarpan’ was also enhanced by long-day initiation or forcing. Elevated N, short-day initiation or long-day forcing enhanced the number of flowers per inflorescence in ‘Gasana’. Elevated N under long-day initiation or under short-day initiation when followed by long-day forcing enhanced the number of flowers per inflorescence in ‘Tarpan’.


    In [22] and [23], D. Ruelle has introduced the notion of Smale space. A Smale space is a hyperbolic dynamical system with local product structure. He has constructed groupoids and its operator algebras from the Smale spaces. After the Ruelle's initial study, I. Putnam in [14] (cf. [9], [15], [16], [17], [26], etc.) constructed various groupoids from Smale spaces and studied their C-algebras. The class of Smale spaces contain two important subclasses of topological dynamical systems as its typical examples. One is the class of shifts of finite type, which are sometimes called topological Markov shifts. The other one is the class of hyperbolic toral automorphisms. The study of the former class from the view point of C-algebras is closely related to the study of Cuntz-Krieger algebras as in [7], [8], [10], [11], [12], etc. That of the latter class is closely related to the study of the crossed product C-algebras of the homeomorphisms of the hyperbolic automorphisms on the torus.

    In this paper, we will focus on the study of the latter class, the hyperbolic toral automorphisms from the view points of C-algebras constructed from the associated groupoids as Smale spaces. Let A=[abcd]GL(2,Z) be a hyperbolic matrix. Let q:R2R2/Z2 be the natural quotient map. We denote by R2/Z2 the two-dimensional torus T2 with metric d defined by

    d(x,y)=inf{zw:q(z)=x,q(w)=y,z,wR2} for x,yT2

    where is the Euclid norm on R2. Then the matrix A defines a homeomorphism on T2 which is called a hyperbolic toral automorphism. It is a specific example of an Anosov diffeomorphism on a compact Riemannian manifold (see [4], [25], etc.). Let λu,λs be the eigenvalues of A such that |λu|>1>|λs|. They are both real numbers. Let vu=(u1,u2),vs=(s1,s2) be the normalized eigenvectors for λu,λs, respectively. The direction along vu expands by A, whereas the direction of vs expands by A1. These directions determine local product structure which makes T2 a Smale space. The groupoid GaA introduced by D. Ruelle [22] of the asymptotic equivalence relation is defined by

    GaA={(x,z)T2×T2limnd(Anx,Anz)=limnd(Anx,Anz)=0} (1)

    with its unit space

    (GaA)(0)={(x,x)T2×T2}=T2. (2)

    The multiplication and the inverse operation on GaA are defined by

    (x,z)(z,w)=(x,w),(x,z)1=(z,x) for (x,z),(z,w)GaA.

    As in [14], the groupoid GaA has a natural topology defined by inductive limit topology, which makes GaA étale. The étale groupoid GaA is called the asymptotic groupoid for the hyperbolic toral automorphism (T2,A). We will first see that the groupoid GaA is realized as a transformation groupoid T2×αAZ2 by a certain action αA:Z2Homeo(T2) associated to GaA, so that the C-algebra C(GaA) of the groupoid GaA is isomorphic to the C-algebra of the crossed product C(T2)×αAZ2 by the induced action αA:Z2Aut(C(T2)). As the action αA:Z2Homeo(T2) is free and minimal having a unique invariant ergodic measure, a general theory of C-crossed product ensures that C(T2)×αAZ2 is a simple AT-algebra having a unique tracial state (cf. [13], [14], [17]).

    Let A=[abcd]GL(2,Z) be a hyperbolic matrix which satisfies det(A)=±1, We denote by Δ(A)=(a+d)24(adbc) the discriminant of the characteristic polynomial of the matrix A, which is positive. We will show the following result.

    Theorem 1.1 (Theorem 2.10 and Proposition 3.1). The C-algebra C(GaA) of the étale groupoid GaA for a hyperbolic matrix A=[abcd] is a simple AT-algebra with unique tracial state τ that is isomorphic to the four-dimensional non-commutative torus generated by four unitaries U1,U2,V1,V2 satisfying the following relations:

    U1U2=U2U1,V1V2=V2V1,V1U1=e2πiθ1U1V1,V1U2=e2πiθ2U2V1,V2U1=e2πiθ3U1V2,V2U2=e2πiθ4U2V2,

    where

    θ1=12(1+adΔ(A)),θ2=cΔ(A),θ3=bΔ(A),θ4=12(1adΔ(A)).

    The range τ(K0(C(GaA))) of the tracial state τ of the K0-group K0(C(GaA)) of the C-algebra C(GaA) is

    τ(K0(C(GaA)))=Z+Zθ1+Zθ2+Zθ3inR. (3)

    We note that the slopes θi,i=1,2,3,4 are determined by the formulas (10), (11) for the slopes of the eigenvectors vu=(u1,u2),vs=(s1,s2).

    Since the étale groupoid GaA is a flip conjugacy invariant and the C-algebra C(GaA) has a unique tracial state written τ, we know that the trace value τ(K0(C(GaA))) is a flip conjugacy invariant of the hyperbolic toral automorphism (T2,A).

    As commuting matrices have common eigenvectors, we know that if two matrices A,BGL(2,Z) commute with each other, then the C-algebras C(GaA) and C(GaB) are canonically isomorphic. Hence two matrices [1110] and [2111] have the isomorphic C-algebras. On the other hand, as the range τ(K0(C(GaA))) of the tracial state of the K0-group K0(C(GaA)) is invariant under isomorphism class of the algebra C(GaA), the C-algebra C(GA1) is not isomorphic to C(GA2) for the matrices A1=[1110] and A2=[3121] (Proposition 4.2).

    For a vector (m,n)R2, we write the vector (m,n)t as [mn] and sometimes identify (m,n) with [mn]. A matrix A=[abcd]GL(2,Z) with det(A)=±1 is said to be hyperbolic if A does not have eigenvalues of modulus 1. Let λu,λs be the eigenvalues of A such that |λu|>1>|λs|. They are eigenvalues for unstable direction, stable direction, respectively. We note that b0,c0 because of the conditions adbc=±1 and |λu|>1>|λs|. Take nonzero eigenvectors vu,vs for the eigenvalues λu,λs such that vu=vs=1. We set vu=(u1,u2),vs=(s1,s2)T2 as vectors. The numbers λu,λs,u1,u2,s1,s2 are all real numbers because of the hyperbolicity of the matrix A. It is easy to see that the slopes u1u2,s1s2 are irrational. We set

    rA:=vu|vs.

    Define two vectors

    v1:=vurAvs,v2:=rAvuvs.

    Lemma 2.1. For two vectors x,zT2, the following three conditions are equivalent.

    (ⅰ) (x,z)GaA.

    (ⅱ) z=x+11r2A(m,n)|v1vu for some m,nZ.

    (ⅲ) z=x+11r2A(m,n)|v2vs for some m,nZ.

    Proof. For two vectors x,zT2 regarding them as elements of R2 modulo Z2, we have (x,z)GaA if and only if

    zx+tvux+svs(modZ2) for some t,sR. (4)

    In this case, we see that tvusvs=(m,n) for some m,nZ so that

    tvusvs|vu=(m,n)|vu, (5)
    tvusvs|vs=(m,n)|vs (6)

    and we have

    t=11r2A(m,n)|v1,s=11r2A(m,n)|v2. (7)

    This shows the implications (ⅰ) (ⅱ) and (ⅲ).

    Assume that (ⅱ) holds. By putting s=11r2A(m,n)|v2, we have the equalities both (5) and (6), so that tvusvs=(m,n). Hence the equality (4) holds and we see that (x,z) belongs to the groupoid GaA. This shows that the implication (ⅱ) (ⅰ) holds, and similarly (ⅲ) (ⅰ) holds.

    Let us define an action αA:Z2Homeo(T2) in the following way. We set

    αA(m,n)(x):=x+11r2A(m,n)|v1vu,(m,n)Z2,xT2.

    For a fixed (m,n)Z2, the map xT2αA(m,n)(x)T2 is the parallel transformation along the vector 11r2A(m,n)|v1vu. Hence αA(m,n) defines a homeomorphism on the torus T2. It is clear to see that αA(m,n)αA(k,l)=αA(m+k,n+l) for (m,n),(k,l)Z2.

    Lemma 2.2. Keep the above notation.

    (ⅰ) If αA(m,n)(x)=x for some xT2, then (m,n)=(0,0). Hence the action αA:Z2Homeo(T2) is free.

    (ⅱ) For xT2, the set {αA(m,n)(x)|(m,n)Z2} is dense in T2. Hence the action αA:Z2Homeo(T2) is minimal.

    Proof. (ⅰ) Suppose that αA(m,n)(x)=x for some xT2, so that 11r2A(m,n)|v1vu=(k,l) for some (k,l)Z2. As the slope of the vector vu is irrational, we have (k,l)=(0,0) and hence (m,n)=(0,0).

    (ⅱ) Let v1=(γ1,γ2). As the slope of vs is irrational and vs|v1=0, the slope γ1γ2 of v1 is irrational, so that the set {mγ1+nγ2|m,nZ} is dense in R. Since (m,n)|v1vu=(mγ1+nγ2)vu and the set {x+tvuT2|tR} is dense in T2, we see that the set

    {x+11r2A(m,n)|v1vu(m,n)Z2}

    is dense in T2.

    The action αA:Z2Homeo(T2) induces an action of Z2 to the automorphism group Aut(C(T2)) of C(T2) by fC(T2)fαA(m,n)C(T2). We write it still αA without confusing.

    If a discrete group Γ acts freely on a compact Hausdorff space X by an action α:ΓHomeo(X), the set {(x,αγ(x))X×X|xX,γΓ} has a groupoid structure in a natural way (cf. [2], [18], [19]). The groupoid is called a transformation groupoid written X×αΓ.

    Proposition 2.3. The étale groupoid GaA is isomorphic to the transformation groupoid

    T2×αAZ2={(x,αA(m,n)(x))T2×T2|(m,n)Z2}

    defined by the action αA:Z2Homeo(T2). Hence the C-algebra C(GaA) of the groupoid GaA is isomorphic to the crossed product C(T2)×αAZ2 of C(T2) by the action αA of Z2.

    Proof. By the preceding discussions, a pair (x,z)T2 belongs to the groupoid GaA if and only if z=αA(m,n)(x) for some (m,n)Z2. Since the action αA:Z2Homeo(T2) is free, the groupoid GaA is identified with the transformation groupoid T2×αAZ2 in a natural way. By a general theory of the C-algebras of groupoids ([2], [18]), the C-algebra C(T2×αAZ2) of the groupoid T2×αAZ2 is isomorphic to the crossed product C(T2)×αAZ2.

    Remark 2.4. Define a map αA:Z2T2 by

    αA(m,n):=11r2A(m,n)|v1vu,(m,n)Z2. (8)

    It is easy to see that the étale groupoid GaA may be written

    GaA=T2×αA(Z2) (9)

    as a transformation groupoid.

    We set

    θ1:=u1s2u1s2u2s1,θ2:=u2s2u1s2u2s1, (10)
    θ3:=u1s1u1s2u2s1,θ4:=u2s1u1s2u2s1. (11)

    Lemma 2.5. The real numbers θi,i=1,2,3,4 satisfy

    θ2θ1=θ4θ3=u2u1,θ1θ3=θ2θ4=s2s1, (12)
    θ1+θ4=1. (13)

    Conversely, if real numbers ζi,i=1,2,3,4 satisfy

    ζ2ζ1=ζ4ζ3=u2u1,ζ1ζ3=ζ2ζ4=s2s1, (14)
    ζ1+ζ4=1, (15)

    then we have ζi=θi,i=1,2,3,4.

    Proof. The identities (12) and (13) are immediate. Conversely, suppose that real numbers ζi,i=1,2,3,4 satisfy (14) and (15). As ζ1=u2u1ζ2=u2u1(s2s1)ζ4, the equality (15) implies

    {u2u1(s2s1)+1}ζ4=1,

    so that

    ζ4=u2s1u1s2u2s1

    and hence

    ζ1=u1s2u1s2u2s1,ζ2=u2s2u1s2u2s1,ζ3=u1s1u1s2u2s1.

    Proposition 2.6. For x=(x1,x2)T2, we have

    αA(1,0)(x1,x2)=(x1+θ1,x2+θ2),αA(0,1)(x1,x2)=(x1+θ3,x2+θ4),

    and hence

    αA(m,n)(x1,x2)=(x1+mθ1+nθ3,x2+mθ2+nθ4)for(m,n)Z2.

    Proof. We have

    αA(m,n)(x1,x2)=(x1,x2)+11r2A(m,n)|vurAvsvu=(x1,x2)+11r2A(m,n)|(u1rAs1,u2rAs2)(u1,u2).

    In particular, for (m,n)=(1,0),(0,1), we have

    αA(1,0)(x1,x2)=(x1+11r2A(u1rAs1)u1,x2+11r2A(u1rAs1)u2),αA(0,1)(x1,x2)=(x1+11r2A(u2rAs2)u1,x2+11r2A(u2rAs2)u2).

    We put ξi=11r2A(uirAsi) for i=1,2 so that

    αA(1,0)(x1,x2)=(x1+ξ1u1,x2+ξ1u2), (16)
    αA(0,1)(x1,x2)=(x1+ξ2u1,x2+ξ2u2). (17)

    We then have

    ξ1=11r2A{u1(u1s1+u2s2)s1}=11r2A{u1(1s21)u2s2s1}=11r2A(u1s2u2s1)s2

    and similarly

    ξ2=11r2A{u2(u1s1+u2s2)s2}=11r2A{u2(1s22)u1s1s2}=11r2A(u2s1u1s2)s1.

    Hence we have ξ1ξ2=s2s1. We also have

    ξ1u1+ξ2u2=11r2A{(u1rAs1)u1+(u2rAs2)u2}=11r2A{u21+u22rA(u1s1+u2s2)}=11r2A(1r2A)=1.

    By Lemma 2.5, we have ξ1u1=θ1,ξ1u2=θ2,ξ2u1=θ3,ξ2u2=θ4, proving the desired assertion from the identities (16) and (17).

    We will next express θi,i=1,2,3,4 in terms of the matrix elements a,b,c,d of A.

    Lemma 2.7. The following identities hold.

    (ⅰ)

    aθ1+bθ2=λuθ1,aθ3+bθ4=λuθ3,cθ1+dθ2=λuθ2,cθ3+dθ4=λuθ4,

    and hence

    aθ1+bθ2+cθ3+dθ4=λu. (18)

    (ⅱ)

    aθ3bθ1=λsθ3,aθ4bθ2=λsθ4,cθ3dθ1=λsθ1,cθ4dθ2=λsθ2,

    and hence

    aθ4bθ2cθ3+dθ1=λs.

    Proof. By the identities

    [θ1θ2]=s2u1s2u2s1[u1u2],[θ3θ4]=s1u1s2u2s1[u1u2],[θ3θ1]=u1u1s2u2s1[s1s2],[θ4θ2]=u2u1s2u2s1[s1s2],

    with θ1+θ4=1, we see the desired assertions.

    Lemma 2.8.

    (ⅰ) (aθ1+bθ2)θ4=(cθ3+dθ4)θ1.

    (ⅱ) (aθ3bθ1)θ2=(cθ4+dθ2)θ3.

    Hence we have

    bθ2=cθ3.

    Proof. (ⅰ) By the first and the fourth identities in Lemma 2.7 (ⅰ), we know the identity (ⅰ). The identities of (ⅱ) is similarly shown to those of (ⅰ). By (ⅰ) and (ⅱ) with the identity θ1θ4=θ2θ3, we get bθ2=cθ3.

    Recall that Δ(A) denotes the discriminant (a+d)24(adbc) of the characteristic polynomial of the matrix A. The real number Δ(A) is positive because of the hyperbolicity of A. By elementary calculations, we see the following lemma.

    Lemma 2.9. The identities

    θ1θ4=θ2θ3,θ1+θ4=1,(aθ1+bθ2)θ4=(cθ3+dθ4)θ1,(aθ3bθ1)θ2=(cθ4+dθ2)θ3

    imply

    (θ1,θ2,θ3,θ4) (19)
    ={(12(1+|ad|Δ(A)),|ad|adcΔ(A),|ad|adbΔ(A),12(1|ad|Δ(A)))or(12(1|ad|Δ(A)),|ad|adcΔ(A),|ad|adbΔ(A),12(1+|ad|Δ(A)))ifad,(12,12cb,12bc,12)or(12,12cb,12bc,12)ifa=d. (20)

    We thus have the following theorem.

    Theorem 2.10. The C-algebra C(GaA) of the groupoid GaA for a hyperbolic matrix A=[abcd] is isomorphic to the simple C-algebra generated by four unitaries U1,U2,V1,V2 satisfying the following relations:

    U1U2=U2U1,V1V2=V2V1,V1U1=e2πiθ1U1V1,V1U2=e2πiθ2U2V1,V2U1=e2πiθ3U1V2,V2U2=e2πiθ4U2V2,

    where

    θ1=12(1+adΔ(A)),θ2=cΔ(A),θ3=bΔ(A),θ4=12(1adΔ(A)). (21)

    Hence the C-algebra C(GaA) is isomorphic to the four-dimensional non-commutative torus.

    Proof. As in Lemma 2.2, the action αA:Z2Homeo(T2) is free and minimal, hence the C-crossed product C(T2)×αAZ2 is simple. The C-crossed product is canonically identified with the C-crossed product ((C(T)C(T))×αA(1,0)Z)×αA(0,1)Z. Let U1,U2 be the unitaries in C(T)C(T) defined by U1(t,s)=e2πit,U2(t,s)=e2πis. Let V1,V2 be the implementing unitaries corresponding to the automorphisms αA(1,0),αA(0,1), respectively. By Proposition 2.6, we know the commutation relations among the unitaries U1,U2,V1,V2 for the slopes θ1,θ2,θ3,θ4 satisfying (20). The second values of (20) go to the first of (20) by substituting V1,U1 with V2,U2, respectively. The forth values of (20) go to the third of (20) by substituting V1,U1 with V1,U1, respectively. When a=d, we have Δ(A)=4bc>0 so that ±cb=cΔ(A),±bc=bΔ(A). Hence the first two of (20) include the second two of (20), so that we may unify (20) into (21).

    Since the C-algebra C(GaA) is isomorphic to a four-dimensional non-commutative torus, we know the following proposition by Slawny [24] (see also Putnam[14]).

    Proposition 2.11 (Slawny [24], Putnam [14]). The C-algebra C(GaA) has a unique tracial state.

    Remark 2.12. (ⅰ) We note that the simplicity of the algebra C(GaA) comes from a general theory of Smale space C-algebras as in [14], [17] as well as a unique existence of tracial state on it. It also follows from a general theory of crossed product C-algebras because the action αA of Z2 to Homeo(T2) is free and minimal. It has been shown that a simple higher dimensional non-commutative torus is an AT-algebra by Phillips [13].

    (ⅱ) Suppose that two hyperbolic matrices A,BGL(2,Z) commute each other. By (8) and (9), the equality αA(Z2)=αB(Z2) holds for the commuting matrices A and B, because they have the same eigenvectors. Hence we know that GaA=GaB, so that the C-algebras C(GaA) and GaB) are isomorphic.

    In this section, we will describe the trace values τ(K0(C(GaA))) of the K0-group of the C-algebra C(GaA) in terms of the hyperbolic matrix A.

    In [20], M. A. Rieffel studied K-theory for irrational rotation C-algebras Aθ with irrational numbers θ, which are called two-dimensional non-commutative tori, and proved that τ(K0(Aθ))=Z+Zθ in R, where τ is the unique tracial state on Aθ. In [6], G. A. Elliott (cf. [3], [13], [21], [24], etc.) initiated to study higher-dimensional non-commutative tori. It is well-known the K-groups of the four-dimensional non-commutative torus as in [6] which says

    K0(C(T2)×αAZ2)K1(C(T2)×αAZ2)Z8

    ([6], cf. [24]). For g=(a1,b1,a2,b2),h=(c1,d1,c2,d2)Z4, we define a wedge product ghZ4 by

    (a1,b1,a2,b2)(c1,d1,c2,d2)=(|a1c1b1d1|,|a1c1b2d2|,|a2c2b1d1|,|a2c2b2d2|)

    where |xyzw|=xwyz. Let Θ=[θjk]4j,k=1 be a 4×4 skew symmetric matrix over R. We regard the matrix Θ as a linear map from Z4Z4 to R by Θ(xy)=Θxy. Then ΘΘ:(Z4Z4)(Z4Z4)=4Z4R is defined by

    (ΘΘ)(x1x2)(x3x4)=12!2!σS4sgn(σ)Θ(xσ(1)xσ(2))Θ(xσ(3)xσ(4))

    for x1,x2,x3,x4Z4. Although we may generally define nΘ:2nZ4R, the wedge product 2nZ4=0 for n>3, so that

    exp(Θ)=1Θ12(ΘΘ)16(ΘΘΘ):evenZ4R

    becomes

    exp(Θ)=1Θ12(ΘΘ).

    Let AΘ be the C-algebra generated by four unitaries uj,j=1,2,3,4 satisfying the commutation relations ujuk=e2πiθjkukuj,j,k=1,2,3,4. The C-algebra AΘ is called the four-dimensional non-commutative torus ([6]). If Θ is non-degenerate, the algebra AΘ has a unique tracial state written τ. By Elliott's result in [6], there exists an isomorphism h:K0(AΘ)evenZ4 such that exp(Θ)h=τ, so that we have

    exp(Θ)(evenZ4)=τ(K0(AΘ)). (22)

    Proposition 3.1. Let τ be the unique tracial state on C(GaA). Then we have

    τ(K0(C(GaA)))=Z+Zθ1+Zθ2+Zθ3inR. (23)

    Proof. Take the unitaries U1,U2,V1,V2 and the real numbers θ1,θ2,θ3,θ4 as in Theorem 2.10. We set the real numbers θjk,j,k=1,2,3,4 such as θjj=θ12=θ21=θ34=θ43=0 for j=1,2,3,4 and θ13=θ4,θ14=θ3,θ23=θ2,θ24=θ1. Let u1=V2,u2=V1,u3=U2,u4=U1 so that we have the commutation relations

    ujuk=e2πiθjkukuj,j,k=1,2,3,4.

    As θ1θ4=θ2θ3, we have

    θ12θ34θ13θ24+θ14θ23=0.

    By (22) or [6] (cf. [3,2.21], [13,Theorem 3.9]), we have

    τ(K0(C(GaA)))=Z+Z(θ12θ34θ13θ24+θ14θ23)+1j<k4Zθjk=Z+Zθ1+Zθ2+Zθ3.

    Remark 3.2. (ⅰ) It is straightforward to see that the skew symmetric matrix Θ=[θjk]4j,k=1 in our setting above is non-degenerate.

    (ⅱ) Suppose that two hyperbolic toral automorphisms (T2,A) and (T2,B) are topologically conjugate. We then know that both the C-algebras C(GaA) and C(GaB) are isomorphic. Since they have unique tracial states τA and τB respectively, we see that

    τA(K0(C(GaA)))=τB(K0(C(GaB))).

    We may also find a matrix MGL(2,Z) such that AM=MB by [1]. We then directly see that the ranges τA(K0(C(GaA))) and τB(K0(C(GaB))) coincide by using the formula (23). Similarly we may directly show that the equality τA(K0(C(GaA)))=τA1(K0(C(GaA1))) by the formula (23).

    In this section, we will present some examples.

    1. A=[1110]. Since a=b=c=1,d=0, we have by Theorem 2.10,

    (θ1,θ2,θ3,θ4)=(12(1+15),15,15,12(515). (24)

    By the formula (23), we have

    τ(K0(C(GaA)))=Z+5+510Z.

    Proposition 4.1. Let A be the matrix [1110]. Put θ=12(1+15). Then the C-algebra C(GaA) is isomorphic to the tensor product AθA5θ between the irrational rotation C-algebras Aθ and A5θ with its rotation angles θ and 5θ respectively.

    Proof. Let U1,U2,V1,V2 be the generating unitaries in Theorem 2.10. Since

    (θ1,θ2,θ3,θ4)=(θ,2θ1,2θ1,1θ)

    by (24), we have

    U1U2=U2U1,V1V2=V2V1,V1U1=e2πiθU1V1,V1U2=e2πi2θU2V1,V2U1=e2πi2θU1V2,V2U2=e2πiθU2V2,

    We set

    u1=U1U22,u2=U2,v1=V1V22,v2=V2.

    It is straightforward to see that the following equalities hold

    u1u2=u2u1,v1v2=v2v1,v1u1=e2πi5θu1v1,v1u2=u2v1,v2u1=u1v2,v2u2=e2πiθu2v2.

    Since the C-algebra C(u1,u2,v1,v2) generated by u1,u2,v1,v2 coincides with C(GaA), we have

    C(GaA)C(u1,v1)C(u2,v2)A5θAθ.

    2. A=[3121]. Since a=3,b=d=1,d=2, we have by Theorem 2.10,

    (θ1,θ2,θ3,θ4)=(3+36,33,36,336)

    and

    λu=aθ1+bθ2+cθ3+dθ4=2+3,λs=aθ4bθ2cθ3+dθ1=23.

    Since θ4=1θ1, θ2=2θ3, θ1=12+θ3, the formula (23) tells us

    τ(K0(C(GaA)))=Z+Zθ1+Zθ2+Zθ3=12Z+36Z.

    Proposition 4.2. Let A1=[1110] and A2=[3121]. Then the C-algebra C(GaA1) is not isomorphic to C(GaA2).

    Proof. Since the C-algebra C(GaA) has the unique tracial state τ, the range τ(K0(C(GaA))) of τ of the K0-group K0(C(GaA)) is invariant under isomorphism class of the C-algebra. As

    τ(K0(C(GaA1)))=Z+5+510Z,τ(K0(C(GaA1)))=12Z+36Z,

    we see that τ(K0(C(GaA1)))τ(K0(C(GaA2))), so that the C-algebra C(GA1) is not isomorphic to C(GA2).

    The author would like to deeply thank the referee for careful reading and lots of helpful advices in the presentation of the paper. This work was supported by JSPS KAKENHI Grant Numbers 15K04896, 19K03537.

    [1] Johnny’s Selected Seeds. 2016. Available from: http://www.johnnyseeds.com/c-247-strawberries.aspx?source=W_fruit_ddcat_092016
    [2] Durner EF (2015) Photoperiod affects floral ontogeny in strawberry (Fragaria x ananassa Duch.) plug plants. Sci Hortic (Amsterdam) 194: 154-159. doi: 10.1016/j.scienta.2015.08.006
    [3] Durner EF, Poling EB, Maas JL (2002) Recent advances in strawberry plug transplant technology. HortTechnology 12:545-550.
    [4] Durner EF, Barden JA, Himelrick DG, et al. (1984) Photoperiod and temperature effects on flower and runner development in day neutral, Junebearing and everbearing strawberries. J Amer Soc Hort Sci 109: 396 400.
    [5] Durner EF (1999) Winter greenhouse strawberry production using conditioned plug plants. HortScience 34: 615-616.
    [6] Deyton D, Sams C, Takeda F, et al. (2009) Off-season greenhouse strawberry production. HortScience 44:1002.
    [7] Paparozzi ET (2013) The challenges of growing strawberries in the greenhouse. HortTechnology 23: 800-802.
    [8] Takeda F (2000) Out-of-season greenhouse strawberry production in soilless substrate. Adv Strawb Res 18: 4-15.
    [9] Takeda F, Hokanson SC (2002) Effects of transplant conditioning on ‘Chandler’ strawberry performance in a winter greenhouse production system. In: S.C. Hokanson and A.R. Jamieson (eds.). Strawberry research to 2001. ASHS Press, Alexandria VA. 132-135.
    [10] Lieten F (1993) Methods and strategies of strawberry forcing in central Europe: Historical perspectives and recent developments. Acta Hortic 348: 158-170.
    [11] Neri D, Baruzzi G, Massetani F, et al. (2012) Strawberry production in forced and protected culture in Europe as a response to climate change. Can J Plant Sci 92: 1021-1036. doi: 10.4141/cjps2011-276
    [12] Yamasaki A (2013) Recent progress if strawberry year-round production technology in Japan. Jpn Agric Res Q 47:37-42. doi: 10.6090/jarq.47.37
    [13] Poling EB, Parker K (1990) Plug production of strawberry transplants. Adv Strawb Prod 9: 37-39.
    [14] Demchak K (2009) Small fruit production in high tunnels. HortTechnology 19: 44-49.
    [15] Kadir S, Carey E, Ennahli S (2006) Influence of high tunnel and field conditions on strawberry growth and development. HortScience 41: 329-335.
    [16] Ballington JR, Poling EB, Olive K (2008) Day-neutral strawberry production for season extension in the midsouth. HortScience 43: 1982-1986.
    [17] Rowley D, Black B, Drost D, et al. (2011) Late-season strawberry production using day-neutral cultivars in high elevation high tunnels. HortScience 46: 1480-1485.
    [18] Takeda F, Newell M (2006) A method for increasing fall flowering in short-day ‘Carmine’ strawberry. HortScience 41: 480-481.
    [19] Black BL, Swartz HJ, Deitzer GF, et al. (2005) The effects of conditioning strawberry plug plants under altered red/far-red light environments. HortScience 40: 1263-1267.
    [20] Durner EF (2016) Enhanced Flowering of The F1 Long-day Strawberry Cultivar ‘Elan’ via Nitrogen and Daylength Manipulation. AIMS Agric Food 1: 4-19. doi: 10.3934/agrfood.2016.1.4
    [21] Bentvelsen GC, Bouw B, Veldhuyzen E, et al. (1996) Breeding strawberries (Fragaria ananassa Duch.) from seed. Acta Hortic 439: 149-153.
    [22] Bentvelsen GC, Bouw B (2006) Breeding ornamental strawberries. Acta Hortic 708: 455-457.
    [23] Bentvelsen GC, Bouw B (2002) Breeding strawberry F1-hybrids for vitamin C and sugar content. Acta Hortic 567: 813-814.
    [24] Bentvelsen GC, Souillat D. Strawberry F1 hybrids in very early greenhouse production with grow light. 2014. Available from: http://hoogstraten.eu/congress/posters_2013/Strawberry%20F1%20hybrids%20in%20very%20early%20greenhouse%20production%20wiht%20grow%20light.pdf
    [25] Heide OM, Stavang JA, Sonsteby A (2013) Physiology and genetics of flowering in cultivated and wild strawberries – a review. J Hortic Sci Biotechnol 88: 1-18. doi: 10.1080/14620316.2013.11512930
    [26] Larson KD (1994) Strawberry. In: Handbook of environmental physiology of fruit crops. Volume I. eds. B Schaffer and P.C. Anderson. CRC Press, Boca Raton, FL. 271-297.
    [27] Sonsteby A, Opstad N, Myrheim U, et al. (2009) Interaction of short day and timing of nitrogen fertilization on growth and flowering of ‘Korona’ strawberry (Fragaria x ananassa Duch.). Sci Hortic (Amsterdam) 123: 204-209. doi: 10.1016/j.scienta.2009.08.009
    [28] Desmet EM, Verbraeken L, Baets W (2009) Optimisation of nitrogen fertilization prior to and during flowering process on performance of short day strawberry ‘Elsanta’. Acta Hortic 842: 675-678.
    [29] Yamasaki A, Yano T (2009) Effect of supplemental application of fertilizers on flower bud initiation and development of strawberry – possible role of nitrogen. Acta Hortic 842: 765-768.
    [30] Sonsteby A, Opstad N, Heide OM (2013) Environmental manipulation for establishing high yield potential of strawberry forcing plants. Sci Hortic (Amsterdam) 157: 65-73. doi: 10.1016/j.scienta.2013.04.014
    [31] Lieten F (2002) The effect of nutrition prior to and during flower differentiation on phyllody and plant performance of short day strawberry ‘Elsanta’. Acta Hortic 567: 345-348.
    [32] Sonsteby A, Heide OM (2007) Quantitative long-day flowering response in the perpetual-flowering F1 strawberry cultivar Elan. J Hortic Sci Biotechnol 82: 266-274. doi: 10.1080/14620316.2007.11512228
    [33] Sonsteby A, Heide OM (2007) Long-day control of flowering in everbearing strawberries. J Hortic Sci Biotechnol 82: 875-884. doi: 10.1080/14620316.2007.11512321
    [34] Wobbrock JO, Findlater L, Gergle D, et al. (2011) The aligned rank transform for nonparametric factorial analyses using only anova procedures. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM. 143-146.
    [35] Le Miere P, Hadley P, Darby J, et al. (1996) The effect of temperature and photoperiod on the rate of flower initiation and the onset of dormancy in strawberry (Fragaria x ananassa Duch.). J Hortic Sci 71: 261-2711.
    [36] Lieten F (2002) The effect of nutrition prior to and during flower differentiation on phyllody and plant performance of short day strawberry ‘Elsanta’. Acta Hortic 567: 345-348.
    [37] van Delm T, Melis P, Stoffels K, et al. (2013) Pre-harvest night-interruption on everbearing cultivars in out-of-soil strawberry cultivation in Belgium. Int J Fruit Sci 13: 217-226. doi: 10.1080/15538362.2012.698176
    [38] Anderson HM, Guttridge CG (1982) Strawberry truss morphology and the fate of high-order flower buds. Crop Res 22: 105-122.
    [39] Hytonen T, Palonen P, Mouhu K, et al. (2004) Crown branching and cropping potential in strawberry (Fragaria x ananassa Duch.) can be enhanced by daylength treatments. J Hortic Sci Biotechnol 79: 466-471.
    [40] Guttridge CG (1985) Fragaria x ananassa. In: Halevy, A.H. (Ed.), Handbook of Flowering, vol 3. CRC Press, Boca Raton, FL. 16-33.
    [41] Mochizuki T (1995) Past and present strawberry breeding programs in Japan. Adv Strawb Res 14: 9-17.
    [42] Mochizuki T, Yoshida Y, Yanagi T, et al. (2009) Forcing culture of strawberry in Japan - production technology and cultivars. Acta Hortic 842: 107-110.
    [43] Strik BC (1985) Flower bud initiation in strawberry cultivars. Fruit Var J 39: 59.
    [44] Battey NH, Le Miere P, Tehranifar A, et al. (1998) Genetic and environmental control of flowering in strawberry. In: K. E. Cockshull, D. Gray, G. B. Seymour, and B. Thomas, eds. Genetic and environmental manipulation of horticultural crops. CAB International, Wallingford, UK. 111-131.
    [45] Anderson HM, Guttridge CG (1982) Strawberry truss morphology and the fate of high-order flower buds. Crop Res 22: 105-122.
    [46] Van den Muijzenberg EWB (1942) De invloed van licht en temperatuur op de periodieke ontwikkeling van de aardbei en de beteekenis daarvan voor de teelt [The influence of light and temperature on the periodic development of the strawberry and its significance in cultivation]. Ph.D. thesis, Laboratorium voor Tuinbouwplantenteelt, Wageningen, the Netherlands. 160.
    [47] Fujimoto K, Kimura M (1970) Studies on flowering of strawberry. III. Effect of nitrogen on flower bud differentiation and development. Abstracts of the Japanese Society for Horticultural Science Spring Meeting 174-175 (In Japanese).
    [48] Furuya S, Yamashita M, Yamasaki A (1988) Effects of nitrogen content on the flower bud initiation induced by chilling under dark condition in strawberries. Bullet Natl Res Inst Veg Ornam Plants Tea Ser D.
    [49] Matsumoto O (1991) Studies on the control of flower initiation and dormancy in the cultivation of strawberry, Fragaria X ananassa Duch. Spec Bullet Yamaguchi Agric Exp Stn 31: 1102.
    [50] Yamasaki A, Yoneyama T, Tanaka F, et al. (2002) Tracer studies on the allocation of carbon and nitrogen during flower induction of strawberry plants as affected by the nitrogen level. Acta Hortic 567: 349-352.
    [51] Yoshida Y (1992) Studies on flower and fruit development in strawberry with special reference to fruit malformation in ‘Ai-Berry’. Memo Fac Agric Kagawa Univ 57: 194.
  • This article has been cited by:

    1. Xiangqi Qiang, Chengjun Hou, Continuous Orbit Equivalence for Automorphism Systems of Equivalence Relations, 2023, -1, 1027-5487, 10.11650/tjm/231105
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5809) PDF downloads(1442) Cited by(5)

Figures and Tables

Tables(11)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog