
Cruise holidays have become increasingly popular in the past two decades, with passenger numbers increasing every year. The global COVID-19 pandemic resulted in several cruise ships being held in quarantine or stranded at sea with mass disruption and cancelled holidays for millions of vacationers. The pandemic highlights the significance of risk perceptions as risk influences travel decision-making. Little research exists on risk perceptions in ocean cruising, or how risk potentially influences tourists' decision-making for a cruise as a holiday. Findings revealed a cruise is perceived as a safe holiday, but health risks are a significant concern. Non-cruisers perceive more risk in getting sick onboard, and cruisers develop strategies to minimize risks, and both groups acknowledge risk is inherent in travel. Findings reveal critical insight into how both cruisers and non-cruisers interpret health and safety risks in cruising, and is a significant empirical contribution to understanding risk in relation to cruising.
Citation: Jennifer Holland. Risk perceptions of health and safety in cruising[J]. AIMS Geosciences, 2020, 6(4): 422-436. doi: 10.3934/geosci.2020023
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Cruise holidays have become increasingly popular in the past two decades, with passenger numbers increasing every year. The global COVID-19 pandemic resulted in several cruise ships being held in quarantine or stranded at sea with mass disruption and cancelled holidays for millions of vacationers. The pandemic highlights the significance of risk perceptions as risk influences travel decision-making. Little research exists on risk perceptions in ocean cruising, or how risk potentially influences tourists' decision-making for a cruise as a holiday. Findings revealed a cruise is perceived as a safe holiday, but health risks are a significant concern. Non-cruisers perceive more risk in getting sick onboard, and cruisers develop strategies to minimize risks, and both groups acknowledge risk is inherent in travel. Findings reveal critical insight into how both cruisers and non-cruisers interpret health and safety risks in cruising, and is a significant empirical contribution to understanding risk in relation to cruising.
Consider a non trivial group A and take an element t not in A. Consider an equation,
s(t)=a1tp1a2tp2...antpn=1, |
where, ajϵA, pj = ±1 and aj≠ 1 if it appears between two cyclically adjacent t having exponents with different signs. The equation s(t)=1 is solvable over the group A if it has a solution in some group extension of A. Precisely, let G be a group containing A, if there exist an injective homomorphism, say λ:A→G, such that, g1gp1g2gp2⋯gngpn=1 in G, where gj=λ(aj) and g∈G then the equation s(t)=1 is solvable over A. A conjecture stated in [2] by Levin asserts that if A is torsion- free, then every equation over A is solvable. Bibi and Edjvet [1], using in particular results of Prishchepov [3], has shown that the conjecture is true for equations of length at most seven. Here we prove the following: Theorem: The singular equation,
s(t)=atbt−1ctdtet−1ftgt−1ht−1=1, |
of length eight is solvable over any torsion-free group.
Levin showed that any equation over a torsion-free group is solvable if the exponent sum p1+p2⋯+pn is same as length of the equation. Stallings proved that any equation of the form a1ta2t⋯amtb1t−1b2t−1⋯bnt−1=1, is solvable over torsion-free group. Klyachko showed that any equation with exponent sum ±1 is solvable over torsion free group.
All necessary definitions concerning relative presentations and pictures can be found in [4]. A relative group presentation P is a triplet <A,t|r> where A is a group, t is disjoint from A and r is a set of cyclically reduced words in A∗<t>. If the relative presentation P is orientable and aspherical then the natural homomorphism A→G is injective. P is orientable and aspherical implies s(t)=1 is solvable. We use weighttest to establish asphericity. The star graph Γ of a relative presentation P=<A,t|r> is a directed graph with vertex set {t,t−1} and edges set r∗, the set of all cyclic permutations of elements of {r,r−1} which begin with t or t−1. S∈r∗ write S=Tαj where αj∈A and T begins and ends with t or t−1. The initial function i(S) is the inverse of last symbol of T and terminal function τ(S) is the first symbol of T. The labeling function on the edges is defined by π(S)=αj and is extended to paths in the usual way. A weight function θ on the star graph Γ is a real valued function given to the edges of Γ. If d is an edge of Γ, then θ(d) = θ(ˉd). A weight function θ is aspherical if the following three conditions are satisfied:
(1) Let R ϵr be cyclically reduced word, say R=tϵ11a1...tϵnnan, where ϵi = ±1 and ai∈ A. Then,
n∑ι=1(1−θ(tϵιιaι...tϵnnantϵ11a1...tϵι−1ι−1aι−1))≥2. |
(2) Each admissible cycle in Γ has weight at least 2.
(3) Each edge of Γ has a non-negative weight.
The relative presentation P is aspherical if Γ admits an aspherical weight function. For convenience we write s(t)=1 as,
s(t)=atbt−1ctdtet−1ftgt−1ht−1=1, |
where a,b,c,e,f,g∈A/{1} and d,h∈A. Moreover, by applying the transformation y=td, if necessary, it can be assumed without any loss that d=1 in A.
(1) Since c(2, 2, 3, 3, 3, 3, 3, 3) = 0, so at least 3 admissible cycles must be of order 2.
(2) ac and ac−1 implies c2 = 1, a contradiction.
(3) af and af−1 implies f2 = 1, a contradiction.
(4) be and be−1 implies e2 = 1, a contradiction.
(5) bg and bg−1 implies g2 = 1, a contradiction.
(6) If two of ac, af and cf−1 are admissible then so is the third.
(7) If two of ac−1, af−1 and cf−1 are admissible then so is the third.
(8) If two of ac,cf and af−1 are admissible then so is the third.
(9) If two of ac−1, af and cf are admissible then so is the third.
(10) If two of be,eg, bg−1 are admissible then so is the third.
(11) If two of be−1, eg and bg are admissible then so is the third.
(1) a=c−1,b=e−1
(2) a=c−1,b=g−1
(3) a=c−1,e=g−1
(4) a=c−1,d=h−1
(5) a=c−1,b=e
(6) a=c−1,b=g
(7) a=c,b=g−1
(8) a=c,e=g−1
(9) a=c,d=h−1
(10) a=c,b=e
(11) a=c,b=g
(12) a=c,e=g
(13) a=f−1,b=g−1
(14) a=f−1,d=h−1
(15) a=f−1,b=g
(16) c=f−1,b=e−1
(17) c=f−1,b=g−1
(18) c=f−1,d=h−1
(19) c=f−1,b=e
(20) c=f−1,b=g
(21) c=f−1,e=g
(22) a=f,d=h−1
(23) a=f,b=e
(24) a=f,b=g
(25) c=f,b=g−1
(26) c=f,e=g−1
(27) c=f,d=h−1
(28) c=f,b=e
(29) b=e,d=h−1
(30) a=c−1,a=f−1,d=h−1
(31) a=c−1,a=f−1,c=f
(32) a=c−1,a=f,c=f−1
(33) a=c−1,c=f−1,d=h−1
(34) a=c−1,c=f,d=h−1
(35) a=c,a=f,c=f
(36) a=c,a=f−1,c=f−1
(37) a=f−1,c=f−1,d=h−1
(38) a=f−1,c=f,d=h−1
(39) a=f,c=f−1,d=h−1
(40) a=f,c=f,d=h−1
(41) b=e−1,b=g−1,d=h−1
(42) b=e−1,b=g−1,e=g
(43) b=e,b=g−1,e=g−1
(44) b=e−1,b=g,e=g−1
(45) b=e−1,b=g,d=h−1
(46) b=g−1,e=g−1,d=h−1
(47) b=g−1,e=g,d=h−1
(48) b=g,e=g−1,d=h−1
(49) b=g,e=g,d=h−1
(50) a=c−1,a=f,c=f−1,d=h−1
(51) a=c−1,a=f−1,c=f,d=h−1
(52) a=c−1,b=e,b=g−1,d=h−1
(53) a=c,a=f−1,c=f−1,d=h−1
(54) a=c,a=f,c=f,d=h−1
(55) b=e−1,b=g−1,e=g,d=h−1
(56) a=c−1,b=e,b=g−1,e=g−1,d=h−1
(57) a=c,a=f,c=f,b=e−1,d=h−1
(58) a=c,a=f,c=f,b=e,d=h−1
(59) c=f,b=e,b=g−1,e=g−1,d=h−1
(60) a=c,a=f−1,c=f−1,b=e−1,b=g−1,d=h−1
(61) a=c,a=f,c=f,b=e−1,b=g−1,d=h−1
(62) a=f−1,c=f,b=e−1,b=g,e=g−1,d=h−1
(63) a=f,c=f−1,b=e,b=g−1,e=g−1,d=h−1
(64) a=f,c=f−1,b=e−1,b=g,e=g−1,d=h−1
(65) a=c−1,a=f,c=f−1,b=e,b=g−1,e=g−1,d=h−1
(66) a=c−1,a=f−1,c=f,b=e−1,b=g,e=g−1,d=h−1
(67) a=c,a=f,c=f,b=e−1,b=g,e=g−1,d=h−1
Recall that P=<A,t|s(t)>=1 where s(t)=atbt−1ctdtet−1ftgt−1ht−1=1 and a,b,c,e,f,g∈A/{1} and d,h∈A. We will show that some cases of the given equation are solvable by applying the weight test. Also the transformation y=td, leads to the assumption that d=1 in A.
In this case the relative presentation P is given as :
P=<A,t|s(t)=atbt−1ctdtet−1ftgt−1ht−1>
= <A,t|s(t)=c−1te−1t−1ct2et−1ftgt−1ht−1=1>.
Suppose, x=tet−1 which implies that x−1=te−1t−1.
Using above substitutions in relative presentation we have new presentation
Q=<A,t,x|t−1c−1x−1ctxftgt−1h=1=x−1tet−1>. Observe that in new presentation Q we have two relators R1=t−1c−1x−1ctxftgt−1h and R2=x−1tet−1. The star graph Γ1 for new presentation Q is given by the Figure 1.
Let γ1↔c−1,γ2↔c,γ3↔1,γ4↔f,γ5↔g,γ6↔h and η1↔1,η2↔e,η3↔1.
Assign weights to the edges of the star graph Γ1 as follows:
θ(γ1)=θ(γ2)=θ(η2)=1 and θ(γ3)=θ(γ4)=θ(γ5)=θ(γ5)=θ(γ6)=θ(η1)=θ(η3)=12. The given weight function θ is aspherical and all three conditions of weight test are satisfied.
(1) Observe that, ∑(1−θ(γj))=6−(1+1+12+12+12)+12)=2 and ∑(1−θ(ηj))=3−(0+1+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contraction. For example, γ3γ6η1 is a cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1atbt−1a−1t2et−1ftb−1t−1h and R2=ta−1t−1. The star graph Γ2 for new presentation Q is given by the Figure 1.
(1) Observe that, ∑(1−θ(γj))=6−(0+1+1+12+1+12)=2 and ∑(1−θ(ηj))=3−(0+1+1)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example γ6γ3η1 is a cycle of length 2, having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbx−1tg−1t−1ftgt−1h and R2=x−1t−1at. The star graph Γ2 for new presentation Q is given by the Figure 1.
(1) Observe that, ∑(1−θ(γj))=6−(0+1+1+12+1+12)=2 and ∑(1−θ(ηj))=3−(0+1+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ5η3γ1η3 is a cycles of length 2 having weight less than 2, which implies b=g−1, a contradiction.
(3) Each admissible cycle has non-negative weight.
In this case we have two relators R1=xbx−1et−1ftg and R2=t−1a−1t2. The star graph Γ3 for the relative presentation is given by the Figure 1.
(1) Observe that ∑(1−θ(γj))=4−(0+1+12+12)=2 and ∑(1−θ(ηj))=4−(0+12+1+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbx−1tbt−1ftgt−1ht−1 and R2=x−1t−1at. The star graph Γ4 for the relative presentation is given by the Figure 1.
(1) Observe that ∑(1−θ(γj))=7−(0+1+1+1+12+12+1)=2 and ∑(1−θ(ηj))=3−(0+12+1+12)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ5η3γ1η3 is a cycle of length 2 having weight less than 2, which implies g=b−1 a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axa−1t2et−1fxh and R2=x−1tb−1t−1. The star graph Γ5 for the relative presentation Q is given by the Figure 2.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+12+1+12+12)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6γ3η3 is a cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight,
In this case we have two relators R1=xg−1xtet−1ftgt−1h and R2=x−1t−1ct. The star graph Γ6 for the relative presentation is given by the Figure 2.
(1) Observe that, ∑(1−θ(γj))=6−(32+12+1+12+12+0)=2 and ∑(1−θ(ηj))=3−(0+1+0)=2
(2) Each admissible cycle having weight less than 2 leads to a contradiction, For example, the cycle γ1η1η3, γ6η1η3γ2 and γ6η1η3γ1 are the cycles of length 2 having weight less than 2, which implies g=1, h=1 and h=g, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xht−1atbt−1atx−1f and R2=x−1tgt−1. The star graph Γ7 for the relative presentation Q is given by the Figure 2.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+1+1+12+0)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each possible admissible cycle having weight less than 2 leads to a contradiction. For Example η2γ5γ1 is a cycle of length 2 having weight less than 2, which implies g=h−1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbxtet−1ftg and R2=x−1t−1at. The star graph Γ8 for the relative presentation is given by the Figure 2.
(1) Observe that, ∑(1−θ(γj))=6−(12+32+0+12+12+1)=2 and ∑(1−θ(ηj))=3−(0+1+0)=2.
(2) Each edge having weight less than 2 leads to a contradiction. For example, γ2η1η3 is a cycle of length 2 having weight less than 2, which implies b=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axatxftgt−1h and R2=x−1tb−1t−1. The star graph Γ9 for the relative presentation is given by the Figure 3.
(1) Observe that, ∑(1−θ(γj))=6−(1+12+12+12+1+12)=2 and ∑(1−θ(ηj))=3−(0+12+12)=2.
(2) Each edge having weight less than 2 leads to a contradiction. For example, γ6γ5η1 is a cycles of length 2 having weight less than 2, which implies that h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axat2et−1fxh and R2=x−1tgt−1. The star graph Γ10 for the relative presentation is given by the Figure 3.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+12+1+12+12)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each edge having weight less than 2 leads to a contradiction. For example, γ6γ3η3 is a cycles of length 2 having weight less than 2, which implies that h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1ctbt−1axfx−1th and R2=x−1t2et−1. The star graph Γ11 for the relative presentation is given by the Figure 3.
(1) Observe that ∑(1−θ(γj))=6−(1+12+0+12+1+1)=2 and ∑(1−θ(ηj))=4−(1+0+12+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbt−1ct2ex−1b−1t−1h and R2=x−1t−1at. The star graph Γ12 for the relative presentation is given by the Figure 3.
(1) Observe that, ∑(1−θ(γj))=6−(12+12+0+1+1+1)=2 and ∑(1−θ(ηj))=3−(0+1+0)=2.
(2) Each edge having weight less than 2 leads to a contradiction. For example, γ6γ3η1 is a cycles of length 2 having weight less than 2, which implies that h=1, a contradiction.
(3) Each edge has non negative weight.
In this case we have two relators R1=t−1xbt−1ct2exg and R2=x−1t−1ft. The star graph Γ13 for the relative presentation is given by the Figure 4.
(1) ∑(1−θ(γj))=6−(12+12+1+1+1+0)=2 and ∑(1−θ(ηj))=3−(0+12+12)=2.
(2) Each edge having weight less than 2 leads to a contradiction. For example, γ6γ4γ1 amd γ6γ5η1η3 are the cycles of length 2 having weight less than 2, which implies g=1 and g=e−1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbt−1ct2ex−1bt−1h and R2=x−1t−1at. The star graph Γ13 for the relative presentation Q is given by the Figure 4.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+1+12+1+0)=2 and ∑(1−θ(ηj))=3−(0+12+12)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6η1γ3 is a cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1ac−1tx−1ctxgt−1h and R2=x−1tb−1t−1c−1t. The star graph Γ14 for the relative presentation Q is given by the Figure 4.
(1) Observe that, ∑(1−θ(γj))=6−(1+12+1+1+12+0)=2 and ∑(1−θ(ηj))=4−(0+1+12+12)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6η1γ4 is a cycles of length 2, having weight less than 2, which implies h=1 a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axtex−1h and R2=x−1tbt−1ct. The star graph Γ15 for the relative presentation Q is given by the Figure 4.
(1) Observe that, ∑(1−θ(γj))=4−(1+0+1+0)=2 and ∑(1−θ(ηj))=4−(1+0+0+1)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ4η4γ2γ1 is a cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−2atbx−1texg and R2=x−1t−1et−1c−1t. The star graph Γ16 for the relative presentation Q is given by the Figure 4.
(1) ∑(1−θ(γj))=4−(1+1+12+12+12)=2 and ∑(1−θ(ηj))=4−(12+12+12+12)=2.
(2) Each admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1atbxtbx−1gt−1h and R2=x−1t−1ct. The star graph Γ17 for the relative presentation Q is given by the Figure 5.
(1) ∑(1−θ(γj))=6−(1+12+1+12+1+12+12)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6η3γ3 is a cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axct2et−1c−1tbt−1h and R2=x−1tbt−1. The star graph Γ18 for the relative presentation Q is given by the Figure 5.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+12+1+12+12)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6γ3η3 is a cycle of length 2 having weight less than 2, which implies h=1 a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1atbxtex−1et−1h and R2=x−1t−1ct. The star graph Γ19 for the relative presentation Q is given by the Figure 5.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+12+1+12+12)=2 and ∑(1−θ(ηj))=3−(0+1+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6η3γ3 is a cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbt−1ct2exg and R2=x−1t−1at. The star graph Γ20 for the Q relative presentation is given by the Figure 5.
(1) Observe that, ∑(1−θ(γj))=6−(1+12+1+1+0+12)=2 and ∑(1−θ(ηj))=3−(0+12+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axctxatget−1h and R2=x−1tbt−1.
The star graph Γ21 for the relative presentation Q is given by the Figure 6.
(1) Observe that, ∑(1−θ(γj))=6−(12+12+1+12+12+1)=2 and ∑(1−θ(ηj))=3−(0+12+12)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6γ3η1, is a cycle of length 2 having weight less than 2, which implies h=1 a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xct2exh and R2=x−1t−1atbt−1. The star graph Γ22 for the relative presentation Q is given by the Figure 6.
(1) Observe that, ∑(1−θ(γj))=4−(12+12+12+12)=2 and ∑(1−θ(ηj))=4−(12+12+12+12)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ23η1 is a cycle of length 2, having weight less than 2, which implies e2=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1atg−1xtexgt−1h and R2=x−1t−1ft>. The star graph Γ23 for the relative presentation is given by the Figure 6.
(1) Observe that, ∑(1−θ(γj))=6−(1+12+12+12+1+12)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6η3γ3 is a cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1atbxtexe−1t−1h and R2=x−1t−1ct. The star graph Γ23 for the relative presentation Q is given by the Figure 6.
(1) Observe that, ∑(1−θ(γj))=6−(1+12+12+12+1+12)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ6η3γ3 has weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−2atbxtexg and R2=x−1t−1ct. The star graph Γ24 for the relative presentation Q is given by the Figure 6.
(1) Observe that, ∑(1−θ(γj))=6−(1+12+1+1+12+0)=2 and ∑(1−θ(ηj))=3−(0+1+0)=2.
(2) Each possible admissible cycle has weight at least 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axt−1cxctgt−1h and R2=x−1t2bt−1. The star graph Γ25 for the relative presentation Q is given by the Figure 7.
(1) ∑(1−θ(γj))=4−(12+12+12+12)=2 and ∑(1−θ(ηj))=4−(12+12+12+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xcta−1t2xftg and R2=x−1t−2atbt−1. The star graph Γ26 for the relative presentation Q is given by the Figure 7.
(1) ∑(1−θ(γj))=6−(1+12+1+12+12+12)=2 and ∑(1−θ(ηj))=5−(12+1+12+12+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbx−1tex−1g and R2=x−1t−1at. The star graph Γ27 for the relative presentation Q is given by the Figure 7.
(1) ∑(1−θ(γj))=5−(1+0+1+1+0)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axctx−1fte−1t−1h and R2=x−1tbt−1. The star graph Γ27 for the new presentation Q is given by the Figure 7.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+1+1+0)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbx−1tex−1gt−1h and R2=x−1t−1at. The star graph Γ27 for the relative presentation Q is given by the Figure 7.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+1+0+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative watch.
In this case we have two relators: R1=xbx−1tet−1c−1tgt−1h and R2=x−1t−1at. The star graph Γ28 for the relative presentation Q is given by the Figure 7.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+1+1+12+12)=2 and ∑(1−θ(ηj))=3−(0+12+12)=2.
(2) Each possible admissible cycle having weight less than 2 leads to a contrdiction. For example, γ5η3γ1 is a cycle of length 2 having weight less than 2, which implies g=b−1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbx−1texg and R2=x−1t−1at. The star graph Γ29 for the relative presentation Q is given by the Figure 8.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+1+0+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbx−1tex−1g and R2=x−1t−1at. The star graph Γ30 for the relative presentation Q is given by the Figure 8.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+1+1+0)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbxtexgt−1h and R2=x−1t−1at. The star graph Γ31 for the relative presentation Q is given by the Figure 8.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+12+12+1)=2 and ∑(1−θ(ηj))=3−(12+0+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbxtex−1gt−1h and R2=x−1t−1at. The star graph Γ32 for the relative presentation Q is given by the Figure 8.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+0+1+1)=2 and ∑(1−θ(ηj))=3−(1+0+0)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbxtex−1g and R2=x−1t−1at. The star graph Γ33 for the relative presentation Q is given by the Figure 9.
(1) Observe that ∑(1−θ(γj))=5−(1+1+0+12+12)=2 and ∑(1−θ(ηj))=3−(12+0+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbt−1ct2ex−1g and R2=x−1t−1at. The star graph Γ38 for the relative presentation Q is given by the Figure 9.
(1) Observe that, ∑(1−θ(γj))=6−(1+12+1+1+12+0)=2 and ∑(1−θ(ηj))=3−(0+12+12)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ2γ−16η1η3γ5 is the cycle of length 2 having weight less than 2, which leads to a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbx−1texg and R2=x−1t−1at. The star graph Γ35 for the relative presentation Q is given by the Figure 9.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+1+1+0)=2 and ∑(1−θ(ηj))=3−(1+0+0)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbx−1tbxb−1 and R2=x−1t−1at. The star graph Γ35 for the relative presentation Q is given by the Figure 9.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+1+0+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbxtexg and R2=x−1t−1at. The star graph Γ36 for the relative presentation Q is given by the Figure 9.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+1+1+0)=2 and ∑(1−θ(ηj))=3−(1+0+0)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axctx−1fx−1 and R2=x−1tbt−1. The star graph Γ37 for the relative presentation Q is given by the Figure 10.
(1) Observe that, ∑(1−θ(γj))=5−(12+12+1+1+0)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axctx−1fx−1h and R2=x−1tbt−1. The star graph Γ37 for the relative presentation Q is given by the Figure 9.
(1) Observe that, ∑(1−θ(γj))=5−(12+12+1+1+0)=2 and ∑(1−θ(ηj))=3−(12+0+12)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axctxfx−1h and R2=x−1tbt−1. The star graph Γ38 for the relative presentation Q is given by the Figure 10.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+1+0+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2, so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axctx−1fte−1t−1h and R2=x−1tbt−1. The star graph Γ39 for the relative presentation Q is given by the Figure 10.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+0+12+1+1)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example, γ5η1η3 is the cycle of length 2 having weight less than 2, which implies h=1, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axctx−1fx and R2=x−1tbt−1. The star graph Γ40 for the relative presentation Q is given by the Figure 10.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+0+12+1+1)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=|t−1axctxfx−1 and R2=x−1tbt−1. The star graph Γ41 for the relative presentation Q is given by the Figure 11.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+0+12+1+1)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axa−1txfx−1 and R2=x−1tbt−1. The star graph Γ41 for the relative presentation Q is given by the Figure 11.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+1+0+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=x−1t−1axctx−1f and R2=x−1tbt−1. The star graph Γ42 for the relative presentation Q is given by the Figure 11.
(1) Observe that, ∑(1−θ(γj))=5−(0+12+12+1+1)=2 and ∑(1−θ(ηj))=3−(12+0+12)=2.
(2) Each admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) each edge has non-negative weight.
In this case we have two relators R1=xatbt−1cx−1f and R2=x−1tbt−2. The star graph Γ43 for the relative presentation Q is given by the Figure 11.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+0+12+1+1)=2 and ∑(1−θ(ηj))=3−(12+12+0)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xt−1axctxf and R2=x−1tbt−1. The star graph Γ44 for the relative presentation Q is given by the Figure 11.
(1) Observe that, ∑(1−θ(γj))=5−(12+12+12+12+1)=2 and ∑(1−θ(ηj))=3−(12+0+12)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbx−1etxg and R2=x−1t−2c−1t. The star graph Γ45 for the relative presentation Q is given by the Figure 12.
(1) Observe that, ∑(1−θ(γj))=4−(0+1+1+0)=2 and ∑(1−θ(ηj))=4−(1+0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbxtex−1g and R2=x−1t−1at. The star graph Γ46 for the relative presentation Q is given by the Figure 12.
(1) Observe that, ∑(1−θ(γj))=5−(1+1+0+0+1)=2 and ∑(1−θ(ηj))=3−(1+0+0)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbxtexg and R2=x−1t−1at. The star graph Γ47 for the relative presentation Q is given by the Figure 12.
(1) Observe that, ∑(1−θ(γj))=5−(1+1+0+1+0)=2 and ∑(1−θ(ηj))=3−(1+0+0)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=x−1t−1axctx−1f and R2=x−1tbt−1. The star graph Γ48 for the relative presentation Q is given by the Figure 12.
(1) Observe that, ∑(1−θ(γj))=5−(0+0+1+1+1)=2 and ∑(1−θ(ηj))=3−(0+0+0)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xc−1x−1tcx−1f and R2=x−1tb−1t−2. The star graph Γ49 for the relative presentation Q is given by the Figure 13.
(1) Observe that, ∑(1−θ(γj))=4−(0+1+1+0)=2 and ∑(1−θ(ηj))=4−(1+0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbxtb−1xg and R2=x−1t−1at. The star graph Γ50 for the relative presentation Q is given by the Figure 13.
(1) Observe that, ∑(1−θ(γj))=5−(12+1+12+1+0)=2 and ∑(1−θ(ηj))=3−(12+0+12=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example γ2η1γ−15 is a cycle of length 2 having weight less than 2 which implies b=g, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=|t−1xbxtbxg and R2=x−1t−1at. The star graph Γ50 for the relative presentation Q is given by the Figure 12.
(1) Observe that, ∑(1−θ(γj))=5−(12+1+12+1+0)=2 and ∑(1−θ(ηj))=3−(12+0+12)=2.
(2) Each admissible cycle having weight less than 2 leads to a contradiction. For example γ2η1γ−15 is a cycle of length 2 having weight less than 2 which implies b=g, a contradiction.
(3) Each edge has non-negative weight.
In this case we have two relators R1=x−1t−1axctxc and R2=x−1tbt−1. The star graph Γ51 for the relative presentation Q is given by the Figure 13.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+0+1+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xat−1x−1atxta−1 and R2=x−1tbt−1t−2. The star graph Γ52 for the relative presentation Q is given by the Figure 13.
(1) Observe that, ∑(1−θ(γj))=6−(0+1+0+1+1+1)=2 and ∑(1−θ(ηj))=4−(0+0+1+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbxtb−1xb−1 and R2=x−1t−1at. The star graph Γ53 for the relative presentation Q is given by the Figure 14.
(1) Observe that, ∑(1−θ(γj))=5−(12+1+12+1+0)=2 and ∑(1−θ(ηj))=3−(12+0+12)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=xbt−1x−1bxtb−1tb−1 and R2=x−1t−1a−1tbt−1. The star graph Γ54 for the relative presentation Q is given by the Figure 14.
(1) Observe that, ∑(1−θ(γj))=6−(12+1+0+1+12+1)=2 and ∑(1−θ(ηj))=4−(1+0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axa−1txax−1 and R2=x−1tbt−1. The star graph Γ55 for the relative presentation Q is given by the Figure 14.
(1) Observe that, ∑(1−θ(γj))=5−(0+1+1+0+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbx−1tb−1xb and R2=x−1t−1at. The star graph Γ56 for the relative presentation Q is given by the Figure 14.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+1+0+1)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1xbx−1tb−1x−1b and R2=x−1t−1at. The star graph Γ57 for the relative presentation Q is given by the Figure 14.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+1+1+0)=2 and ∑(1−θ(ηj))=3−(0+0+1)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In this case we have two relators R1=t−1axatx−1ax and R2=x−1tbt−1. The star graph Γ58 for the relative presentation Q is given by the Figure 15.
(1) Observe that, ∑(1−θ(γj))=5−(1+0+1+0+1)=2 and ∑(1−θ(ηj))=3−(1+0+0)=2.
(2) Each possible admissible cycle has weight atleast 2 so θ admits an aspherical weight function.
(3) Each edge has non-negative weight.
In all the cases, 3 conditions of weight test are satisfied.
Hence the equation s(t)=atbt−1ctdtet−1ftgt−1ht−1=1 where a,b,c,e,f,g∈A/{1} and d,h∈A is solvable.
In this article, we reviewed some basic concepts of combinatorial group theory (like torsion-free group, equations over groups, relative presentation, weight test) and discussed the two main conjectures in equations over torsion-free groups. We investigated all possible cases and solved the singular equation of length eight over torsion free group by using weight test. This result will be useful in dealing with equations over torsion-free groups.
The authors wish to express their gratitude to Prince Sultan University for facilitating the publication of this article through the Theoretical and Applied Sciences Lab.
All authors declare no conflicts of interest in this paper.
[1] | Awoniyi O (2020) The Petri-Dish Effect. Disaster Med Public Health Prep 2020. |
[2] | Mizumoto K, Chowell G (2020) Transmission potential of the novel coronavirus (COVID-19) onboard the Diamond Princess Cruises Ship, 2020. Infect Dis Model 5: 264-270. |
[3] | Rocklö v J, Sjö din H, Wilder-Smith A (2020) COVID-19 outbreak on the Diamond Princess cruise ship: estimating the epidemic potential and effectiveness of public health countermeasures. J Travel Med 27: 1-7. |
[4] | Cruise Lines International Association (2019) State of the Industry Outlook. Cruise Lines International Association. Available from: https://cruising.org/-/media/research-updates/research/state-of-the-cruise-industry.ashx |
[5] |
Papathanassis A (2019) The growth and development of the cruise sector: a perspective article. Tour Rev 75: 130-135. doi: 10.1108/TR-02-2019-0037
![]() |
[6] | Seatrade Cruise News (2018) A golden age for the Med but ports need to prepare for the new generation of large ships. Seatrade Cruise News. Available from: http://www.seatrade-cruise.com/news/news-headlines/a-golden-age-for-the-med-but-ports-need-to-prepare-for-the-new-generation-of-large-ships.html. |
[7] |
Bowen C, Fidgeon P, Page SJ (2014) Maritime tourism and terrorism: Customer perceptions of the potential terrorist threat to cruise shipping. Curr Issues Tour 17: 610-639. doi: 10.1080/13683500.2012.743973
![]() |
[8] |
Floyd MF, Pennington-Gray L (2004) Profiling risk perceptions of tourists. Ann Tour Res 31: 1051-1054. doi: 10.1016/j.annals.2004.03.011
![]() |
[9] |
Sö nmez SF, Graefe AR (1998) Determining future travel behavior from past travel experience and perceptions of risk and safety. J Travel Res 37: 171-177. doi: 10.1177/004728759803700209
![]() |
[10] |
Petrick JF, Li X, Park SY (2007) Cruise passengers' decision-making processes. J Travel Tour Mark 23: 1-14 doi: 10.1300/J073v23n01_01
![]() |
[11] | Gibson P (2006) Cruise Operations Management. Elsevier: Oxford. |
[12] | Lupton D (1999) Risk. London, Routledge. |
[13] |
Aven T (2012) The risk concept—historical and recent development trends. Reliab Eng Syst Saf 99: 33-44. doi: 10.1016/j.ress.2011.11.006
![]() |
[14] |
Slovic P (2016) Understanding perceived risk: 1978-2015. Environ Sci Policy Sustain Dev 58: 25-29. doi: 10.1080/00139157.2016.1112169
![]() |
[15] | Priest GL (1990) The new legal structure of risk control. Daedalus 119: 207-227. |
[16] | Kungwani P (2014) Risk management—An analytical study. IOSR J Bus Manag 16: 83-89. |
[17] |
Douglas M, Wildavsky A (1982) How can we know the risks we face? Why risk selection is a social process. Risk Anal 2: 49-58. doi: 10.1111/j.1539-6924.1982.tb01365.x
![]() |
[18] | Fuchs G, Reichel A (2004) Cultural differences in tourist destination risk perception: An exploratory study. Tourism 52: 21-37. |
[19] |
Kim DJ, Ferrin DL, Rao HR (2008) A trust-based consumer decision-making model in electronic commerce: The role of trust, perceived risk, and their antecedents. Decis Support Syst 44: 544-564. doi: 10.1016/j.dss.2007.07.001
![]() |
[20] | Bauer R (1960) Consumer behaviour as risk taking. In: Hancock RS (Ed.), Dynamic Marketing for a Changing World. Chicago: American Marketing Association, 389-398. |
[21] | Tarlow PE (2006) Terrorism and tourism. In: Wilkes J, Pendergast D, Leggat P (Eds.), Tourism in turbulent times: Towards safe experiences for visitors (Advances in Tourism Research). Abingdon, UK, Routledge, 79-92. |
[22] | Lin PJ, Jones E, Westwood S (2009) Perceived risk and risk-relievers in online travel purchase intentions. J Hosp Mark Manag 18: 782-810. |
[23] |
Lepp A, Gibson H (2003) Tourist roles, perceived risk and international tourism. Ann Tour Res 30: 606-624. doi: 10.1016/S0160-7383(03)00024-0
![]() |
[24] |
Yang ECL, Khoo-Lattimore C, Arcodia C (2017) A systematic literature review of risk and gender research in tourism. Tour Manag 58: 89-100. doi: 10.1016/j.tourman.2016.10.011
![]() |
[25] | Cohen E (1972) Toward a sociology of international tourism. Soc Res 39: 164-182. |
[26] |
Gibson H, Yiannakis A (2002) Tourist roles: Needs and the lifecourse. Ann Tour Res 29: 358-383. doi: 10.1016/S0160-7383(01)00037-8
![]() |
[27] |
Roehl WS, Fesenmaier DR (1992) Risk perceptions and pleasure travel: An exploratory analysis. J Travel Res 30: 17-26. doi: 10.1177/004728759203000403
![]() |
[28] | Fuchs G, Reichel A (2011) An exploratory inquiry into destination risk perceptions and risk reduction strategies of first time vs. repeat visitors to a highly volatile destination. Tour Manag 32: 266-276. |
[29] |
Reisinger Y, Mavondo F (2005) Travel anxiety and intentions to travel internationally: Implications of travel risk perception. J Travel Res 43: 212-225. doi: 10.1177/0047287504272017
![]() |
[30] |
Karl M (2018) Risk and uncertainty in travel decision-making: Tourist and destination perspective. J Travel Res 57: 129-146. doi: 10.1177/0047287516678337
![]() |
[31] |
Dowling GR (1986) Perceived risk: The concept and its measurement. Psychol Mark 3: 193-210. doi: 10.1002/mar.4220030307
![]() |
[32] |
Pizam A, Jeong GH, Reichel A, et al. (2004) The relationship between risk-taking, sensation-seeking, and the tourist behavior of young adults: A cross-cultural study. J Travel Res 42: 251-260. doi: 10.1177/0047287503258837
![]() |
[33] |
Kahneman D, Tversky A (1974) Judgment under uncertainty: Heuristics and biases. Science 185: 1124-1131. doi: 10.1126/science.185.4157.1124
![]() |
[34] |
Gray JM, Wilson MA (2009) The relative risk perception of travel hazards. Environ Behav 41: 185-204. doi: 10.1177/0013916507311898
![]() |
[35] |
Money RB, Crotts JC (2003) The effect of uncertainty avoidance on information search, planning, and purchases of international travel vacations. Tour Manag 24: 191-202. doi: 10.1016/S0261-5177(02)00057-2
![]() |
[36] |
Williams AM, Baláž V (2015) Tourism risk and uncertainty: Theoretical reflections. J Travel Res 54: 271-287. doi: 10.1177/0047287514523334
![]() |
[37] |
Tversky A, Kahneman D (1973) Availability: A heuristic for judging frequency and probability. Cogn Psychol 5: 207-232. doi: 10.1016/0010-0285(73)90033-9
![]() |
[38] |
Migacz S, Durko A, Petrick J (2016) It was the best of times, it was the worst of times: The effects of critical incidents on cruise passengers' experiences. Tour Mar Environ 11: 123-135. doi: 10.3727/154427315X14513374773445
![]() |
[39] | Baker D (2016) The cruise industry: Past, present and future. J Tour Res 14: 145-153. |
[40] |
Finucane ML, Alhakami A, Slovic P, et al. (2000) The affect heuristic in judgments of risks and benefits. J Behav Decis Mak 13: 1-17. doi: 10.1002/(SICI)1099-0771(200001/03)13:1<1::AID-BDM333>3.0.CO;2-S
![]() |
[41] |
Slovic P, Peters E (2006) Risk perception and affect. Curr Dir Psychol Sci 15: 322-325. doi: 10.1111/j.1467-8721.2006.00461.x
![]() |
[42] |
Sö nmez SF, Graefe AR (1998) Influence of terrorism risk on foreign tourism decisions. Ann Tour Res 25: 112-144. doi: 10.1016/S0160-7383(97)00072-8
![]() |
[43] | Baker D (2013) Cruise passengers' perceptions of safety and security while cruising the Western Caribbean. Rosa Dos Ventos 5: 140-154. |
[44] |
Le TH, Arcodia C (2018) Risk perceptions on cruise ships among young people: Concepts, approaches and directions. Int J Hosp Manag 69: 102-112. doi: 10.1016/j.ijhm.2017.09.016
![]() |
[45] |
Lois P, Wang J, Wall A, et al. (2004) Formal safety assessment of cruise ships. Tour Manag 25: 93-109. doi: 10.1016/S0261-5177(03)00066-9
![]() |
[46] |
Eliopoulou E, Papanikolaou A, Voulgarellis M (2016) Statistical analysis of ship accidents and review of safety level. Saf Sci 85: 282-292. doi: 10.1016/j.ssci.2016.02.001
![]() |
[47] |
Dawson J, Johnston M, Stewart E (2017) The unintended consequences of regulatory complexity: The case of cruise tourism in Arctic Canada. Mar Policy 76: 71-78. doi: 10.1016/j.marpol.2016.11.002
![]() |
[48] | Lück M, Maher PT, Stewart EJ (2010) Cruise tourism in Polar Regions: Promoting environmental and social sustainability? London: Earthscan Ltd. |
[49] |
Maher PT, Johnston ME, Dawson JP, et al. (2011) Risk and a changing environment for Antarctic tourism. Curr Issues Tour 14: 387-399. doi: 10.1080/13683500.2010.491896
![]() |
[50] |
Dahl E (2014) Medical cruise challenges in Antarctica. J Travel Med 21: 223-224. doi: 10.1111/jtm.12118
![]() |
[51] | Brosnan IG (2011) The diminishing age gap between polar cruisers and their ships: A new reason to codify the IMO Guidelines for ships operating in polar waters and make them mandatory? Mar Policy 35: 261-265. |
[52] |
Schrö der-Hinrichs JU, Hollnagel E, Baldauf M (2012) From Titanic to Costa Concordia—A century of lessons not learned. WMU J Marit Affairs 11: 151-167. doi: 10.1007/s13437-012-0032-3
![]() |
[53] | Greenberg MD, Chalk P, Willis HH, et al. (2006) Maritime terrorism: Risk and liability. Centre for Terrorism Risk Management Policy: Rand Corporation. |
[54] | Button K (2016) The Economics and Political Economy of Transportation Security. Cheltenham, UK: Edward Elgar Publishing. |
[55] | Klein R, Lück M, Poulston J (2017) Passengers and risk: health, wellbeing and liability. In: Dowling R, Weeden C (Eds.), Cruise Ship Tourism (2nd ed.), Wallingford: CABI, 106-123. |
[56] | Panko TR, Henthorne TL (2019) Crimes at sea: A review of crime onboard cruise ships. Int J Saf Secur Tour Hosp 20: 1-23. |
[57] |
Chien PM, Sharifpour M, Ritchie BW, et al. (2017) Travelers' health risk perceptions and protective behavior: a psychological approach. J Travel Res 56: 744-759. doi: 10.1177/0047287516665479
![]() |
[58] |
Liu-Lastres B, Schroeder A, Pennington-Gray L (2018) Cruise line customers' responses to risk and crisis communication messages: An application of the risk perception attitude framework. J Travel Res 58: 849-865. doi: 10.1177/0047287518778148
![]() |
[59] |
Kura F, Amemura-Maekawa J, Yagita K, et al. (2006) Outbreak of Legionnaires' disease on a cruise ship linked to spa-bath filter stones contaminated with Legionella pneumophila serogroup 5. Epidemiol Infect 134: 385-391. doi: 10.1017/S095026880500508X
![]() |
[60] | Baker DM, Stockton S (2013) Smooth sailing! Cruise passengers demographics and health perceptions while cruising the Eastern Caribbean. Int J Bus Soc Sci 4: 8-17. |
[61] |
Tang C, Weaver D, Shi F, et al. (2019) Constraints to domestic ocean cruise participation among higher income Chinese adults. Int J Tour Res 21: 519-530. doi: 10.1002/jtr.2279
![]() |
[62] | Cruise Lines International Association. (2019) 2018 Global Passenger Report. Available from: https://cruising.org/-/media/research-updates/research/clia-global-passenger-report-2018.ashx |
[63] | Westwood S (2007) What lies beneath? Using creative, projective and participatory techniques in qualitative tourism inquiry. In: Ateljevic I, Pritchard A, Morgan N (Eds.), The critical turn in tourism studies: Innovative research methodologies. Oxford: Elsevier, 293-316. |
[64] |
Coulter RA, Zaltman G, Coulter KS (2001) Interpreting consumer perceptions of advertising: An application of the Zaltman Metaphor Elicitation Technique. J Advert 30: 1-21. doi: 10.1080/00913367.2001.10673648
![]() |
[65] |
Braun V, Clarke V (2006) Using thematic analysis in psychology. Qual Res Psychol 3: 77-101. doi: 10.1191/1478088706qp063oa
![]() |
[66] | GP Wild (International) Limited (2017) Report on Operational Incidents 2009 to 2016. Available from: https://www.cruising.org/docs/default-source/research/report-on-operational-incidents-2009-to-2016.pdf?sfvrsn=0. |
[67] | Henthorne TL, George BP, Smith WC (2013) Risk perception and buying behavior: An examination of some relationships in the context of cruise tourism in Jamaica. Int J Hosp Tour Adm 14: 66-86. |
[68] |
Simpson PM, Siguaw JA (2008) Perceived travel risks: The traveller perspective and manageability. Int J Tour Res 10: 315-327. doi: 10.1002/jtr.664
![]() |
[69] | Rotter JB (1966) Generalized expectancies for internal versus external control of reinforcement. Psychol Monogr Gen Appl 80: 1-28. |