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Citation: Erno Widayanto, Agoes Soehardjono, Wisnumurti Wisnumurti, Achfas Zacoeb. The effect of vibropressing compaction process on the compressive strength based concrete paving blocks[J]. AIMS Materials Science, 2020, 7(3): 203-216. doi: 10.3934/matersci.2020.3.203
[1] | Pengliang Xu, Xiaomin Tang . Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, 2021, 29(4): 2771-2789. doi: 10.3934/era.2021013 |
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[5] | Shanshan Liu, Abdenacer Makhlouf, Lina Song . The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras. Electronic Research Archive, 2022, 30(8): 2748-2773. doi: 10.3934/era.2022141 |
[6] | Hongliang Chang, Yin Chen, Runxuan Zhang . A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29(3): 2457-2473. doi: 10.3934/era.2020124 |
[7] | Margarida Camarinha . A natural 4th-order generalization of the geodesic problem. Electronic Research Archive, 2024, 32(5): 3396-3412. doi: 10.3934/era.2024157 |
[8] | Jinguo Jiang . Algebraic Schouten solitons associated to the Bott connection on three-dimensional Lorentzian Lie groups. Electronic Research Archive, 2025, 33(1): 327-352. doi: 10.3934/era.2025017 |
[9] | 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 |
[10] | Ying Hou, Liangyun Chen, Keli Zheng . Super-bimodules and O-operators of Bihom-Jordan superalgebras. Electronic Research Archive, 2024, 32(10): 5717-5737. doi: 10.3934/era.2024264 |
The Schrödinger-Virasoro algebra is an infinite-dimensional Lie algebra that was introduced (see, e.g., [10]) in the context of non-equilibrium statistical physics. In [21], the author give a representation of the Schrödinger-Virasoro algebra by using vertex algebras, and introduced an extension of the Schrödinger-Virasoro algebra. To be precise, for
{Li,Hj,Ii|i∈Z,j∈ε+Z} |
and Lie brackets
[Lm,Ln]=(m−n)Lm+n,[Lm,Hn]=(12m−n)Hm+n,[Lm,In]=−nIm+n,[Hm,Hn]=(m−n)Im+n,[Hm,In]=[Im,In]=0. |
The Lie algebra
Post-Lie algebras were introduced around 2007 by B. Vallette [22], who found the structure in a purely operadic manner as the Koszul dual of a commutative trialgebra. Post-Lie algebras have arose the interest of a great many authors, see [4,5,12,13]. One of the most important problems in the study of post-Lie algebras is to find the post-Lie algebra structures on the (given) Lie algebras. In [13,18,20], the authors determined all post-Lie algebra structures on
In this paper, we shall study the graded post-Lie algebra structures on the Schrödinger-Virasoro algebra. We only study the twisted Schrödinger-Virasoro algebra
Throughout this paper, we denote by
The paper is organized as follows. In Section 2, we give general results on post-Lie algebras and some lemmas which will be used to our proof. In Section 3, we completely characterize the graded post-Lie algebra structures on Schrödinger-Virasoro algebra
We will give the essential definitions and results as follows.
Definition 2.1. A post-Lie algebra
[x,y]▹z=x▹(y▹z)−y▹(x▹z)−⟨x,y⟩▹z, | (1) |
x▹[y,z]=[x▹y,z]+[y,x▹z] | (2) |
for all
Suppose that
τ(x▹1y)=τ(x)▹2τ(y),∀x,y∈L. |
Remark 1. The left multiplications of the post-Lie algebra
Lemma 2.2. [15] Denote by
Der(S)=Inn(S)⊕CD1⊕CD2⊕CD3 |
where
D1(Ln)=0,D1(Hn)=Hn,D1(In)=2In,D2(Ln)=nIn,D2(Hn)=0,D2(In)=0,D3(Ln)=In,D3(Hn)=0,D3(In)=0. |
Since the Schrödinger-Virasoro algebra
Lm▹Ln=ϕ(m,n)Lm+n, | (3) |
Lm▹Hn=φ(m,n)Hm+n, | (4) |
Lm▹In=χ(m,n)Im+n, | (5) |
Hm▹Ln=ψ(m,n)Hm+n, | (6) |
Hm▹Hn=ξ(m,n)Im+n, | (7) |
Im▹Ln=θ(m,n)Im+n, | (8) |
Hm▹In=Im▹Hn=Im▹In=0, | (9) |
for all
We start with the crucial lemma.
Lemma 3.1. There exists a graded post-Lie algebra structure on
ϕ(m,n)=(m−n)f(m), | (10) |
φ(m,n)=(m2−n)f(m)+δm,0μ, | (11) |
χ(m,n)=−nf(m)+2δm,0μ, | (12) |
ψ(m,n)=−(n2−m)h(m), | (13) |
ξ(m,n)=(m−n)h(m), | (14) |
θ(m,n)=mg(m)+δm,0na, | (15) |
(m−n)(f(m+n)(1+f(m)+f(n))−f(n)f(m))=0, | (16) |
(m−n)δm+n,0μ(1+f(m)+f(n))=0, | (17) |
(m2−n)(h(m+n)(1+f(m)+h(n))−f(m)h(n))=0, | (18) |
nδm+n,0a(1+f(m)+g(n))=0, | (19) |
n(m+n)(g(m+n)(1+f(m)+g(n))−f(m)g(n)) =δn,0m2a(f(m)−g(m)), | (20) |
(m−n)δm+n,0a(1+h(m)+h(n))=0, | (21) |
(m−n)(g(m+n)(1+h(m)+h(n))−h(m)h(n))=0. | (22) |
Proof. Suppose that there exists a graded post-Lie algebra structure satisfying (3)-(9) on
x▹y=(adψ(x)+α(x)D1+β(x)D2+γ(x)D3)(y)=[ψ(x),y]+α(x)D1(y)+β(x)D2(y)+γ(x)D3(y) |
where
Lm▹Ln=[ψ(Lm),Ln]+β(Lm)nIn+γ(Lm)In=ϕ(m,n)Lm+n, | (23) |
Lm▹Hn=[ψ(Lm),Hn]+α(Lm)Hn=φ(m,n)Hm+n, | (24) |
Lm▹In=[ψ(Lm),In]+α(Lm)2In=χ(m,n)Im+n, | (25) |
Hm▹Ln=[ψ(Hm),Ln]+β(Hm)nIn+γ(Hm)In=ψ(m,n)Hm+n, | (26) |
Hm▹Hn=[ψ(Hm),Hn]+α(Hm)Hn=ξ(m,n)Im+n, | (27) |
Hm▹In=[ψ(Hm),In]+α(Hm)2In=0, | (28) |
Im▹Ln=[ψ(Im),Ln]+β(Im)nIn+γ(Im)In=θ(m,n)Im+n, | (29) |
Im▹Hn=[ψ(Im),Hn]+α(Im)Hn=0, | (30) |
Im▹In=[ψ(Im),In]+α(Im)2In=0. | (31) |
Let
ψ(Lm)=∑i∈Za(m)iLi+∑i∈Zb(m)iHi+∑i∈Zc(m)iIi,ψ(Hm)=∑i∈Zd(m)iLi+∑i∈Ze(m)iHi+∑i∈Zf(m)iIi,ψ(Im)=∑i∈Zg(m)iLi+∑i∈Zh(m)iHi+∑i∈Zx(m)iIi |
where
The "if'' part is a direct checking. The proof is completed.
Lemma 3.2. Let
g(n),h(n)∈{0,−1}for everyn≠0. | (32) |
Proof. By letting
Lemma 3.3. Let
g(Z)=h(Z)=0org(Z)=h(Z)=−1. |
Proof. Since
a(1+g(−1))=0. | (33) |
By letting
(m2−n)(h(m+n)(1+h(n))=0, | (34) |
n(m+n)(g(m+n)(1+g(n))=0, | (35) |
(m−n)(g(m+n)−h(m)h(n)+h(m)g(m+n)+h(n)g(m+n))=0. | (36) |
We now prove the following four claims:
Claim 1. If
By (34) with
Claim 2. If
By (34) with
Claim 3. If
By (35) with
Claim 4. If
By (35) with
Now we consider the values of
Case i. If
Case ii. If
Case iii. If
Case iv. If
Lemma 3.4. Let
(i)
(ii)
(iii)
Proof. By
h(m+n)(h(n)+1)=0 if m⩽1,m2−n≠0, | (37) |
g(m+n)(g(n)+1)=0 if m⩽1,n≠0,m+n≠0, | (38) |
g(m+n)(1+h(m)+h(n))=h(m)h(n) if m≠n. | (39) |
We first prove the following six claims:
Claim 1. If
By (37) with
Claim 2. If
By (37) with
Claim 3. If
By (37) with
Claim 4. If
By (37) with
Next, similar to Claims 1 and 3, we from (38) obtain the following claims.
Claim 5. If
Claim 6. If
Now we discuss the values of
Case i. When
By Claim 1 we have
Case ii. When
By Claim 2 we have
Case iii. When
By Claims 3 and 4 we have
It is easy to check that the values of
Lemma 3.5. Let
(i)
(ii)
(iii)
for some
(iv)
Proof. Take
h(0)(1+f(−n)+h(n))=f(−n)h(n), for all n≠0, | (40) |
a(1+f(−n)+g(n))=0, for all n≠0, | (41) |
a(1+h(−n)+h(n))=0, for all n≠0, | (42) |
g(0)(1+h(−n)+h(n))=h(−n)h(n), for all n≠0. | (43) |
Note that
h(n)(h(m+n)+1)=0 for all m>0,m2−n≠0; | (44) |
h(m+n)(h(n)+1)=0 for all m<0,m2−n≠0; | (45) |
g(n)(g(m+n)+1)=0 for all m>0,n≠0,m+n≠0; | (46) |
g(m+n)(g(n)+1)=0 for all m<0,n≠0,m+n≠0; | (47) |
g(m+n)(1+h(m)+h(n))=h(m)h(n) for all m≠n. | (48) |
For any
Claim 1. If
In fact, by (44) with
Claim 2. If
This proof is similar to Claim 1 by using (44) and (45). Also, similar to Claims 1 and 2, by (46) and (47) we can obtain the following two claims:
Claim 3. If
Claim 4. If
According to (32), by Claims 1 and 2,
(1)
(2)
(3)
(4)
In view of the above result, the next proof will be divided into the following cases.
Case i. When
By taking
Case ii. When
By taking
Case iii. When
By (48) we see that
Case iv. When
Note that
Lemma 3.6. Let
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Proof. The proof of the "if" direction can be directly verified. We now prove the "only if" direction. In view of
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When
When
When
Lemma 3.7. Let (P(ϕi,φi,χi,ψi,ξi,θi),▹i), i=1,2 be two algebras with the same linear space as S and equipped with C-bilinear products x▹iy such that
Lm▹iLn=ϕi(m,n)Lm+n,Lm▹iHn=φi(m,n)Hm+n,Lm▹iIn=χi(m,n)Im+n,Hm▹iLn=ψi(m,n)Hm+n,Hm▹iHn=ξi(m,n)Im+n,Im▹iLn=θi(m,n)Im+n,Hm▹iIn=Im▹iHn=Im▹iIn=0 |
for all m,n∈Z, where ϕi,φi,χi,ψi,ξi,θi, i=1,2 are complex-valued functions on Z×Z. Furthermore, let τ:P(ϕ1,φ1,χ1,ψ1,ξ1,θ1)→P(ϕ2,φ2,χ2,ψ2,ξ2,θ2) be a linear map determined by
τ(Lm)=−L−m,τ(Hm)=−H−m,τ(Im)=−I−m |
for all
{ϕ2(m,n)=−ϕ1(−m,−n);φ2(m,n)=−φ1(−m,−n);χ2(m,n)=−χ1(−m,−n);ψ2(m,n)=−ψ1(−m,−n);ξ2(m,n)=−ξ1(−m,−n);θ2(m,n)=−θ1(−m,−n). | (49) |
Proof. Clearly,
τ(Lm▹iLn)=−ϕi(m,n)L−(m+n),τ(Lm▹iHn)=−φi(m,n)H−(m+n),τ(Lm▹iIn)=−χi(m,n)I−(m+n),τ(Hm▹iLn)=−ψi(m,n)H−(m+n),τ(Hm▹iHn)=−ξi(m,n)I−(m+n),τ(Im▹iLn)=−θi(m,n)I−(m+n) |
for
The remainder is to prove that
τ(Lm▹1Ln)=−ϕ1(m,n)L−(m+n)=ϕ2(−m,−n)L−(m+n)=τ(Lm)▹2τ(Ln),τ(Lm▹1Hn)=−φ1(m,n)H−(m+n)=φ2(−m,−n)H−(m+n)=τ(Lm)▹2τ(Hn),τ(Lm▹1In)=−χ1(m,n)I−(m+n)=χ2(−m,−n)I−(m+n)=τ(Lm)▹2τ(In),τ(Hm▹1Ln)=−ψ1(m,n)H−(m+n)=ψ2(−m,−n)H−(m+n)=τ(Hm)▹2τ(Ln),τ(Hm▹1Hn)=−φ1(m,n)I−(m+n)=φ2(−m,−n)I−(m+n)=τ(Hm)▹2τ(Hn), |
τ(Im▹1Ln)=−θ1(m,n)I−(m+n)=ϕ2(−m,−n)I−(m+n)=τ(Im)▹2τ(Ln) |
and
Theorem 3.8. A graded post-Lie algebra structure on
where
Proof. Suppose that
Conversely, every type of the
Finally, by Lemma 3.7 with maps
The Rota-Baxter algebra was introduced by the mathematician Glen E. Baxter [2] in 1960 in his probability study, and was popularized mainly by the work of Rota [G. Rota1, G. Rota2] and his school. Recently, the Rota-Baxter algebra relation were introduced to solve certain analytic and combinatorial problem and then applied to many fields in mathematics and mathematical physics (see [6,7,19,23] and the references therein). Now let us recall the definition of Rota-Baxter operator.
Definition 4.1. Let
[R(x),R(y)]=R([R(x),y]+[x,R(y)])+λR([x,y]),∀x,y∈L. | (50) |
Note that if
In this section, we mainly consider the homogeneous Rota-Baxter operator
R(Lm)=f(m)Lm, R(Hm)=h(m)Hm, R(Im)=g(m)Im | (51) |
for all
Lemma 4.2. (see [1]) Let
Theorem 4.3. A homogeneous Rote-Baxrer operator
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
for all
Proof. In view of Lemma 4.2, if we define a new operation
Lm▹Ln=[R(Lm),Ln]=(m−n)f(m)Lm+n, | (52) |
Lm▹Hn=[R(Lm),Hn]=(m2−n)f(m)Hm+n, | (53) |
Lm▹In=[R(Lm),In]=−nf(m)Im+n, | (54) |
Hm▹Ln=[R(Hm),Ln]=−(n2−m)h(m)Hm+n, | (55) |
Hm▹Hn=[R(Hm),Hn]=(m−n)h(m)Im+n, | (56) |
Im▹Ln=[R(Im),Ln]=mg(m)Im+n | (57) |
and
A similar discussion to Lemma 3.1 gives
(m−n)(f(m+n)−f(n)f(m)+f(m)f(m+n)+f(n)f(m+n))=0,(m2−n)(h(m+n)−f(m)h(n)+f(m)h(m+n)+h(n)h(m+n))=0,n(m+n)(g(m+n)(1+f(m)+g(n))−f(m)g(n))=0,(m−n)(g(m+n)−h(m)h(n)+h(m)g(m+n)+h(n)g(m+n))=0. |
From this we conclude that Equations (10)-(22) hold with
The natural question is: how we can characterize the Rota-Baxter operators of weight zero on the Schrödinger-Virasoro
Definition 4.4. A pre-Lie algebra
(x▹y)▹z−x▹(y▹z)=(y▹x)▹z−y▹(x▹z),∀x,y,z∈A. | (58) |
As a parallel result of Lemma 4.2, one has the following conclusion.
Proposition 1. (see [8]) Let
Using a similar method on classification of Rota-Baxter operators of weight
We would like to express our sincere thanks to the anonymous referees for their careful reading and valuable comments towards the improvement of this article.
[1] | Shackel B (2003) The challenges of concrete block paving as a mature technology. Proceedings of 7th International Conference Concrete Block Paving, 1-9. |
[2] | Nevill AM, Brooks JJ (2010) Concrete Technology, 2nd Eds., England: Pearson Education Limited. |
[3] |
Penteado CSG, de Carvalho EV, Lintz RCC (2016) Reusing ceramic tile polishing waste in paving block manufacturing. J Clean Prod 112: 514-520. doi: 10.1016/j.jclepro.2015.06.142
![]() |
[4] |
Wattanasiriwech D, Saiton A, Wattanasiriwech S (2009) Paving blocks from ceramic tile production waste. J Clean Prod 17: 1663-1668. doi: 10.1016/j.jclepro.2009.08.008
![]() |
[5] |
Uygunolu T, Topcu IB, Gencel O, et al. (2012) The effect of fly ash content and types of aggregates on the properties of pre-fabricated concrete interlocking blocks (PCIBs). Constr Build Mater 30: 180-187. doi: 10.1016/j.conbuildmat.2011.12.020
![]() |
[6] |
Gencel O, Ozel C, Koksal F, et al. (2012) Properties of concrete paving blocks made with waste marble. J Clean Prod 21: 62-70. doi: 10.1016/j.jclepro.2011.08.023
![]() |
[7] | Agyeman S, Obeng-ahenkora NK, Assiamah S, et al. (2019) Exploiting recycled plastic waste as an alternative binder for paving blocks production. Case Stud Constr Mater 11: e00246. |
[8] | Udawattha C, Galabada H, Halwatura R (2017) Mud concrete paving block for pedestrian pavements. Case Stud Constr Mater 7: 249-262 |
[9] |
De Silva P, Sagoe-Crenstil K, Sirivivatnanon V (2007) Kinetics of geopolymerization: role of Al2O3 and SiO2. Cement Concrete Res 37: 512-518 doi: 10.1016/j.cemconres.2007.01.003
![]() |
[10] |
Arslan B, Kamas T (2017) Investigation of aggregate size effects on the compressive behavior of concrete by electromechanical and mechanical impedance spectroscopy. Procedia Struct Integr 5: 171-178. doi: 10.1016/j.prostr.2017.07.093
![]() |
[11] | Ling T, Nor H, Mudiyono R (2006) The effect of cement and water cement ratio on concrete paving block. Constr Build Environ 3: 26-27. |
[12] | Baskaran K, Gopinath K (2013) Study on applicability of ACI and DOE mix design methods for paving blocks. Annual Transactions of Institution of Engineers, Sri Lanka, 127-134. |
[13] |
Xuan D, Zhan B, Poon CS (2016) Development of a new generation of eco-friendly concrete blocks by accelerated mineral carbonation. J Clean Prod 133: 1235-1241. doi: 10.1016/j.jclepro.2016.06.062
![]() |
[14] | Djamaluddin AR, Caronge MA, Tjaronge MW, et al. (2020) Evaluation of sustainable concrete paving blocks incorporating processed waste tea ash. Case Stud Constr Mater 12: e00325. |
[15] |
Sulistyana P, Widoanindyawati V, Pratamab MMD (2014) The influence of compression applied during production to the compression strength of dry concrete: An experimental study. Procedia Eng 95: 465-472. doi: 10.1016/j.proeng.2014.12.206
![]() |
[16] | ACI Committee 309 (2011) Behavior of fresh concrete during vibration. |
[17] |
Xiao YJ, Liu R, Song HP, et al. (2015) The characteristics of perlite sound absorption board formed by vibration molding. Open Mater Sci J 9: 39-42. doi: 10.2174/1874088X01509010039
![]() |
[18] | Boral limited (2006) DS2006 compaction of concrete. Available from: https://www.boral.com/news-announcements/management-presentations. |
[19] | Badan Standardisasi Nasional (1996) Bata beton (paving block). SNI 03-0691-1996. |
[20] | Iffat S (2015) Relation between density and compressive strength of hardened concrete. Concrete Res Lett 6: 182-189. |
[21] | Wersall C (2016) Frequency optimization of vibratory rollers and plates for compaction of granular soil. Available from: http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A929931&dswid=1941. |
[22] |
Koh HB, Yeoh D, Shahidan S (2017) Effect of re-vibration on the compressive strength and surface hardness of concrete. IOP Conference Series: Materials Science and Engineering, 271: 012057. doi: 10.1088/1757-899X/271/1/012057
![]() |
[23] | Arslan ME, Yozgat E, Pul S, et al. (2011) Effects of vibration time on strength of ordinary and high performance concrete. Proceedings of the 4th WSEAS international conference on Energy and development-environment-biomedicine, 270-274. |
[24] | Kovalska A, Auzins J (2011) Investigation of vibropressing process technology. Proceedings of the 10th International Scientific Conference, 26: 408-412. |
1. | Zhongxian Huang, Biderivations of the extended Schrödinger-Virasoro Lie algebra, 2023, 8, 2473-6988, 28808, 10.3934/math.20231476 | |
2. | Ivan Kaygorodov, Abror Khudoyberdiyev, Zarina Shermatova, Transposed Poisson structures on not-finitely graded Witt-type algebras, 2025, 31, 1405-213X, 10.1007/s40590-024-00702-8 |
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