We prove that every countable left‐ordered group embeds into a finitely generated left‐ordered simple group. Moreover, if the first group has a computable left‐order, then the simple group also has a computable left‐order. We also obtain a Boone–Higman–Thompson type theorem for left‐orderable groups with recursively enumerable positive cones. These embeddings are Frattini embeddings, and isometric whenever the initial group is finitely generated. Finally, we reprove Thompson's theorem on word problem preserving embeddings into finitely generated simple groups and observe that the embedding is isometric. A landmark result in combinatorial group theory and computability on groups is the Boone–Higman theorem. It states that a finitely generated group has decidable word problem if and only if it embeds into a simple subgroup of a finitely presented group  of piecewise homeomorphisms of flows of the suspension, see Definition T. In particular, it is finitely generated if H is so. Just as in .The Φt that sends  to  is a homeomorphism and defines a flow Φ on Σ, the suspension flow, so that  is a dynamical system as well. The orbits of the suspension flow are homeomorphic to the real line.The map H(φ) the group of homeomorphisms of Σ that preserves the orbits of the suspension flow, and by H0(φ) the subgroup of H(φ) that, in addition, preserves the orientation on each orbit.We denote by 3.2C be a clopen subset of X and let J⊂R be of diameter <1. The embedding of C×J into X×R descends to an embedding into Σ that we denote by πC,J.Let C⊂X and subset J of diameter <1 in R, the map πC,J is a chart for the suspension, whose image is denoted by UC,J. If z is in the interior of UC,J, then πC,J is a chart at z.Definition 3.1(Dyadic chart) Let C be a clopen subset of X, and let J be a dyadic interval of length <1 in R. Then πC,J:C×J↪Σ is called dyadic chart.For every clopen Definition 3.2(Dyadic map) A dyadic map is a map f of real numbers such that f(x)=λx+c, where λ is a power of 2 and c is a dyadic rational.Definition 3.3(↦.FJ denotes the group of piecewise dyadic homeomorphisms of J with finitely many breakpoints.Definition 3.4πC,J be a dyadic chart and let f∈FJ. Then fC,J is the map in T(φ) whose restriction to UC,J is given byFC,J be the subgroup of T(φ) generated by the elements fC,J for all f∈FJ.Let We recall that T(φ) is infinite, simple, left‐ordered and finitely generated  and I1:=.Lemma 3.7(. Let C(J)⊂bij(J) denote the subgroup of all piecewise homeomorphisms with dyadic breakpoints on J. The subgroup of C(J) of orientation preserving bijections is denoted by C+(J).Example 3.9C(J).Every countable group embeds into Let us fix a half‐open interval Example 3.10J, and therefore into C+(J).Every countable left‐orderable group embeds into the group of orientation preserving homeomorphisms of J is dense in J, the next lemma is a basic property of the (piecewise) continuity.Lemma 3.11C(J) is uniquely determined by its values on non‐dyadic rational points on J. Moreover, every function from C(J) is continuous at non‐dyadic rational points.Every function in Since the set of non‐dyadic rational points of To construct respective embeddings into finitely generated simple groups, we propose the following extension of the construction in .3.4G of C(J).Definition 3.12τΣ,J:G→bij(Σ) as follows: for each g∈G, let τΣ,J(g):=gΣ,J, where gΣ,J is defined bygΣ,J is the identity map elsewhere.Define Let us fix a subgroup Definition 3.13T)(The group  Let G be a subgroup of C(J). We define T as the subgroup of bij(Σ) generated by τΣ,J(G) and T(φ).We extend Definition Lemma 3.14G embeds into T by g↦gΣ,J. Moreover, if G is finitely generated, then T is finitely generated as well.The group T and the fact that T(φ) is finitely generated. For the first statement, it is enough to notice that, by definition, gΣ,J is an identity map if and only if g=1.□The second statement follows from the definition of Definition 3.15((Non‐dyadic) rational points) A point ∈Σ is called a rational point on Σ if t∈Q. If, in addition, t is not dyadic, we say that  is a non‐dyadic rational point.Lemma 3.16Σ.There exists a dense and recursive set of non‐dyadic rational points in X:={x1,x2,…}⊂X that is dense in X, for example, the set of proper ternary fractions. Moreover, for all i∈N, let Ri⊂Σ be defined asLet us choose a recursive countable subset R=∪i=1∞Ri. Note that each of Ri is a recursive set. Therefore, since X is also recursive by our choice, then we get that R is recursive as well.□We denote Lemma 3.17G is a subgroup of C(J), then the elements of T are uniquely defined by their values on any (countable) dense set of non‐dyadic rational points of Σ. Moreover, the elements of T are continuous at non‐dyadic rational points of Σ.If Σ. Let R⊂Σ be such a set. Let us define X⊂X such that for each x∈X there exists t∈Q such that ∈R. Since R is dense, X is dense as well.By Lemma X⊂X is dense in X, by Lemma {∣x∈X,t∈R} is dense in Σ. Therefore, the elements of T are uniquely defined by their restrictions to the Φ‐orbits of the elements  for x∈X. Now, the lemma follows from the combination of this observation with Lemma □Note that since 3.5To prove simplicity results, we use the following standard tool.Y be a set, and H a group acting faithfully on Y. Then the rigid stabilizer of a subset U⊂Y is the subgroup of H whose elements move only points from U. We denote the rigid stabilizer of U by RiSt(U).Let T(φ) in  be an element of Thompson's group such that J∩f(J)=∅. Then fX× is in T(φ) and separates UX×J from UX,f(J). By the previous claim, fX,J∈N. Therefore, the first derived subgroup of the rigid stabilizer of the interior of UX×J is in N by Lemma τΣ,J(G) is in the rigid stabilizer of the interior of UX,J. Thus τΣ,J(G)′ is in N. As G is assumed to be perfect, this yields the claim.□Let T=⟨τΣ,J(G),T(φ)⟩.□Now, to conclude the proof of Lemma 3.6Lemma 3.20G⩽C+(J), then the group T is left‐orderable. Moreover, if G is finitely generated and consists of computable functions, then there exists a left‐order on T with recursively enumerable positive cone.If G⩽C+(J), the action of T on Φ‐orbits of elements of X⊂Σ is orientation preserving.First of all, note that since R={,,…} be a fixed, recursively enumerated and dense subset of non‐dyadic rationals in Σ. The existence of such sets is by Lemma Let f∈T, define f>1 if for the smallest index k∈N such that f≠ we have f= such that qk>sk. Therefore, by Lemma f≠1 either f>1 or f−1>1 and for f1,f2>1, f1f2>1. By Lemma T.Now for R is recursive. Therefore, to check whether f>1, we can consecutively compute the values f,f,…. We stop at the first k such that f≠. By Lemma f≠1, and since G consists of computable maps, this procedure recursively enumerates the positive cone of the above defined left‐order.□Recall that the set Corollary 3.21T(φ) has a left‐order with recursively enumerable positive cone.The group 4J is a fixed interval that is strictly contained in ↦. is called a chart. The maps hi are local representations of h.Each of the triples Remark 4.2 tables of Thompson in )+1,t↦f|(t−1)+1), }.Definition 4.6G‐dyadic maps) Let h∈T, and let {}1⩽i⩽n be a chart representation of h such that for every hi, 1⩽i⩽n, one of the following takes place:(I)hi is a dyadic map on Ii;(II)hi=fΛ is the composition a dyadic map f and a G‐dyadic map Λ. Then, the representation {}1⩽i⩽n is called canonical. Also, charts for which hi corresponds to (I) or (II) are called charts of type (I) or (II), respectively.4.2Definition 4.9(Inverse) Let {}1⩽i⩽n be a chart representation of h∈T. Then {}1⩽i⩽n is a chart representation of h−1 and is called the inverse of the initial one.The following operations can be applied to go from one chart representation to another.Definition 4.10(Refinements) The following operations on charts are called refinement.(1)I=I0∪I1, then  can be replaced by ,f|I0) and ,f|I1).If (2)J=J0∪J1, then  can be replaced by (C×f−1(J0),C×J0,f|f−1(J0)) and (C×f−1(J1),C×J1,f|f−1(J1)).If (3)C=C0∪C1, then  can be replaced by  and .If Definition 4.11(Reunions) A reunion is the inverse operation of a refinement.Definition 4.12(Shifts) A shift (of order m∈Z) is replacing a triple  by (φm(C)×(I−m),φm(C)×(J−m),t↦f(t+m)−m).Remark 4.13T. In particular, chart representations of elements from T are not unique.A chart representation obtained by a refinements, reunion, or a shift on its charts corresponds to the same element from Remark 4.14G are computable, the operations Since the functions in Lemma 4.15T is finitely generated and each of the generators can be represented by a canonical chart representation, which can be algorithmically determined.T(φ) given by Lemma G can be represented by a canonical chart representation by definition.□Indeed, the generators of Lemma 4.16The inverse, refinements and shifts preserve the canonicity of chart representations.We will prove only that shift operations on charts of type (II) preserve the canonicity of chart representations, as the rest of statements of the lemma are straightforward.m is applied, is . Then a shift of order m would transform it into the chart (φm(Ci)×(Ii−m),φm(Ci)×(Ji−m),Λ∼), where Λ∼:Ii−m→Ji−m is defined as Λ∼(x)=Λ(x+m)−m.Suppose that the initial chart of type (II), on which a shift operation of order Λ=fg1f1…gnfn is decomposed as in Definition Λ∼=f∼g1f1…gnf∼n, where f∼n(x)=fn(x+m) and f∼(x)=f(x)−m. The chart Λ∼ is also a G‐dyadic map. Therefore, (φm(Ci)×(Ii−m),φm(Ci)×(Ji−m),Λ∼) satisfies the definition of charts of type (II) from Definition □Suppose that Definition 4.17(Composition) Let {}1⩽i⩽n and {}1⩽i⩽m be chart representations such that ⋃Ii=⋃Ji′=. Then we say that the chart representationRemark 4.18f,f′∈T, respectively, then the composition chart representation corresponds to ff′.Note that if, in Definition Remark 4.19Note that if the two chart representations in Definition Lemma 4.20h∈T be given by a canonical chart representation {}1⩽i⩽n. Then there is an algorithm to determine a canonical chart representation {}1⩽j⩽n′ of h such that ⋃Jj′=.Let 1⩽i⩽n:We describe the algorithm. For Ji⊂ do nothing, go to i+1;if Ji∩ and Ji∖ is non‐empty, let Ji1=Ji∩, Ji2=Ji∖ and apply a refinement  from Definition Lemma 4.21h∈T be given by a canonical chart representation {}1⩽i⩽n. Then there is an algorithm to determine a canonical chart representation {}1⩽j⩽n′ of h such that ⋃Ij′=.Let □The proof is analogous to the proof of Lemma Lemma 4.22f,f′∈T, given by their canonical chart representations, computes a canonical representation of ff′.There is an algorithm that for any two elements {}1⩽i⩽n and {}1⩽i⩽m of, respectively, f and f′ such that ⋃Ii=⋃Ji′=. Then their composition will be a canonical chart representation of ff′ , given as a word in finite set of generators, outputs a canonical chart representation of f. In particular, every element from T has a canonical chart representation.There exists an algorithm that for any input From the previous two lemmas we get:T have canonical chart representations, see Lemma □It follows from Remark 4.3T.Lemma 4.24h∈T and let {}1⩽i⩽n be a chart representation of h. Then h=1 in T if and only if, for all 1⩽i⩽n, we have hi=id (and Ji=Ii).Let The following observations are useful for studying the groups 1⩽i⩽n. Recall that h maps  to  for ∈Ci×Ii. As h=1, =. Thus, for any x∈Ci and t∈Ii, there is m∈Z such that (φm(x),t−m)=). We conclude that hi(t)=t−m and φm(x)=x. But φ is a minimal subshift, that is, every orbit of φ is dense. In particular, m=0. Therefore, hi=id. This yields one side of the assertion. The inverse assertion is trivial.□Let Lemma 4.25G‐dyadic map is equal to the identity, then the word problem in T is decidable.If there is an algorithm to decide whether a h∈T one can algorithmically find a canonical chart representation for h. By Lemma h=1 if and only if for any canonical chart representation of h the corresponding charts of types (I), and (II) are identity charts. If hi a local representation in a chart of type (I), hi is a piecewise dyadic map with finitely many breakpoints, so that we can algorithmically check whether hi=id.By Lemma h=fΛ:I→I is a local representation in a chart of type (II), where f:I0→I is dyadic and Λ:In→I0 a G‐dyadic map. We note that h=id on I if, and only if, Λf=id on I0. In fact, Λf is a G‐dyadic map, so that, by assumption, we can algorithmically check whether h=id.□Now suppose that Corollary 4.26T(φ) is computably left‐orderable. In particular, the word problem in T(φ) is decidable.T(φ) has a recursively enumerable positive cone. Thus, by Lemma T(φ) is computably left‐orderable.□By Lemma 5Theorem 5.1G embeds into a finitely generated perfect group H. In addition:(1)G is computable, then H has decidable word problem;if (2)G is left‐ordered, then H is left‐ordered;if (3)G is computably left‐ordered, then the left order on H is computable;if (4)the embedding is a Frattini embedding. Moreover, in case (2) and (3), the order on H continues the order on G.Every countable group Our next goal is to prove the following.G is assumed to be finitely generated, assertion (1) is proved in ∈Σ. As the action of T(φ) on Σ preserves the Φ–orbits, T(φ) acts on the Φ‐orbit of z0, the action is orientation‐preserving, and its orbits are dense. Finally, recall that the Φ‐orbit of z0 is homoemorphic to R. We fix such a homeomorphism. This induces an action of T(φ) on R. We fix this action of T(φ).Let us fix an action of C0 denote the group of functions from R to G of bounded support. The action of T(φ) on R induces an action σ of T(φ) on C0 such that for every h∈C0 and f∈T(φ),Let permutational wreath productG≀RT(φ) is defined as the semi‐direct product C0⋊σT(φ), where, for  and ∈C0⋊σT(φ), :=(h1σ(f1)(h2),f1f2).The g∈G, we define the following function g¯ in C0:G¯:={g¯∣g∈G}.Definition 5.2(Splinter groups) The splinter group is the subgroup of the permutational wreath product G≀RT(φ) generated by G¯ and T(φ). We denote it by Sp.For every T(φ) and G are finitely generated. We note the following.Lemma 5.3G embeds into Sp with image G¯. Moreover, Sp is finitely generated.The group Recall that Lemma 5.4Sp is perfect.The splinter group Lemma 5.5G¯ is in the first derived subgroup Sp′ of Sp.The group To prove Lemma f1 and f2∈T(φ) be such that f1 maps ∈(G≀⟨z⟩)⟨s⟩. Moreover, the element , regarded as a map ⟨s⟩→G≀⟨z⟩, has support ⊆{1}. In addition, (1) is a map ⟨z⟩→G such that )(1)=gi.As it is shown in Section g,h∈G, their images in ⟨c,s⟩ are conjugate. Let g¯ and h¯ be the images of g and h in ⟨c,s⟩, respectively. In particular, g¯ and h¯ are elements of the form . Let ∈⟨c,s⟩⩽(G≀⟨z⟩)≀⟨s⟩ be such that g¯−1=h¯. Then we getsupp(fg¯snf−1)=supp(g¯)={1}. On the other hand,n=0, hence g¯(1) is conjugate to h¯(1) in G≀⟨z⟩. Repeating this argument one more time with respect to the pair g¯(1),h¯(1)∈G≀⟨z⟩ and using the fact that (g¯(1))(1)=g and (h¯(1))(1)=h, we get that g is conjugate to h in G. Since g,h∈G are arbitrarily chosen elements from G, we get that the embedding from . Since every left‐ordered group embeds as a subgroup into Homeo+(J), we have the following.Proposition 6.1G embeds into a finitely generated left‐ordered group H. In addition, the order on H continues the order on G.Every countable left‐ordered group Let G be countable left‐orderable group. Then, by Theorem G embeds into a finitely generated perfect left‐orderable group G1. On its own turn, since G1 is left‐orderable, it embeds into Homeo+(J). Let G2⩽Homeo+(J) such that G2 is isomorphic to G1. Let H=T, see Definition H has the required properties.□Let G is finitely generated), as required by Remark Homeo+(J).We now construct an embedding as in the previous proposition, that, in addition, is Frattini and isometric m∈Z, there exists at most one x∈QI such that {x}d=m and {g(x)}d⩽0.For each (2)x1≠x2∈QI and {x1}d={x2}d, then {g(x1)}d≠{g(x2)}d.If Let g is a bijection from I to J, then we say that g is strongly permuting the dyadic parts if it maps rational points to rational points and its restriction g∣QI:QI→QJ satisfies Definition Remark 6.5g:QI→QJ is strongly permuting the dyadic parts, then, for each m∈Z, the set {{g(x)}d∣x∈QI,{x}d=m} is unbounded from above.If If 0<i⩽n:fi:Ii→Ji such that fi(x)=λix+ci, where λi is a power of 2 and, for all i∉{0,n}, ci≠0; andbijective dyadic maps gi:Ji→Ii−1, whose restrictions to QJi strongly permute the dyadic parts.bijective maps Lemma 6.6Λ=g1f1g2f2…gnfn, then, for large enough N∈N, the setΛ≠id.If Let us consider, for n.We will prove the lemma by induction on n=1. Then Λ(x)=g1(λ1x+c1). If c1=0 or N>{c1}d and {x1}d={x2}d=N, then, by Lemma {λ1x1+c1}d={λ1x2+c1}d. The statement now follows as g1 strongly permutes the dyadic parts (see Remark Let n>1. Then Λ(x)=g1(λ1Λ2(x)+c1), where Λ2=g2f2…gnfn and c1≠0. By inductive assumption, for any large enough N, there exists a sequence {xi}i=1∞ such that {xi}d=N and limi→∞{Λ2(xi)}d=∞. By Lemma i, {λ1Λ2(xi)+c1}d={c1}d, hence, the lemma follows as g1 strongly permutes the dyadic parts.□Next let 6.2J be a fixed closed interval in R with non‐empty interior. We prove:Proposition 6.7G be a countable group.Let G is left‐orderable, then there is an embedding Ψ:G↪Homeo+(J) such that, for all g∈G∖{1}, the map Ψ(g):J→J is strongly permuting the dyadic parts and does not fix any rational interior point of J.If G is computably left‐orderable, then, in addition, all the maps Ψ(g) can be taken to be computable.If Let QJ={q0,q1,…} such that the natural order on QJ is computable with respect to this enumeration.As in the proof of Proposition Φ:G→QJ.Lemma 6.8G is enumerated and densely left‐ordered, then there is an enumeration G={gi1,gi2,…} and an order preserving bijection Θ:G→QJ such that:(1)j, {Θ(gij)}d∈N and {Θ(gij)}d∉{{Θ(gik)}d∣1⩽k<j};for odd (2)j, gij∉{gikgil−1gim∣1⩽k,l,m<j};for even (3)G is computably left‐ordered, then the enumeration G={gi1,gi2,…} and the map j↦Θ(gij) are computable.if If We first strengthen Lemma G={1=g0,g1,g2,…} and QJ={0=r0,r1,r2,…} be fixed (recursive) enumerations. We define Θ:g0↦r0 and Θ:gij↦rij, where  and  are permutations of  and , respectively, defined recursively as follows.Let Step2n+1. Let G2n={gi1,…,gi2n} and Q2n={ri1,…,ri2n} be already defined. Let us define gi2n+1 as the element of the smallest index that is not in G2n. Suppose that gis<gi2n+1<git and that no element from G2n is in between gis and git. Then define ri2n+1∈QJ to be of the smallest index such that:(O1)ri2n+1∉Q2n and ris<ri2n+1<rit;(O2){ri2n+1}d∈N and {ri2n+1}d∉{{rij}d∣1⩽j⩽2n}.Step2n+2. Let G2n+1:={gi1,…,gi2n+1} and Q2n+1={ri1,…,ri2n+1} be already defined. Let us define ri2n+2 as the rational of the smallest index that is not in Q2n+1. Suppose that ris<ri2n+2<rit and that no element from Q2n+1 is in between ris and rit. Then let us define gi2n+2∈G as the element of the smallest index such that:(E1)gi2n+2∉G2n+1 and gis<gi2n+2<git; and(E2)gi2n+2∉{gikgil−1gim∣1⩽k,l,m⩽2n+1}.Θ defined this way is order preserving by (O1) and (E1). Condition (O2) yields assertion (1), and (E2) yields assertion (2). Finally, as the procedure is algorithmic, we also obtain assertion (3).□The bijection Proof of Proposition 6.7G is dense. Let the enumeration G={g0,g1,…} and Θ:G→QJ satisfy the assertions of Lemma Θ:G→QJ induces the embedding ρGΘ:G↪Homeo+(J) according to ρGΘ(g)(Θ(h))=Θ(gh) for g,h∈G. Denote Ψ=ρGΘ. Let h∈G∖{1}. By Lemmas Ψ(h) is strongly permuting the dyadic parts.By Remark QJ such that Θ(gi)=ri and let ri≠rj∈QJ. We define rk=Ψ(h)(ri)=Θ(hgi) and rl=Ψ(h)(rj)=Θ(hgj), so that gk=hgi and gl=hgj.To this end, we enumerate i≠j such that {ri}d={rj}d and {rk}d,{rl}d∉N. Since {rk}d,{rl}d∉N, the indices k and l are even. Then, since gk=hgi=(hgj)(gj−1)(gi) and gl=hgj=(hgi)(gi−1)(gj), by (2) of Lemma i or j. Let j>i. Then, since {ri}d={rj}d, we get the index j is even. Since gj=gi(hgi)−1(hgj), again by (2) of Lemma We first show property (1) of Definition ri≠rj∈QJ such that {ri}d={rj}d and suppose that {rl}d={rk}d. Without loss of generality, l>i,j,k . Then, since {rk}d={rl}d, by (1) of Lemma l has to be even. Therefore, since Θ(hgj)=rl and l is even, by (2) of Lemma hgj∉{gmgn−1gp∣1⩽m,n,p<j}. On the other hand, since l>i,j,k, we get hgj=(hgi)(gi−1)(gj), a contradiction.Next, we prove property (2) of Definition □This completes the proof of Proposition 6.3G be countable left‐orderable group. Then, by Theorem G embeds into a finitely generated perfect left‐orderable group G1. Moreover, this embedding is a Frattini embedding.Let J:=. Since G1 is left‐orderable, there is an embedding Ψ:G1↪Homeo+(J). Let G2=Ψ(G1). By Proposition G2⊆Homeo+(J) strongly permute the dyadic parts and do not fix any rational interior point of J.Let G2–dyadic maps, see Definition Lemma 6.9Λ be a G2–dyadic map. If G2 has decidable word problem, then there is an algorithm to decide whether Λ=id.Let For the definition of n>0 and, for all 0⩽i⩽n, let Ji⊂J and let gi:Ji→Ii−1 be the restriction of an element of G2 such that gi≠id. Moreover, let fi:Ii→Ji, given by fi(x)=λix+ci, be dyadic maps such that fi≠id.Let 0<i<n, Ji⊂ and, by definition, λi is a power of 2, we get that ci≠0 for 0<i<n. Then, by Lemma Λ:=g1f1g2f2…gnfn≠id.Since, for all n=1 and f1=id, then Λ=g1∈G. Then we decide using the algorithm for the word problem in G.□If Lemma 6.10G1↪T is isometric.The embedding Combining Lemma Lemma 6.11G1↪T is a Frattini embedding.The embedding h,g∈G2 and t∈T. We assume that ht−1gt=1.Let t by a canonical chart representation  such that ⋃iJi=; and represent t−1 by . We recall that ti(Ii)=Ji.We represent k be a index such that 1/2 is in the closure of Jk and such that (after applying a chart refinement if necessary) Jk⊆J. As g is fixing 1/2, there is Jk′⊆Jk such that g(Jk′)⊆Jk and such that 1/2 is in the closure of Jk′. We let Ik′=tk−1(Jk′) and Ik′′=tk−1gtk(Ik′).Let  is in a chart representation of t−1gt. Up to applying the algorithm of Lemma Ik′′ is in . Moreover, up to applying a chart refinement if necessary, we may assume that either Ik′′∩J is empty or consists of one point (1/4 or 1/2), or Ik′′⊆J.Then the triple Ik′′∩J is empty or consists of one point, then  is in a chart representation of ht−1gt. Thus tk−1gtk=id on Ik′ by Lemma g acts as the identity on Jk′. Since non‐trivial elements of G2 do not fix any rational interior points of J, the element g=1.If ,htk−1gtk) is in a chart representation of ht−1gt. Thus htk−1gtk=id on Ik′ by Lemma h(Ik′′)=Ik′. This is only possible if Ik′⊆J. But then Proposition tk acts as an element of G2. Since non‐trivial elements of G2 do not fix any rational interior points of J, this implies that g and h are conjugate in G1.□Otherwise, the triple We can now conclude Theorems Proof of Theorems 1, 2 and Remark 1.2H=T.We let H is finitely generated left‐orderable and simple by Lemmas G embeds into H, and the order on H extends the order on G. Moreover, by Lemma G is a Frattini embedding.The group G is computably left‐ordered, we may in addition assume that G1 is computably left‐ordered, see Theorem g∈G1, Ψ(g) is computable. Therefore, by Lemma H is recursively enumerable. Moreover, by Lemma H has decidable word problem. By Lemma H is computable.□If Proof of Theorem 4G be finitely generated left‐orderable group with a recursively enumerated positive cone. If G has decidable word problem, then the left‐order on G is computable by Lemma G embeds into a finitely generated computably left‐ordered simple group H. In particular, the word problem in H is decidable. Thus H can be defined by a recursively enumerable set of relations. By  and J= and that is identity outside of ∪i=1∞Di. In particular, the map ξl:Dk→Dm is dyadic.Remark 7.3ξl:J→J, l∈N, are computable and continuous at non‐dyadic points. Also, they have finitely many (dyadic) breakpoints outside of any open neighborhood of 1/3.The maps For every λ:G1→C(J) by λ(g(l))=ξl, for all l∈N.Remark 7.4λ is an embedding of G1 into computable maps in C(J).The map Let us define Λ=g1f1g2f2…gnfn:In→I0, where fi:Ii→Ji and gi:Ji→Ii−1, be a G2‐dyadic map as in Definition Ji⊆J=. Correspondingly, we say that a chart of type (II), see Definition special if the local representation in this chart is of the form f0Λ, where Λ is a special G2–dyadic map.Let fi, gi, 1⩽i⩽n, are computable.Lemma 7.5K1,K2,…,Ks⊆ fixes 1/3, then it acts as an element of G2.Special local representations are of the form By Lemma G2 is decidable, this implies that there exists an algorithm that decides whether a special G2‐dyadic map represents the identity map. This, combined with Lemmas Corollary 7.8T is decidable.The word problem in Since the word problem for 7.2G↪T is Frattini.Lemma 7.9G1↪T is a Frattini embedding.The embedding The embedding We adapt the proof of Lemma h,g∈G2 and t∈T. We assume that ht−1gt=1.Let t by a canonical chart representation {} such that ⋃iJi=; and represent t−1 by {}. We recall that ti(Ii)=Ji.We represent k be a index such that 1/3 is in the closure of Jk and such that (after applying a chart refinement if necessary) Jk⊆J. As g is fixing 1/3, there is Jk′⊆Jk such that g(Jk′)⊆Jk and such that 1/3 is in the closure of Jk′. We let Ik′=tk−1(Jk′) and Ik′′=tk−1gtk(Ik′).Let  is in a chart representation of t−1gt. Up to applying the algorithm of Lemma Ik′′ is in . Moreover, up to applying a chart refinement if necessary, we may assume that either Ik′′∩J is empty or consists of one point 1/3, or Ik′′⊆J.Then the triple Ik′′∩J is empty or consists of one point 1/3, then  is in a chart representation of ht−1gt. Thus tk−1gtk=id on Ik′ by Lemma g acts as the identity on Jk′. As 1/3 is in the closure of Jk′, this implies that g=1.If ,htk−1gtk) is in a chart representation of ht−1gt. Thus htk−1gtk=id on Ik′ by Lemma tkhtk−1g:Jk′→Jk′ has to be the identity as well. As g is fixing 1/3, tkhtk−1 has to fix 1/3.Otherwise, the triple tk was dyadic ), it would have to fix 1/3, so that tk=id by Lemma tk is of type (II), we may assume that tk is special, see Lemma tk acts as an element of G2.If g and h are conjugated by elements of G2. This implies that g and h are conjugated in G1.□Thus Corollary 7.10G↪T is Frattini.The embedding Combining Lemma Lemma 7.11G1↪T is an isometric embedding.The embedding X for G1, and denote the generating set of T(φ) given by Lemma Y. We denote the union of the bijective images of X and Y in T by Z and recall that Z generates T. We assume that all generating sets are symmetric. We denote by |.|A the word metric with respect to the generating set A.We fix a finite generating set g∈G2 and let t=z1⋯zm be a reduced word in the alphabet Z that represents g−1∈T, so that tg=1. In addition, we assume that m=|g|Z. We represent every generator zi by a canonical chart representation, see Lemma  for t. Recall that the maps ti are compositions ti=h1⋯hmi, where each map hj is a local representation in the canonical chart representation of a generator in Z and mi⩽m. In addition, up to applying the algorithm of Lemma t, we may assume that ⋃Ii=.Let Ik be an interval such that 1/3 is in the closure of Ik and such that (after applying a chart refinement if necessary) Ik⊆J.Let (Ci×g−1(Ik),Ci×Ji,tig) is in a canonical chart representation of the identity. By Lemma tig is the identity mapping. In particular, g−1(Ik)=Ji, so that Ji⊆J and 1/3 is in the closure of Ji.Then ti is not dyadic ) by Lemma ti=fg1f1⋯gnfn is a chart of type (II), then, by Lemma ti is special. Thus ti=g1⋯gn∈G2  is isometric.If Combining Lemma Proof of Theorem 7.1T follows from Lemma T is decidable provided that it is decidable in G2. By Corollary G↪T is Frattini. By Corollary G is finitely generated. Therefore, the embedding G↪T satisfies Theorem □The simplicity of Journal of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not‐for‐profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.The


Historic records suggest that a virulent form of treponematosis, sexually transmitted syphilis was introduced to Asia from Europe by the da Gama crew, who landed in India in 1498. Our objective is to assess the gross pathology of human skeletal remains from the Tang dynasty of China to test the presence of treponemal infection in East Asia before 1498. We interpret this paleopathological evidence in the context of site ecology and sociocultural changes during the Tang dynasty.We examined the gross pathology of 1598 human skeletons from Xingfulindai (AD 618 to AD 1279) archeological site located on the Central Plain of China. Using the modified diagnostic criteria defined by Hackett's classical work, we classify the pathology as consistent, strongly suggestive, or pathognomonic for treponemal infection.Twelve adult individuals from Xingfulindai had bone lesions suggestive of systemic pathology. Two of these individuals displayed a combination of lesion patterns pathognomonic of treponemal disease and one had lesions consistent with treponematosis. The radiocarbon dates for the bone samples from these skeletons place them before AD 1200.The location of Xingfulindai in a continental climatic zone is not typical for yaws and bejel ecology, because these strains occur in the tropics, or in hot, dry environments, respectively. The urban setting, where there is documented evidence for increased interaction between multiple ethnic groups and a developed institution of courtesans during the Tang dynasty, favors sexually transmitted syphilis as the more likely diagnosis. This study supports an earlier spread of syphilis to China than 1498. Syphilis in China before Vasco da Gama? Together with the tropical skin disease pinta, linked to T. carateum, these infectious diseases are collectively referred to as treponematoses , yaws, and endemic syphilis (bejel) are caused by three subspecies of the spirochete bacteria Historically, three alternative models for the introduction of treponemal infection to continental China can be found in the literature: the Vasco da Gama model Bi, , the TraT. p. ssp pertenue (yaws) in Vietnam about 4000 years ago. They consider the migration of farmers from Southern China into the region as a route of transmission for this disease. Finally, it is conceivable that different forms of treponemal diseases have been present for a long time but evaded historic documentation in some regions by being confused with other diseases  tomb area  is located in the suburb of Xi'an City, Shaanxi Province , rhinopharyngitis mutilans (gangosa) and saber shin, as well as generalized subperiosteal proliferative lesions, substantial sclerosis and osteitis, and involvement of the medullary cavity of the long bones have all been recorded in patients suffering from yaws, bejel, and syphilis  suis Grin, . Therefois Grin, .Treponema cases. These categories were assigned based on Hackett  on the occipital and parietal bones  had cranial hyperostosis or widespread periostosis or osteitis on multiple long bones suggestive of systemic pathology , which produces bright red bumpy fruits reminiscent of syphilis sores. He noted that the disease started spreading from the Lingbiao area of Guangzhou. His text also described a treatment for the disease, including the use of mercury ointment:Consistent with the Vasco da Gama model, the earliest direct description of ST syphilis can be found in “Compendium of Materia Medica” 本草纲目 (c1578‐1596) by Li Shizhen 李時珍 Li, , which i“杨梅疮，古方不载，亦无病者，近时起于岭表，传及四方，盖岭表风土卑炎，岚瘴熏蒸，饮啖辛热，男女淫猥。湿热之邪积蓄既深，发为毒疮。遂致互相传染，然皆淫邪之人病之。“In the past, no books mentioned syphilis (myrica sores) nor how to cure it. Syphilis probably started in the Guangxi and Guangdong area and began to spread. The fields there are low‐lying and the weather is hot, humid, and foggy. Thus, people there like to eat spicy food. Because of this, it makes symptoms worse, causing toxic sores which begins to infect others, but only infects those who had sexual behavior.”医用轻粉，银朱劫剂，五、七日即愈。其性燥烈，善逐痰涎。涎乃脾之液，此物入胃，气归阳明，故涎被劫，随火上升，从喉颊齿缝而出，故疮即干痿而愈。若服之过剂，及用不得法，则毒气窜入经络筋骨之间，莫之能出。痰涎既去，血液耗涸，筋失所养，营卫不从，变为筋骨挛痛，发为痈毒疳漏。久则生虫为癣，手足皱裂，遂成废痼。“Use medical calomel (mercury chloride) or cinnabar (mercury sulfide) to cure it. It usually takes five to seven days to cure. These types of drugs are so strong that they will easily make the spleen produce saliva. When the drugs go into the stomach, it will produce good energy and send it to yangming. Then, the energy will go up with the saliva that is produced from the spleen via the throat to the mouth, and it will gradually make the sores smaller, eventually curing them. However, if the dose is too large, then it can worsen the symptoms because the overdose can be toxic and produce bad energy to go into the meridian system. What's even worse, the bad energy has no way to get out, which will result in having less saliva, low blood pressure, weak bones, malnutrition, and then causes bone pain. This ultimately results in ulcer and gum disease. If the patient ignores this for a while, it will cause skin ulcers, and their hands and feet will begin cracking and develop a chronic illness.”其类有数种，治则一也。其证多属厥阴，阳明二经，而兼乎他经。邪之所在，则先发出，如兼少阴太阴，则发于咽喉。兼太阳、少阳，则发于头耳之类。 “Although there are many types of syphilis, the principles are the same. Syphilis will send bad energy to jueyin and yangming. People will feel discomfort in different parts of their body, depending on where the bad energy moves to. For example, if the energy goes to shaoyin and taiyin, the patient will feel discomfort in their throat. If it moves to taiyang and shaoyang, then the patient will feel discomfort around the head and ears.” In this passage, “jueyin,” “yangming,” “shaoyin,” “shaoyang,” “taiyang,” and “taiyin” are the circulatory tracks of traditional Chinese medicine, which can be thought as the major flows or meridians of the qi energy, with distinct locations in the body Needham, :48. For Smilax glabra) is a good remedy to cure yangming disease, because it tastes sweet and light, which helps smooth the “fire”—bad energy. […] Presently, the soufeng jiedu (搜风解毒) soup is used to treat syphilis. Curing patients with serious symptoms by drinking soufeng jiedu will take a month; those with light symptoms are cured within half a month. To make soufeng jiedu soup use one liang  of 土茯苓 tufuling (Smilax glabra) and 1.5 g of 薏苡仁 yiyiren (Job's tears: Coix lacryma‐jobi), 金银花 jinyinhua (Japanese honeysuckle Lonicera japonica), 防风 fangfeng (Saposhnikovia divaricate), 木瓜 mùguā , 木通 mutong (chocolate vine Akebia quinata), and 白鲜皮 báixiǎnpí  each respectively, as well as 1.2 g of 皂荚子 zaojiazi (Acacia). For people who do not have enough good energy, one can add 人参 rencan  2.1 g; people who have low blood pressure can add 当归 danggui  2.1 g). Cook all of the herbs with two bowls of water and make soup. Drink the soup three times a day and avoid drinking tea, eating beef, lamb, and chicken; maintain abstinence. This is the secret formula.”惟土茯苓气平，味甘而淡，为阳明本药。[…] 今医家有搜风解毒汤治杨梅疮，不犯轻粉。病深者月余，浅者半月而愈。服轻粉而筋骨挛痛，瘫痪不能动履者，服之亦效。其方用土茯苓一两，薏苡仁、金银花、防风、木瓜、木通、白鲜皮各五分，皂荚子四分。气虚，加人参七分；血虚，加当归七分。惟忌茶、肉、法面，房劳。盖秘方也。 “Powdered tu fuling . The text described two types of sores,  Morgan, . However4.2caries sicca pathognomonic or strongly suggestive of treponematosis has been described on a skull from Fujian Province dated to the Song dynasty (AD 960–1279)  Zhang,   Zhang, . However) Zhang, . Two pos) Zhang,   Zhang, . In this) Zhang,  proposed) Zhang, . Because) Zhang,  findings) Zhang,  dating tThe Island Southeast Asia and Pacific model developed by Buckley and Oxenham , reviews4.3T. pallidum it is important to examine the environmental and sociocultural conditions that favored the emergence and spread of the new forms of treponemal disease. The Xingfulindai cases of treponemal infection, with direct radiocarbon dates that place skeletons prior to 1400s, and their association with the large urban population of Chang'an, present us with such an opportunity. According to the New Book of Tang (chapter 41.27:1669), the capital city of Chang'an had 362,921 households with a total population of 960,188 people within its limits and ruled over 20 provinces. The population of Chang'an was multiethnic, with a notable influx of people from the northwestern frontier area of imperial China as well as Persia , was extremely active, promoting an increased level of interaction, bringing into contact previously isolated areas Waugh, . As suchIf these pathological cases from Xingfulindai represent individuals with sexually transmitted syphilis, it is important to investigate how this would have affected the community by considering the known transmission rate of this disease, and the functional effects on people. Unfortunately, unbiased research on sexual transmission of syphilis is rare ; data curation (lead); formal analysis ; funding acquisition ; investigation (lead); project administration ; resources ; supervision (lead); visualization . Guoshuai Gao: Data curation ; investigation . Xiangyu Zhang: Investigation (supporting). Bo Gao: Data curation (supporting); investigation (supporting); resources (supporting). Chenggang Duan: Data curation (supporting); investigation (supporting). Hong Zhu: Formal analysis (supporting); supervision (supporting). Aida R. Barbera: Formal analysis (supporting); investigation (supporting); writing – original draft (supporting). Siân Halcrow: Conceptualization ; funding acquisition ; methodology (supporting); writing – original draft (supporting). Kate Pechenkina: Conceptualization (lead); formal analysis (lead); investigation ; methodology (lead); project administration ; resources (supporting); supervision ; visualization ; writing – original draft .Figure S1 Supplementary figuresClick here for additional data file.Table S1 Supplementary tablesClick here for additional data file.


Monobius Alatawi and Kamran, where all the leg tarsi in females have one midventral seta. Moreover, the genus Tillandsobius Bolland is synonymized with the genus Tycherobius Bolland and the genus Neophyllobius Berlese is categorized in two new subgeneric divisions. For the first time, a key to all known species of the genus Neophyllobius is provided. The ambiguities in the ventral idiosoma setal notation are highlighted and discussed.The present study erects a new genus, Monobius Alatawi and Kamran, is hereby proposed for the two already described species, viz; M. electrus (Żmudziński) and M. meyerae (Bolland). In addition, the monospecific genus Tillandsobius Bolland is synonymized with the genus Tycherobius Bolland due to variations in the setae number of tibiae I–IV. Further, the genus Neophyllobius Berlese is categorized in two new subgeneric divisions as Neophyllobius Berlese and Monophyllobius Mirza. The number and position of the midventral setae on tarsi I–IV are considered as strong diagnostic generic and subgeneric diagnostic characters. The present study also includes the key to all known species of the genus Neophyllobius. The morphological characters of ten poorly described Neophyllobius species were studied in detail through published literature. The ambiguities in the ventral idiosoma setal notation are highlighted and discussed. It is concluded that two intercoxal setae 3a–4a are always present on small platelets, paired aggenital setae (ag) are present anteriorly and paired genital setae (g) present posteriorly on genital shield. In addition, five records of new species for Saudi Arabia are reported along with re–descriptions of three species.A new genus,  Neophyllobius Berlese ..NeophyllNeophyllobius has one or two midventral setae on tarsi I–IV, if two setae are present, they are in a longitudinal line. While the genera Tillandsobius and Tycherobius always have two midventral setae on tarsi I–II, consistently not in a longitudinal line and variously spaced (Tillandsobius (one species) and Tycherobius (25 species) are closely related.These three genera were also differentiated based on the position of midventral setae on tarsi I–II. The genus y spaced . In thisNeophyllobius, Tillandsobius and Tycherobius. However, the number and position of the mid–ventral setae on tarsi I–II, remain a constant and persistent generic diagnostic character. Therefore, the monospecific genus Tillandsobius is hereby synonymized with the genus Tycherobius. In addition, we propose a new genus, Monobius Alatawi and Kamran gen. nov., for the two species  namely, M. meyerae (Bolland) and M. electrus (Żmudziński), described originally in the genus Neophyllobius. The diagnoses of the genera Neophyllobius and Tycherobius are modified and provided below. The morphological characters of these three genera, including the new genus, are summarized in Based on the evidence provided above, the number of setae on tibiae I–IV does not represent a strong morphological character to differentiate the three genera urn:lsid:zoobank.org:act:3B429EFF-A148-46A5-8BE4-32127A03E707Type species: Neophyllobius electrus Żmudziński, 2020:3 [, 2020:3 .Diagnosis: Leg tarsi I–II with one mid-ventral seta present on distal half and a proximal solenidion, tarsi I–IV with 9–9–7–7 tactile setae.Remarks: The new genus Monobius, is morphologically closer to the genera Neophyllobius and Tycherobius  and distinct from Acamerobia, Camerobia, Bisetolobius and Decaphyllobius . It can be further distinguished from Neophyllobius and Tycherobius due to the presence of one midventral seta distally on leg tarsi I–II vs. two midventral setae on leg tarsi I–II in later two genera.Etymology: The generic epithet is derived from the diagnostic character of one midventral seta on all leg tarsi (mono = one)Monobius includes two species M. meyerae [M. electrus [Neophyllobius.The new genus  meyerae  and M. eelectrus . Both spNeophyllobius meyerae Bolland, 1991:63 [ 1991:63 .Remarks: The species M. meyerae (Bolland) is referred to the new genus Monobius due to presence of one midventral seta on all leg tarsi. The species can be differentiated from the second species of the genus M. electrus (Żmudziński) based on number of setae on femur III (2 vs. 3), number of dorsal body setae (15 vs. 14) and state of pdx setae (present vs. absent).Distribution: South AfricaNeophyllobius electrus Żmudziński, 2020:3 [, 2020:3 .Remarks: The species M. electrus (Żmudziński) is referred to the new genus Monobius due to presence of one midventral seta on all leg tarsi.Distribution: Fossil preserved in Baltic Amber, PolandTycherobius Bolland, 1986: 205 [Tillandsobius Bolland, 1986: 205; synonym nov.Diagnosis: Two midventral setae on tarsi I–II, not present in a longitudinal line,Remarks: The monospecific genus Tillandsobius was erected by Bolland [Tycherobius only by difference in number of setae on tibiae I–IV. As mentioned earlier, tibial setal counts are variable among the species of the genus Tycherobius and cannot be considered as a generic diagnostic character , tarsi I–IV (10–10–7–7 vs. 7–7–8–8), femur II–III (3–2 vs. 4–3) and differences in length of dorsal body setae .Distribution: Florida, USANeophyllobius Berlese, 1886:19 [Type species: Neophyllobius elegans Berlese, 1886:19 [Diagnosis: Two midventral setae on tarsi I–II, present in a longitudinal line.Neophyllobius, 114 species have two midventral setae on all leg tarsi. In contrast, 14 species have one midventral seta on leg IV and seven species with no such information available [Neophyllobius is categorized in two new subgenera; Neophyllobius Berlese and Monophyllobius Mirza, based on the number of midventral setae on leg tarsi III–IV.Among the species of the genus vailable ,8,9,16. urn:lsid:zoobank.org:act:D34B5F98-1F09-45C8-8C64-62CA683C154EType species: Neophyllobius elegans Berlese, 1886:19 [Diagnosis: Leg tarsi III–IV always with two midventral setaeNumber of species included: 114urn:lsid:zoobank.org:act:8648D2D3-7F38-442A-8C15-680FB8D560A7Type species: Neophyllobius texanus McGregor, 1950:66 [ 1950:66 .Diagnosis: Leg tarsus III often and tarsus IV always with one midventral setaEtymology: The subgeneric epithet refers to the presence of one midventral seta on leg tarsi III and IV (mono = one)Number of species included: 14N. combreticola, N. lorestanicus, N. denizliensis, T. emadi and C. southcotii. The species N. combreticola along with two previously reported species, viz; N. muscantribii and N. fissus are redescribed in detail. In addition, the species N. lorestanicus and N. denizliensis were previously misidentified as N. communis and N. hispanicus, respectively.The present research reported five new records of camerobiid mites from Saudi Arabia, viz; Neophyllobius combreticola Bolland, 1991:196 [1991:196 ; Beyzavi1991:196 .Redescription n = 3)Female  and one sword-like seta, tarsus with two eupathidia (acmζ and sulζ), two simple setae (ba and va) and one small solenidion (ω)  .Dorsum  present, genito-anal valves with a pair of genital setae (g) and three pairs of anal setae ps1, ps2 and ps3 –1(κ)–1–1, tibiae 9(φ)–8(φ)–8(φ)–7(φ), tarsi 10(ω)–10(ω)–8–8. All tarsi with ambulacrum bearing a pair of claws and an empodium with two rows of tenent hairs, all genu setae long and whip like, dorsal seta on genu I 290–300 Female  and one sword–like seta, tarsus with two eupathidia (acmζ and sulζ), two simple setae (ba and va) and one small solenidion (ω).athosoma : 51–55 lDorsum , 4c 43–44, intercoxal setae length: 1a 32–35, 3a 34–46, 4a 25–27 (ag) 14–15, genito-anal valves with a pair of genital setae (g) 10–13 and three pairs of anal setae ps1, ps2 and ps3 –1(κ)–1–1, tibiae 9(φ)–8(φ)–8(φ)–7(φ), tarsi 10(ω)–10(ω)–8–8. All tarsi with ambulacrum bearing a pair of claws and an empodium with two rows of tenent hairs, all genu setae long and whiplike, dorsal seta on genu I 218–220 , Al–Qatif, SA, 26°34.36.7′ N, 49°59.07.1′ E, 25 January 2014 and 11 December 2013, coll. Kamal Alsahwan; one female, date palm (Arecaceae), Al–Imam University, Riyadh, SA, 24°48.785′ N, 46°42.142′ E, 19 March 2011, coll. Kamal Alsahwan.Previous Distribution: TrinidadNeophyllobius muscantribii Bolland, 1991:73 [ 1991:73 .Neophyllobius eragrostidis Bolland, 1991:73 [Eragrostis (plant), syn. by du Toit, Theron and Ueckermann, 1998 [ 1991:73 ; Eragrosnn, 1998 .Redescription n = 3)Female  and one sword-like seta, tarsus with two eupathidia (acmζ and sulζ), two simple setae (ba and va) and one small solenidion (ω).athosoma : 54–60 lDorsum  14–15 present, genito–anal valves with a pair of genital setae (g) 7–9 and three pairs of anal setae   .Leg (κ)–1(κ)–1–1, tibiae 9(φ)–8(φ)–8(φ)–7(φ), tarsi 10(ω)–10(ω)–8–8. All tarsi with ambulacrum bearing a pair of claws and an empodium with two rows of tenent hairs, dorsal seta on genu I (38–41) twice as long as the segment , Asir, SA, 19°15.884′ N, 42°05.261′ E, 29 October 2019, coll. M. Kamran and H. M. S. Mushtaq, 2 females, Kavi plant and Sider like, Al–Baha, SA, 19°59.807′ N, 41°25.715′ E, 25 April 2013, coll. Kamal Alsahwan.Previous Distribution: South AfricaNeophyllobius lorestanicus Khanjani, Hoseini, Yazdanpanah and Masoudian, 2014: 441 [014: 441 .Remarks: The species N. lorestanicus is a new record for the camerobiid mite fauna of Saudi Arabia and no distinct morphological differences were found between Saudi Arabian specimens of N. lorestanicus and the original description. It belongs to the Neophyllobius subgenus nov. based on presence of two in-line midventral setae on tarsi I–IV.Material Examined: Two females, date palm (Arecaceae), Irqa, Riyadh, SA, 24°41.130′ N, 46°34.550′ E, 11 December 2013, coll. Kamal Alsahwan; one female, Acacia spp, (Fabaceae). Al–Ahsa, SA, 25°55.371′ N, 48°59.037′ E, 22 October 2020, coll. J. H. Mirza, N. A. Elgoni, H. M. Sajid and M. W. Khan.Previous distribution: IranNeophyllobius denizliensis Akyol, 2020:88 [Remarks: The species N. denizliensis is a new record for the camerobiid mite fauna of Saudi Arabia. It was previously misidentified as N. hispanicus [Neophyllobius subgenus nov. based on presence of two in-line midventral setae on tarsi I–IV. No morphological differences were found between Saudi Arabian specimens and original description of the N. denizliensis.spanicus . It beloMaterial Examined: One female, Tamarix spp., (Tamaricaceae), Tabuk, SA, 28°30.653′ N, 36°28.168′ E, September 29, 2020, coll. J. H. Mirza, H. M. S. Mushtaq and E. M. Khan; one female, Phoenix dactylifera (Arecaceae), Irqa, Riyadh, SA, 13 December 2020, coll. Kamal Alsahwan; one female, grasses (Poaceae). Al–Imam University, Riyadh, SA, 24°48.785′ N, 46°42.142′ E, 7 April 2016, coll. M. Kamran.Previous distribution: TurkeyTycherobius Bolland, 1986:205 [1986:205 .Type species: Neophyllobius lombardinii Summers and Schlinger, 1955 [er, 1955 .Tycherobiusemadi Khanjani, Hajizadeh, Ostovan and Asali FayazTycherobius emadi Khanjani, Hajizadeh, Ostovan and Asali Fayaz, 2013:134 [2013:134 ; Hoseini2013:134 .Remarks: The species Tycherobius emadi is first time reported from Saudi Arabia and is morphologically similar to the original description [cription .Material Examined: One female, soil, and leaf debris, Tabuk, SA, 26°58.271′ N, 49°40.220′ E, 19 October 2019, coll. M. Kamran and H. M. S. Mushtaq.Previous distribution: IranCamerobia Southcott, 1957: 306 [957: 306 .Type species: Camerobia australis Southcott, 1957 [tt, 1957 Camerobia southcotti GersonCamerobia southcotti Gerson, 1972: 502 [972: 502 .Remarks: The species Camerobia southcotti is a new record for the camerobiid mite fauna in Saudi Arabia and is morphologically similar to the original description [cription .Material Examined: Three females, unidentified wild host plant, Jeddah, SA, 26°58.271′ N, 49°40.220′ E, 25 April 2016, coll. M. Kamran and J. H. Mirza.Previous distribution: IsraelThe following 10 species are not included in the key either due to lack of available literature or incomplete descriptions/illustrations. The available but scarce information from different literatures about these species are summarized in the Neophyllobius elegansBerleseNeophyllobius elegans Berlese, 1886:19 [ 1886:19 ; Canestr 1886:19 ; Bolland 1886:19 .Bolland  reportedTable 2Neophyllobius guajavaeChatterjee and GuptaNeophyllobius guajavae Chatterjee and Gupta, in Gupta, 2002:38 [ 2002:38 .N. natalensis.The description and illustrations provided are very poor and didn’t help to place the species in the diagnostic key provided in present work. Although, the original authors have compared the species with N. guajavae belongs to the new subgenus Monophyllobius based on tarsi III–IV, each, with one midventral seta (as only illustrated) and closely resembling N. variegata. The dorso-central setae in both the species are short where c1 is 1/2 and 1/3 the length of c1–d1 in N. guajavae and N. variegata, respectively.Based on the present work, the species Neophyllobius hyderabadensisIndra, Rao and ThakurNeophyllobius hyderabadensis Indra, Rao and Thakur, 1980:121 [1980:121 .N. hyderabadensis . Hence, it was not possible to provide any conclusive remarks on this species.The original published description and illustration were not found. The first author contacted Mr. Mahran Zeity who published a new camerobiid species from India, but was not able to get any information about Neophyllobius mexicanusMcGregorNeophyllobius mexicanus McGregor, 1950:49 [ 1950:49 ; de Leon 1950:49 ; Bolland 1950:49 .The original description and illustration provided by McGregor  were notNeophyllobius ornatusWomersleyNeophyllobius ornatus Womersley, 1940:248 [1940:248 ; Bolland1940:248 ; Fan and1940:248 .N. fissus, N. aegyptium, N. niloticus due to the femur IV with one seta.The status of this species is unresolved. The type specimen is lost as reported by Bolland  and Fan N. ornatus could belong to group of eight species  where all leg tarsi have two mid-ventral setae and femur I–IV with 4–3–2–1 setae.Based on available information and classification of species in the present work, Neophyllobius saxatilisHalbertNeophyllobius saxatilis Halbert, 1923:384 [1923:384 ; van Eyn1923:384 ; Bolland1923:384 .van Eyndhoven  reportedN. saxatlilis could be placed among 48 species of the new subgenus Neophyllobius having all leg tarsi with two in-line midventral setae, femur I–IV with 4–3–2–2 setae, palp tarsus having two setae and two eupathidia, one solenidion on leg tarsi I–IV, dorsal setae c1 reaching the bases of setae d1, setae d1 not reaching the bases of setae f1.In the present study, the Neophyllobius summersiMcGregorNeophyllobius summersi McGregor, 1950:67 [ 1950:67 ; Bolland 1950:67 .Neophyllobius species, to date, do not contain such palp chaetotaxy. Bolland [N. mangiferus to be close to N. summersi and differed from the latter on the basis of length of dorso-central setae and genu I–IV setae. However, it was not possible to include N. summersi, in the present diagnostic key, even with its unique character of palpfemur.McGregor  first ti Bolland  added so Bolland  considerNeophyllobius vanderwieliOudemansNeophyllobius vanderwieli Oudemans, 1926:121 [1926:121 ; Bolland1926:121 .N. kamalii, N. karabagiensis and N. sycomorus) which share the same character.Bolland  mentioneNeophyllobius sp.This unidentified species was reported from Yemen by Ueckermann et al. . The autNeophyllobius sp.N. dichantii, but no morphological data was providedThis unidentified species was mentioned by Ripka et al. , which wNeophyllobius, with 50 new species and 35 species redescriptions. A detailed genus diagnosis was given including; the ventral idiosoma with three pairs of ventral setae, two pairs of genital setae and three pairs of anal setae. In the remarks, the author stated that pregenital setae were never found in any other species of Neophyllobius, described until that time, in contrast to what was illustrated by Kuznetsov and Livshits [Bolland ,16 proviLivshits ; presenc1a is found on the coxa I in all species of Neophyllobius and it was counted along with coxal setal count. The similar concept is evident in the genus diagnosis provided by Bolland [Neophyllobius described to date. If the three pairs of ventral setae include the coxal seta 1a, then it supports the absence of aggenital seta (ag) in adult females as reported by Bolland [1a, then the total number of setae on the ventral idiosoma according to Bolland [It is difficult to discern the ventral idiosomal chaetotaxy from the genus diagnosis provided by Bolland  as that  Bolland  and all  Bolland . However Bolland , add up Neophyllobius, one each of Tycherobius (T. dazkiriensis), Bisetolobius (B. varius) species are available till date. For the ease of understanding, we here consider them as four different ventral idiosoma chaetotaxy (VIC) descriptions. All these cases have a total of four pairs of setae excluding coxal and anal setae. The species representing these cases are also presented in the 3a, 4a, ag and g setae are present. The VIC #2 is represented by 10 species where paired 3a is absent. The VIC #3 is represented by 14 species of Neophyllobius and a species, T. dazkiriensis, where paired setae 4a is absent. The VIC #4 includes 13 species of Neophyllobius and a species B. varius, where paired aggenital setae (ag) is absent. It is noteworthy that, in the cases 2–4, where either one of the paired setae was absent, two pairs of genital setae were described and illustrated , N. lorioi , N. saberi  and N. zolfigolii .Adding to this confusion, four different ventral idiosoma descriptions and illustrations represented by 55 ustrated . In addiNeophyllobius, few descriptions are available for immature stages, of which most are poor, which makes it further difficult to understand the setal ontology. Paredes–León et al. [N. cibyci. It was reported that on the ventral idiosoma, the intercoxal seta 3a is present in larval stages where the setae 4a appear in protonymphs and aggenital setae (ag) is only present in adult females with one pair of genital (g) setae. The redefinition of the family Camerobiidae provided by Fan and Walter [4a, ag and g present in females.In the genus n et al.  studied Neophyllobius, the Tycherobius (other than the exception mentioned above) have similar situation as that presented in VIC#1. The females of the other two genera, Camerobia  and Acamerobia (one species) also follow the similar ventral idiosoma setal notation.In its support, the species of closely related genus of Camerobia, Decaphyllobius, Neophyllobius and Tycherobius have been reported. As a case study, the ventral idiosoma of all these species are presented ; which were two pairs of intercoxal setae 3a–4a always present on small platelets, a pair of aggenital seta (ag), present on ventral integument and on or off the anterior margin of genital shield and a pair of genital setae (g) present on sides of genital opening. We support this notation based on the evidence discussed above and recommend future works to follow it.Based on the understanding from all the published literature on the family Camerobiidae and after carefully observing the relative position of ventral setae in Saudi Arabian camerobiid species, we reach the conclusion that four pairs of setae were present on the ventral idiosoma, excluding setae 1a and three pairs of anal setae  setae ………………………………………………………………………N. yunusi Bolland *8 Tarsus II with 10(ω) setae …………………………………………………………N. fani8‘ Tarsus II with 9 ….………………………………………………………………………… 9291 Some of the dorso–central setae reach or cross the bases of setae next in line  setae, genu IV setae not reaching base of corresponding tarsi ………………………………………………………………………… N. ayvalikensis Akyol93‘ Tarsi II with 9(+ω) setae, genu IV setae reaching base of corresponding tarsi ………………………………………………………….............1c twice the length of 1b, genus I seta passes tibiae and genu II and IV seta passing tarsi ends ………………………………………………………..…. N. lorioi Smiley and Moser94 Femur setae very short, femur I setae 3 and 4 nearly at one level, coxal setae 1c nearly as long as 1b, none of the genu setae reaching tibia end ……………………………………………………………. N. acaciae Bolland94‘ Femur setae longer, coxal seta N. lobatus De Leon95 Femur I–IV strongly serrulate at their proximal posterior margins …………………………………………………………………………………95‘ Femur I–IV not strongly serrulate at their bases ……………………………… 9696 Femur I setae 3 and 4 not based on one or nearly one level ………………….. 9796‘ Femur I setae 3 and 4 based on one or nearly one level ……...........................11397 Genu I–III and femur I–IV setae strongly lanceolate ………………………….. 9897‘ Genu I–III setae not lanceolate …………………………………………………... 991c longer than 1b…………………………………………………………………. N. ceratoniae Bolland98 Distal setae on femur IV not passing genu border, coxal seta 1c shorter than 1b…………………………………………………………………N. lanceolatus Bolland98‘ Distal setae on femur IV passing genu border, coxal seta 99 Genu I setae not reaching second row of tibia setae ………………………... 10099‘ Genu I setae reaching or longer than second row of tibia setae …………….. 104100 Genu II setae not reaching second row of tibia setae …………………………. 101N. binisetosus Bolland100‘ Genu II setae reaching second row of tibia setae ……….. N. atriplicis Bolland101 Genu III setae not whip like and plumed at the top ……….. 101‘ Genu III setae whip like …………………………………………………………. 102d1, e1 and f1 much longer than other dorsal setae, coxal setae 1c much shorter than 1b …… …………………………………………………………………….N. ambulans Meyer102 Genu II setae not reaching the first row of tibiae setae, dorso–central setae 102‘ Genu II setae reaching at least the first row of tibiae setae …......................... 103N. cavumarboris Meyer and Ryke103 Dorso–central setae d1 and e1 much longer than other dorsal setae ……… …………………………………………………. N. camelli Khanjani and Ueckermann103‘ Dorso–central setae c1 longer than other dorsal setae …………………………………………….. 104 Genu II setae not whip like ….…………………………………………………. 105104‘ Genu II setae whip like ………………………………………………………….. 107105 Proximal setae on femur IV reaching the base of distal setae ......................... 106N. armeniaca Bolland105‘ Proximal setae on femur IV not reaching the base of distal setae………………………………………………………………N. abiegnus Khaustov and Abramov106 Genu II seta long whiplike, crossing first row of setae on respective tibiae ……………………………………………… N. ayyildizi Koç and Madanlar106‘ Genu II seta short, not reaching first row of setae on respective tibiae …………………………………………………..… 107 Genu I and II setae reach till second row of tibia setae ……………………… 108107‘ Genu I and II setae are passing tibial border …………………………………. 1091c:1b = 6:18 …………………………………………………………………… N. iramus De Leon108 Dorsal setae small, not pointed, body small, coxa 1 seta 1c:1b = 6:17 ………………………………………………………………………… N. equalis De Leon108‘ Dorsal setae more broad and occupied with very strong stings, coxa I seta N. setosus Bolland109 Dorsal setae specially spinosed ………………………………………………………………………… 109‘ Dorsal setae not specially spinosed ………………………………………………………………………… 110110 First seta on femur II not reaching the third, proximal femur I seta not the longest ……………………………………………………………................................................. 111N. pruni Bolland110‘ First seta on the femur II the longest femur seta which passes easily the third, proximal femur I seta the longest …………………………………………… e1 longest among the dorso–central setae……………………………………111 Setae N. askalensis Doğan and Ayyildız……………………………………………………e1 not the longest among the dorso–central setae …………………….. 112111‘ Setae N. viticola Bolland112 Proximal seta on femur I equal in length with the second, distal seta on femur IV long, passes genu border, first seta on femur I reaches the bases of the fourth ……................................................................................................. N. combreticola112‘ Proximal seta on femur I shorter than the second, distal seta on femur IV short, not passing genu border, first seta on femur I does not reach the fourth seta at all ………………………………………………………………………… 113 Dorsal setae broad, setae d1 strong, genu II seta as long as genu, genu I and II setae not reaching first row of tibiae setae ………………………………………………… 114113‘ Dorsal setae smaller, genu I and II setae at least reaching first row of tibiae setae ……………………………………………………………………………………... 115N. curtipilis De Leon114 Genual setae distinctly spinose …………………………….. N. spatulus De Leon114‘ Genual setae linear and faintly spinose …………………….. N. sierrae McGregor115 Genu I setae reach till first row of tibiae setae, palp femur swollen, palp genu short ………………………………………………………………………… 115‘ Genu I setae longer, palp femur longer ……………………………………… 116N. americanus Banks116 Two eupathidia on the palp tarsus based very different in level, dorsal setae small with many spicules ……………………………………………….. 116‘ Two eupathidia on the palp tarsus on similar level, dorsal setae broader with less spicules …………………………………………………………………………………... 117117 Genu III–IV setae are passing tarsus end ……………………………………… 118117‘ Genu III–IV setae not reaching beyond end of respective legs ……………… 120pdx and c1 grouped on a small finely striated platelet ……………………………………………………. N. tescalicola Parades–Leon et al.118 Dorso–central setae pdx and c1 not grouped on a platelet …………………… 119118‘ Dorso–central setae c1 and d1 are the same as the distance between setae c1–d1 and d1–e1 respectively ………………………………………………. N. farrieri De Leon119 Lengths of dorsal setae c1 and d1 are distinctly longer than the distance between setae c1–d1 and d1–e1 respectively ……………………………………… N. cibyci Parades–Leon et al.119‘ Dorsal setae sce shorter sci ……………………………………………………… 121120 Setae sce equal in length to sci …………………………ΩN. consobrinus De Leon120‘ Setae vi, ve and sci subequal in length ……………ΩN. inequalis De Leon121 Lateral setae vi and ve ………… ΩN. piniphilus Bolland121‘ Lateral setae sci distinctly longer than e1 passes the bases of h1 ………….……. N. aegyptium Soliman and Zaher122 Setae e1 do not reach the bases of h1 ……………………………………… 123122‘ Setae 123 Genu I seta short not reaching second row of setae on corresponding tibiae ……………………………………… 124123‘ Genu I seta long, extend beyond corresponding tarsus base ……………… 126N. ferrugineus FanTarsi I–II with 8 setae each ……………………………………124‘ Tarsi I–II with 10 setae each ……………………………………… 1251c twice as long as coxal seta 1b …. N. lalbaghensis Zeity and Gowda125 Coxal seta 1c 1.5 times as long as coxal seta 1b … N. womersleyi Fan and Walter125‘ Coxal seta e1–f1 two times longer as distance d1–e1, setae f1 the longest ………………………………………………………………… N. niloticus Bolland126 Distance e1–f1 sub–equal to d1–e1 ……………………………………………. 127126‘ Distance N. bamiensis Khanjani et al.127 Palp femur with both setae short, not reaching end of palp genu; long setae of genu I–III not reaching end of respected legs …………………. 127‘ Palp femur with one of two setae reaches end of tarsus ……………………. 128d1 reaching base of e1, coxal setae 1c longer than 2c in length ………………………………………………………………………………... N. fissus128 Dorsal setae d1 extend beyond the bases of e1, coxal setae 1c almost equal to 2c in length ……………………………………………………………………N. punctulatus Fan128‘ Dorsal setae * = species described based on male holotype.N. bisetalis and N. spatulus with two setae on coxae II as mentioned by Bolland and Swift [+ = two species, viz; nd Swift  and Bollnd Swift , respectα = The character of genu I–IV without solenidion is mentioned in the original description .N. sycomorus. However, Bolland [N. sycomorus with the illustrations of male and female.β = The male specimens were not reported at the time of original description for  Bolland  providedN. consobrinus and N. inequalis suggesting later could be a deutonymph of former. However, these species are placed in the diagnostic key based on available information. but types of each species require re–examination.Ω = These three species have minute differences among them. Bolland and Swift  have als


The authors regret that there were two errors in the original article. In the “Experimental details” section on page 115205, “1 M sodium sulfide at 70–80 °C for 24 h” should have read “0.5 M sodium sulfide at 70–80 °C for 24 h”. Additionally, The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.


Scientific Reportshttps://doi.org/10.1038/s41598-021-93324-3, published online 07 July 2021Correction to: The original version of this Article contained errors.In Table 2 legend, the symbol of “picomol” was incorrectly given as “nanomol”.−1), salinity (S in yr−1), Total Alkalinity (TA in µmol kg−1 yr−1), salinity-normalized alkalinity (nTA in µmol kg−1 yr−1), total dissolved inorganic carbon (DIC in µmol kg−1 yr−1), salinity-normalized dissolved inorganic carbon (nDIC in µmol kg−1 yr−1), in situ pH in total scale (pHT yr−1), total hydrogen ion concentrations ([H+]T in nanomol kg−1 yr−1), ion carbonate concentration excess over aragonite saturation (exCO3 = in µmol kg−1 yr−1), and anthropogenic CO2.”“Average trends obtained with the seasonally detrended data the in situ temperature (T in °C yrnow reads:−1), salinity (S in yr−1), Total Alkalinity (TA in µmol kg−1 yr−1), salinity-normalized alkalinity (nTA in µmol kg−1 yr−1), total dissolved inorganic carbon (DIC in µmol kg−1 yr−1), salinity-normalized dissolved inorganic carbon (nDIC in µmol kg−1 yr−1), in situ pH in total scale (pHT yr−1), total hydrogen ion concentrations ([H+]T in picomol kg−1 yr−1), ion carbonate concentration excess over aragonite saturation (exCO3 = in µmol kg−1 yr−1), and anthropogenic CO2.”“Average trends obtained with the seasonally detrended data the in situ temperature (T in °C yrAdditionally, the article contains a repeated error where the symbol for “pmol” was incorrectly given as “nmol” in the Results section, under the subheading ‘Acidifcation drivers’, in Figure Furthermore, in Figure The original Article and accompanying Supplementary Information file have been corrected.Supplementary Information.


Rn into hypercubes of two differnet side lengths p or q. This generalizes the Pythagorean tiling in R2. We also show that this tiling is unique up to symmetries, which proves a variation of a conjecture by Bölcskei from 2001. For positive integers p and q, this tiling also provides a tiling of (Z/(pn+qn)Z)n.We construct a unilateral lattice tiling of  Keller  for {a,a+1,a+2,…,b−1,b}⊆Z and ei for the ith standard unit vector in Rn.We write 2Theorem 2p,q∈R+ with q<p and n∈N⩾2. There exists a lattice tiling of Rn into hypercubes of side length p or q.Let We split Theorem Remark 2After a preprint of this article appeared on arXiv, the author was informed by Mihalis Kolountzakis that Theorem A be the following n×n matrix:Let Lemma 1n∪n−1× is a system of representatives of Rn/AZn up to null sets.Theorem We split the proof of Lemma Claim 1Rn has a representative in .Every element Proof of Claim 2A1:Consider the following matrix a1,..,an be the columns of A and b1,…,bn the columns of A1. We obtain A1 in the following way.b1=an.bk=bk−1−ak−1, for k∈.Let l∈ such that xi<p for all i<l and xl⩾p, then x−bl is a representative of x in .□Now let Claim 3x=T∈ be such that xi⩾q−p for all i<l and xl<q−p. Then x+bl is a representative of x in  such that xi<p for all i<l and xl⩾p. From the definition of x, it follows that l<n and x−bl is a representative of x in n∪n−1×=C=⋃i∈NQi up to null sets, which concludes the proof.□We now prove Lemma Remark 3n∪n−1× cannot be representatives of the same element in Rn/AZn as this would imply that there were neighborhoods of the two points corresponding to the same set in Rn/AZn of non‐zero Lebesgue measure.Two different points in the interior of Lemma 2i∈ and let ei be the ith standard unit vector, then pei∉AZn and qei∉AZn.Let To show that the tiling is unilateral, the followimg Lemma is sufficient.n with difference pei and therefore they are both different representatives of Rn/AZn, so pei∉AZn. Analogously, there are two points in the interior of ∈Bn′ is in the stabilizer of the tiling ⇔ S is of the following form.si,j=si+1,j+1 for i<n,j<n.si,n=−si+1,1 and sn,j=−s1,j+1 for i<n,j<n.sn,n=s1,1.The group of symmetries of the tiling has to keep the cube invariant, so it is a supgroup of the hyperoctahedral group S be as above and i such that si,1=±1. Without loss of generality, si,1=1. Now‘⇐’: Let v:=qei±pej such that v≠ai and i≠j. Thenai−v is not in AZn because it is a vector connecting two inner points of  there exists i∈ such that Saj=±ai. Now suppose si,j=±1, then Saj=±ai and therefore□‘⇒’: Let Theorem 4p,q∈N with gcd=1. Then pn+qn=min{l∈N|lei∈AZn} for all i∈. In particular, there exists a unilateral lattice tiling of (Z/(pn+qn)Z)n into hypercubes of side length p or q.Let On the periodicity of the tiling along the coordinate axis, we have the following result:i∈. From Cramer's rule, we know that the solution of Ax=ei in Qn is of the form x=k/det(A)=k/(pn+qn) for a k=T∈Zn and therefore (pn+qn)ei∈AZn. More specifically, ki=det(Ai)/(pn+qn), where Ai=. Calculating the determinant with the Laplace expansion along the last column, we get det(Ai)=qn−1 and since gcd=1 the statement follows.Let Remark 4q=2 and p=1, the tiling gives a lower bound on the maximal number of hypercubes of side length two that can be packed in (Z/(2n+1)Z)n, which coincides with the optimal solution .21.q=2 a4Rn into hypercubes of two sizes is equivalent to the tiling described in the previous sections. It is known that such a tiling does not exist if we only use cubes of one size  and so x:=T∈BZn with |xi|⩽p for all i∈. Now x connects two inner points of the big cube, a contradiction.□Assume the contrary. Then there exists Lemma 4A small and a big cube that properly touch share a corner.T be a big cube and S a small cube that properly touch. Without loss of generality, let T=n and S properly touches T in {q}×Rn−1. We write S=×⨂j=2n where xi∈ for all i∈.Let S touches T in a full side but S and T do not share a corner, then xi∈ for all i∈ and without loss of generality xn∉{0,p−q}. There have to be big cubes T1,T2 that touch T in {q}×⨂j=2n−1× and {q}×⨂j=2n−1×, respectively, for a ε>0. Therefore, T1=×⨂j=2n−1× and T2=×⨂j=2n−1× where yi,zi∈ for all i∈. Now E:=×⨂j=2n−1× cannot be filled by any cubes, because it touches a small cube and it touches T1 and T2 in opposite full sides of distance p  for all i∈. Let T3 be the big cube properly touching S in {q}×⨂j=2n−1× and let T4 be the big cube properly touching S in {q+p}×⨂j=2n for a ε>0. We write T4=×⨂j=2n with wi∈ for all i∈. If wn<xn, then E1:=×⨂j=2n−1× cannot be filled by any cubes, because it touches a small cube and it touches T and T4 in opposite full sides of distance p. Otherwise E2:=×⨂j=2n−1× cannot be filled by any cubes, because it touches a small cube and it touches T3 and T4 in opposite full sides of distance p ××⨂j=i+1n−1Ii(j)× with Ii(j)∈{,}. Since Ti and Tj do not properly intersect, we have Ii(j)= or Ij(i)=. From Lemma Ti′:=⨂j=1i−1Ji(j)××⨂j=i+1n−1Ji(j)× withTi′ and Tj cannot properly intersect, Ji(j)= or Ij(i)= and consequently {Ii(j),Ij(i)}={,}.Lemma 6i∈ such that Ti=⨂j=1i−1××⨂j=i+1n−1×.There exists exactly one Now assume Tn:=n−1× is also a cube in the tiling. Let T′ be such that T′ touches T in n−1×{q} then T′=⨂j=1n−1× with xj∈×. Since S is the unique small cube properly touching T in Rn−1×{q}, F:=× is filled by big cubes properly touching T. Now consider E:=)×. E is touching F and T′′ in full opposite sides of distance p so it can only by filled by small cubes and since small cubes cannot touch, by at most one small cube T and x2:=T are inner points of E but their midpoint xm:=(x1+x2)/2=T is an inner point of Tn so E cannot be filled by a single convex set.□Assume without loss of generality Claim 2x1=p.It holds that Proof of Claim 2x1>p. Then E′:=×n−2× is touching T′ and S in full opposite sides of distance at most q−p so E′ cannot be filled.□Assume T′=×n−2× and since two different cubes of the form ⨂j=1i−1××⨂j=i+1n−1× would intersect, T′ is the unique cube of this form.□Therefore S be a small cube in the tiling and without loss of generality let S:=n. Let T:={T1−,…,Tn−,T1+,…,Tn+} be the set of cubes properly touching S such that S touches Ti− in Ri−1×{−p/2}×Rn−i and Ti+ in Ri−1×{p/2}×Rn−i for all i∈ and C:={c1−,…,cn−,c1+,…,cn+} their corresponding center point. We can now define a function ν:T→T that assigns every cube T∈T the cube T′ such that  are in the same relation as  from Lemma i. We also use ν as a function ν:C→C with the obvious meaning. Consequently, for all i∈ there exists j∈∖{i} such that ν(vi−)−vi−=qei±pej and for symmetry reasons ν(vi+)−vi+=−qei∓pej. Now let G:= be the directed graph with vertex set V:=T and edge set E:={)|T∈T} =Ti∓.Now again let G has out‐degree one so G contains a cycle. Let C:= be a cycle in G with center points , T¯1=T¯k and 3⩽k⩽2n+1. For i∈, let ji∈ such that T¯i∈{Tji−,Tji+}. Then c¯i can be written as{ji,ji+1}⊎Si−⊎Si+= and xi,xi+1∈{−1,1} andvi:=c¯i+1−c¯i=−xiqeji+xi+1peji+1, then ∑i=1k−1vi=0 because ck=c1.Every vertex in Tj−∈C⇔Tj+∈C for all j∈ and C is an even cycle. Without loss of generality let T¯1=T1−, then for symmetry reasons T¯k/2+1=T1+ andTherefore ej1=ejk/2+1=e1 and {ej1,…,ejk/2} are pairwise different. Consequently, k/2=n which concludes the proof.□where {b1,b2,…,bn}⊆BZn whereC:=, then SC=A where S=i,j∈∈Bn′ andNow consider the sets of vectors det(C)=det(A)=det(B)=pn+qn and therefore CZn=BZn and A and B describe the same tiling up to symmetries.Now Mathematika is owned by University College London and published by the London Mathematical Society. All surplus income from the publication of Mathematika is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.


The most frequently used model for simulating multireader multicase (MRMC) data that emulates confidence-of-disease ratings from diagnostic imaging studies has been the Roe and Metz (RM) model, proposed by Roe and Metz in 1997 and later generalized by Hillis (2012), Abbey et al. (2013), and Gallas and Hillis (2014). A problem with these models is that it has been difficult to set model parameters such that the simulated data are similar to MRMC data encountered in practice. To remedy this situation, Hillis (2018) mapped parameters from the RM model to Obuchowski–Rockette (OR) model parameters that describe the distribution of the empirical AUC outcomes computed from the RM model simulated data. We continue that work by providing the reverse mapping, i.e., by deriving an algorithm that expresses RM parameters as functions of the OR empirical AUC distribution parameters.We solve for the corresponding RM parameters in terms of the OR parameters using numerical methods.An algorithm is developed that results in, at most, one solution of RM parameter values that correspond to inputted OR parameter values. The algorithm can be implemented using an R software function. Examples are provided that illustrate the use of the algorithm. A simulation study validates the algorithm.The resulting algorithm makes it possible to easily determine RM model parameter values such that simulated data emulate a specific real-data study. Thus, MRMC analysis methods can be empirically tested using simulated data similar to that encountered in practice. The resulting data are called multireader multicase (MRMC) data. These studies are typically used to compare different imaging modalities with respect to reader performance. Often measures of reader performance are functions of the estimated receiver-operating-characteristic (ROC) curve, such as the area under the ROC curve (AUC). The Obuchowski and Rockette method (OR)The most frequently used model for simulating MRMC data that emulate confidence-of-disease ratings from such studies has been the model first proposed by Roe and MetzBecause RM model parameters are expressed in terms of the latent rating data distribution, in contrast to MRMC analysis results that are almost always expressed in terms of parameters that describe the distribution of the reader performance outcomes, it has been difficult to set RM model parameter values such that the simulated data exhibit characteristics that are similar to MRMC data encountered in practice. To remedy this situation, Gallas and HillisAn outline of this paper is as follows. In Sec. 22.12.1.1Let Using the RM notation, the model is given as Roe and Metz constrain the sum of the error variance and variance components involving case to be equal to one: Without loss of generality, Roe and Metz impose the constraints 2.1.2Although Roe and Metz only consider simulations for equal test DV distributions for each reader, the model can be easily modified to allow for test DV distributions that differ in their median AUC values by not setting Note that the RM model that allows for test-dependent AUCs is completely defined by seven parameters: 2.1.3In practice, estimated binormal-model nondiseased and diseased distribution variances for a reader-test combination are often different, with diseased subjects typically having more variable test results. Thus to better emulate real data, HillisSimilar to the original RM model,The algorithm discussed in this paper will be for the RMH model, which includes the original RM model2.2Obuchowski and Rockette–,,These error variance–covariance parameters are typically estimated by averaging corresponding conditional-on-readers estimates computed using the jackknife,The Defining 33.1The RMH-to-OR mapping, previously derived by Hillis,In this section, we discuss the main points of the OR-to-RMH algorithm when the goal is to emulate data from a real study with the RMH model; i.e., to determine RMH parameter values such that the expected values of the OR parameter estimates from the simulated MRMC samples are described by the unspecified, which we assume throughout this section. Two other options for The 3.1.1The OR-to-RMH algorithm requires inputted values for It is possible that there are several unspecified, that maps Let However, it is possible for the OR-to-RMH algorithm to return a solution such that Eq. (18) holds only approximately, i.e., The lower and upper limits for For example, 0.01,0.5  and b=2,5  #creates data frame with 5 rows, each = RM_values> RM_Table34 <- c> RM_Table34 <- c> RM_Table34 <- c> RM_Table34 <- c> print(RM_Table3)n0 n1 delta1 delta2 var_R var_TR var_C1 69 45 2.392224 2.957029 0.1223413 0.005180485 0.47169641.1 138 90 2.392224 2.957029 0.1223413 0.005180485 0.47169641.2  45 69 2.392224 2.957029 0.1223413 0.005180485 0.47169641.3  69 45 2.645200 2.645200 0.1223413 0.005180485 0.47169641.4  69 45 1.275900 1.275900 0.1223413 0.005180485 0.4716964var_TC var_RC var_error b b_method mean_to_sig11 0.1222262 0.1091448 0.2969327 0.656081 unspecified 4.5635531.1 0.1222262 0.1091448 0.2969327 0.656081 unspecified 4.5635531.2 0.1222262 0.1091448 0.2969327 0.656081 unspecified 4.5635531.3 0.1222262 0.1091448 0.2969327 0.656081 unspecified 4.5635531.4 0.1222262 0.1091448 0.2969327 0.656081 unspecified 4.563553mean_to_sig2 Pr1_improper Pr2_improper1 5.64101 0.003896242 7.862956e-051.1 5.64101 0.003896242 7.862956e-051.2 5.64101 0.003896242 7.862956e-051.3 5.64101 0.003896242 7.862956e-051.4 5.64101 0.003896242 7.862956e-05> OR_values_Table3 <- RMH_to_OR(RM_Table3)> printn0 n1 AUC1 AUC2 var_R var_TR error_var1 69 45 0.8970000 0.9410000 0.001540000 2.080000e-04 0.00078800021.1 138 90 0.8970000 0.9410000 0.001540000 2.080000e-04 0.00039125761.2 45 69 0.8970000 0.9410000 0.001540000 2.080000e-04 0.00063444271.3 69 45 0.9190000 0.9190000 0.001644069 7.426773e-05 0.00078900631.4 69 45 0.7500034 0.7500034 0.007014410 3.019443e-04 0.0023458109cov1 cov2 cov3 corr1 corr21 0.0003412041 0.0003388401 0.0002356121 0.4330000 0.43000001.1 0.0001703301 0.0001691406 0.0001176498 0.4353401 0.43229971.2 0.0002800701 0.0002778178 0.0001940871 0.4414426 0.43789271.3 0.0003644012 0.0003363961 0.0002513892 0.4618483 0.42635421.4 0.0012240655 0.0012083227 0.0009406161 0.5218091 0.5150981corr3 b mean_to_sig1 mean_to_sig2 Pr1_improper1 0.2990000 0.656081 4.563553 5.641010 0.0038962421.1 0.3006966 0.656081 4.563553 5.641010 0.0038962421.2 0.3059174 0.656081 4.563553 5.641010 0.0038962421.3 0.3186150 0.656081 5.046146 5.046146 0.0007838341.4 0.4009769 0.656081 2.433985 2.433985 0.326185605Pr2_improper1 7.862956e-051.1 7.862956e-051.2 7.862956e-051.3 7.838340e-041.4 3.261856e-019.3> VanDyke_OR_orig_values <- data.frame> Table4_OR1 <- VanDyke_OR_orig_values #creates data frame with 3 rows,> # each the same as VanDyke_OR_orig_values> Table4_OR2 <- data.frame,+ b_input = c, mean_sig_input = c)> Table4_OR <- cbind> print[1] "Original OR parameter values"> print(Table4_OR)n0 n1 AUC1 AUC2 var_R var_TR var_error corr1 corr2 corr31 69 45 0.897 0.941 0.00154 0.000208 0.000788 0.433 0.43 0.2991.1 69 45 0.897 0.941 0.00154 0.000208 0.000788* 0.433 0.43 0.2991.2 69 45 0.897 0.941 0.00154 0.000208 0.000788* 0.433 0.43 0.299b_method b_input mean_sig_input1 unspecified NA NA1.1 mean_to_sigma NA 5.21.2 specified 1 NAmean_to_sigma = mean_to_sigma or specified it is not necessary to specify a value for var_error, or the value can be NA*Note that with > Table4_RMH <- OR_to_RMH(Table4_OR)> print[1] "Table 4 RMH parameter values"> print(Table4_RM)n0 n1 delta1 delta2 var_R var_TR var_C1 69 45 2.392224 2.957029 0.12234134 0.005180485 0.47169641.1 69 45 2.303940 2.847902 0.11347812 0.004805176 0.46746761.2 69 45 1.855834 2.293997 0.07362882 0.003117776 0.4498198var_TC var_RC var_error b b_method1 0.1222262 0.1091448 0.2969327 0.6560810 unspecified1.1 0.1220955 0.1089342 0.3015027 0.6929693 mean_to_sigma1.2 0.1215947 0.1080172 0.3205683 1.0000000 specifiedmean_to_sig1 mean_to_sig2 Pr1_improper Pr2_improper1 4.563553 5.641010 0.003896242 7.862956e-051.1 5.200000 6.427723 0.001778344 2.748745e-051.2 Inf Inf 0.000000000 0.000000e+00> Table5_true_values <- RM_to_OR(Table4_RM)> print[1] "Table 4 True OR values"> printn0 n1 AUC1 AUC2 var_R var_TR var_error cov11 69 45 0.897 0.941 0.00154 0.000208 0.0007880002 0.00034120411.1 69 45 0.897 0.941 0.00154 0.000208 0.0007664249 0.00033186201.2 69 45 0.897 0.941 0.00154 0.000208 0.0006584975 0.0002851294cov2 cov3 corr1 corr2 corr3 b1 0.0003388401 0.0002356121 0.433 0.43 0.299 0.65608101.1 0.0003295627 0.0002291610 0.433 0.43 0.299 0.69296931.2 0.0002831539 0.0001968908 0.433 0.43 0.299 1.0000000mean_to_sig1 mean_to_sig2 Pr1_improper Pr2_improper1 4.563553 5.641010 0.003896242 7.862956e-051.1 5.200000 6.427723 0.001778344 2.748745e-051.2 Inf Inf 0.000000000 0.000000e+009.49.4.1> VanDyke_OR_altered_values_a <- data.frame> RM_values = OR_to_RMWarning message: In OR_to_RM.defaultn0 n1 delta1 delta2 var_R var_TR var_C var_TC var_RC var_error b1 69 45 NA NA NA NA NA NA NA NA NAb_method mean_to_sig1 mean_to_sig2 Pr1_improper Pr2_improper1 unspecified NA NA NA NAx1 x2 x3 x4 x5 x6 x71 1.264641 1.563224 NA NA NA NA NA9.4.2> VanDyke_OR_altered_values_b <- data.frame> RM_values <- OR_to_RMWarning message: In OR_to_RM.defaultn0 n1 delta1 delta2 var_R var_TR var_C var_TC var_RC var_error b1 69 45 NA NA NA NA NA NA NA NA NAb_method mean_to_sig1 mean_to_sig2 Pr1_improper Pr2_improper1 unspecified NA NA NA NAx1 x2 x3 x4 x5 x6 x71 1.264641 1.563224 0.06838082 NA NA NA NA9.4.3> VanDyke_OR_altered_values_c <- data.frame> RM_values <- OR_to_RMWarning message: In OR_to_RM.default  Try changing (reduce or increase) the value of var_error.( b) Try using one of the other two b_method options, which should always work.> printn0 n1 delta1 delta2 var_R var_TR var_C var_TC var_RC var_error b1 69 45 NA NA NA NA NA NA NA NA NAb_method mean_to_sig1 mean_to_sig2 Pr1_improper Pr2_improper1 unspecified NA NA NA NAx1 x2 x3 x4 x5 x6 x71 1.264641 1.563224 0.06838082 0.07127637 NA NA NA


Huan Yuan, Yan Xu, Yi Luo, Jia‐Rong Zhang, Xin‐Xin Zhu, Jian‐Hui Xiao, Aging Cell, 2022, 21(9):e13686. https://doi.org/10.1111/acel.13686In the published version of Yuan et al. , the autIn the Discussion, “Figure S4” should have been “Figure S5” and the revised sentence should read as follows:Figure S5), indicating that the accelerated aging model was successfully established.”“Consistent with the previous findings , long‐term injection of d‐gal reduced the activity of T‐AOC, GSH‐Px, and SOD and increased the content of MDA, AGEs, and RAGEs in the blood, kidney, liver, and heart  and (b–k). The revised caption is given below.In Figure 3 legend, p16FIGURE 3 CaM/CaMKII and Nrf2/HO‐1/NQO1 signaling pathways were attenuated after YWHAE knockdown. (a) Changes in 14‐3‐3ε, t‐Nrf2, n‐Nrf2, HO‐1, NQO1, CaM, p‐CaMKII, and CaMKII expression in hAMSCs upon different treatments. (b–k) Relative expression levels of 14‐3‐3ε, n‐Nrf2, t‐Nrf2, CaM, CaMKII, p‐CaMKII, HO‐1, and NQO1. n = 3. n‐Nrf2, nuclear Nrf2; t‐Nrf2, total Nrf2; p‐CaMKII, phosphorylated CaMKII; Control, control group; H2O2, senescent group; GA‐D, GA‐D treatment group; Mock‐vehicle, GA‐D treatment group plus empty carrier; h‐14‐3‐3ε, GA‐D treatment group plus YWHAE overexpression; sh‐14‐3‐3ε, GA‐D treatment group plus YWHAE knockdown; x¯ ± SD, mean ± standard deviation. *p < .05, **p < .01.In Figure 5 legend, there is an error in citing part figure labels. “(c)” should be “(i)”; “(d–f)” should be “(j–l)”; “(g–l)” should be “(c–h)” and the revised caption is provided below.FIGURE 5 GA‐D enhanced the defense against oxidative stress on the sera in D‐gal‐caused aging mice. (a) The experimental design. (b) The swimming test of mice in the different groups. (c–h) Activity of T‐AOC, SOD, MDA, GSH‐Px, AGEs, and RAGE in the sera of mice exposed to different treatments. (i) Histopathological organization of the liver, kidney, and heart tissues in different mice groups, analyzed via hematoxylin–eosin (HE) staining. Scale bar: 200 μm. (j–l) The histological scores of liver, kidney, and heart tissues in different mouse groups. n = 6. AGEs, advanced glycation end products; GSH‐Px, glutathione peroxidase; igGG‐H, in vivo high‐dose GA‐D treatment group; igGG‐L, in vivo low‐dose GA‐D treatment group; igGG‐M, in vivo medium‐dose GA‐D treatment group; MDA, malondialdehyde; MG, model group ; NG, normal group; RAGEs, receptor for advanced glycation end products; SG, solvent group ; SOD, superoxide dismutase; T‐AOC, total antioxidant capacity; x¯ ± SD, mean ± standard deviation. *p < .05, **p < .01.In addition, the published version of Supplementary MaterialClick here for additional data file.


Therefore, we replace the wordings for all special cases considered in [“exact lossless” in the Abstract and Sections 1.2, 1.3 and 4.1 with “simplified lossless”.“exact rate region” in Sections 1.2, 1.3, 3.2, 4, 4.1, 4.2 and 6 with “simplified rate region”.“The … rate region … when … is the set of …” in Lemmas 1–4 with “The … rate region … when … includes the set of …”.“the lossless rate regions” in Section 4.3 with “the further simplified lossless rate regions”.“the exact lossless rate region” in Section 4.3.1 with “the further simplified lossless rate region”.“the lossless rate region” in Sections 4.3.2 and 4.4 with “an inner bound for the lossless rate region”.“one exact region” in the Abstract with “one achievable region”.“evaluate the exact rate region” in Section 1.2 with “evaluate a simplified rate region”.“the rate region for” in Section 1.3 with “a rate region for”.“the exact lossless” in Section 4.3.2 with “a simplified lossless”.Special case results given in Lemmas 1–4 and evaluated in Section 4.4 as an example are unfortunately not necessarily the exact rate regions for all channel models, although they are valid inner bounds. In more detail, the inner bounds given in Theorems 1 and 2 can be simplified for functions that are partially invertible with respect to Furthermore, below Remark 1, delete the sentence “i.e., rate regions for which inner and outer bounds match”. Since [The author states that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.