We derive some simple relations that demonstrate how the posterior convergence rate is related to two driving factors: a “penalized divergence” of the prior, which measures the ability of the prior distribution to propose a nonnegligible set of working models to approximate the true model and a “norm complexity” of the prior, which measures the complexity of the prior support, weighted by the prior probability masses. These formulas are explicit and involve no essential assumptions and are easy to apply. We apply this approach to the case with model averaging and derive some useful oracle inequalities that can optimize the performance adaptively without knowing the true model. In Jiang , there adu2-divergence for any u ∈  (defined later), which is more general than the squared Hellinger distances d−1/22 (corresponding to u = −1/2). A recent work by Norets  and 𝒫 is the support of the prior π.Consider a “posterior sequence” of densities penalized divergence” dt2 ≡ infK⊂𝒫[supq∈Kd2 + n−1ln(1/π(K))], for some divergence d of a prior π from the true density p0.Here we define a “u ∈ . This is because the dt divergence is monotonically increasing in t. For any du− so that 1/u is not an integer, we can use a more stringent divergence du′− with the −u′ being the next larger value from the integer range {−1/2, −1/3, −1/4,…} to bound the convergence rate in du−.The result can be extended to the continuously valued penalized divergence” of the form dt2 ≡ infK[supq∈Kdt2 + n−1ln(1/π(K))], related to a prior π. This first part supq∈Kdt2 describes the maximal divergence of a set K (proposed by a prior π) from p0. We can understand this part as the approximation error of the prior π when it is used to propose densities to approximate a true density p0. The second part penalizes an unlikely set K with a small prior π(K). Combining the two parts, we can perhaps try to interpret d2 as the approximation error (away from p0) by a not-too-unlikely set proposed by a prior π. This “penalized divergence” is a critically important driving factor for determining the convergence rates in the previous results. It is noted that although this factor corresponds to the approximation ability of π, it already has a complexity penalty built in it implicitly. This is from the penalty against a small prior; the second part is n−1ln(1/π(K)), which is, roughly speaking, about d/n, where d is the number of parameters proposed by the prior π  ∝ δd).In this and other works, we notice that we often encounter in the convergence rate results a quantity similar to the “d/n, up to some logarithm factors, where d is the dimension of the parameters involved in the prior. It is noted, however, that with model averaging the higher dimensional model can be downweighted by the model prior, so that effectively one can make d to be of order 1 for this complexity factor, so that the convergence rate will be controlled by the first factor  alone.The other factor behind the convergence rate is related to the complexity of the model, which is proportional to π = πmπ(df∣m), jointly over a model index m ∈ M in a set of nonoverlapping models M, and density f ∈ 𝒫m (the support of prior given model m)  ∝ ∑m∈M∫f∈𝒫mI ∈ A)Πi=1nf(Yi)πmπ(df∣m) for an event A. In this case, let L1 balls of radius (1/n)u1/ needed to cover the prior-support 𝒫m under model m. Then we have the following, under the iid assumption.The convergence rate result in p for iid data satisfying u ∈ {1/2,1/3,1/4,…} and any 0 < t < ∞, with probability 1, for almost all large sample size n, one has 𝒫 and 𝒫m are supports of the mixing prior π = ∑mπmπ(df∣m) and the model-m prior π(df∣m), respectively.Consider a “posterior sequence” of densities m for the bound on the right hand side. Again, the convergence rate is displayed explicitly, and we will try to explain the driving factors of the convergence rate later. This is unlike the previous works where one has to conjecture a rate ϵ and check that it satisfies many conditions.This is an oracle inequality that achieves the best performance of all models π, developing an idea pioneered by Walker u1/ over all such covers ∪jBj of 𝒫, where each Bj is an L1 ball of radius nu−1/. We may name it as the “ℓu-norm prior complexity” for covering the prior support. An unbounded prior support may still be coverable by infinitely many Bj's, so that Nu(π) is finite, even with an infinite covering number Nu(π) is the “ℓu-norm complexity” of this prior π defined in this remark.The complexity in Nu(π) defined in the previous remark in parametric models, where densities p ∈ 𝒫 are parameterized by a d dimensional parameter θ, and a prior on θ induces a prior π on the densities in 𝒫. A more rigorous treatment is given in the example of L1 distance between two densities pθ1 and pθ2 by the maximal norm |·|∞: ∫ | p1 − p2 | ≤cd|θ1−θ2|∞ for some constant c > 1. Then, to cover the parameter space, we can use ℓ∞ ball Aj's in the parameter space with radius h = (cdnu1/)−1, so that the corresponding densities cover the L1-ball Bi with the required radius nu−1/. These sets Aj, with small volumes vol(Aj) = (2h)d, can be used to form a fine partition of the parameter space, so that the norm ∑j∈Nπ(Bj)u = ∑j∈Nπ(θ ∈ Aj)u = [∑j∈Nπ(θj)uvol(Aj)]vol(Aj)u−1 ≈ [∫π(θ)udθ]((2h)d)u−1, where π(θj) is the prior density function evaluated at some intermediate point in the set Aj. The sum in the square bracket is a Riemann sum over a fine grid, which we will assume to be approximated by an integral under some regularity conditions, even if the domain may be unbounded. Therefore, we have an upper bound of the norm complexity as n. Assume that the prior density is Lu integrable in the parameter space, and the norm |π|u scales as (const)d as in the case of an iid prior π(θ) = ∏j=1dπ1(θj). Then the complexity term in the bound of d.We now describe heuristically how to bound the “norm complexity” Nu(π) is the “ℓu-norm complexity” of this prior π, which in this case should be the infimum of the ℓu norm [∑m∈M∑j∈𝒩m{πmπ(Bj∣m)}u]u1/ over all such covers ∪m,jBmj of 𝒫, where each Bmj is an L1 ball of radius nu−1/, and under each model m, ∪j∈𝒩mBmj represents a cover of its prior support 𝒫m using possibly infinitely many balls. The defining expression of Nu(π) can also be related to the norm complexities of all the conditional priors π given the model choices: Nu(π) = [∑m∈M{πmNu(π(·∣m))}u]u1/. With model averaging using some suitable weights πm, this term Nu(π) and its effect on the convergence rate no longer diverge with the complexity of the model, in contrast to the conclusion of dt2. An example below (in its second part) is used to illustrate this.Similar to This is a simple binary regression example intended for illustration. We will see that model averaging can be used to derive nearly optimal convergence rates that are adaptive to the assumptions on the true model. In the first part, we will illustrate how to bound the penalized divergence with a uniform prior with a bounded support. In the second part, we will illustrate how to bound the norm complexity when the prior has an unbounded support.Z∣X ~ Bin), X ∈ Unif, where the true conditional mean function μ(X) is denoted as μ0(X)∈ , which is bounded away from 0 and 1, for any value of X. We consider an m-piecewise constant working model for the mean function μ. Suppose the prior is π = πmπ where m indicates the m-piecewise constant model μ(x) = ∑j=1mθjI. We consider an independent uniform prior π = ∏j=1mI. Consider a binary regression model We will consider two different setups of the true model.μ0 has continuous derivative bounded by D. We call this a “dense” setup since we may need a large piecewise constant model (with large m increasing with sample size n) to approximate this quite arbitrary true mean function μ0.In the first setup, the true u = −1/2 and t = 1. In the present case, we have d12 = ∫01dx(μ1 − μ2)2/(μ2(1 − μ2)), ≤(δ(1 − δ))−1∫01dx(μ1 − μ2)2, if p1 = μ1z(1 − μ1)z1− and p2 = μ2y(1 − μ2)z1−. We will sometimes use the mean function μ's to denote the corresponding distances as d12 and so on.To apply μ0(x), there exists a μ* in the support of π(·∣m) so that |μ0 − μ* | <D/m everywhere. In fact, we can take μ* = ∑j=1mθj*I where θj* = μ0((j − 1)/m)∈. Now let μ be close to μ*, in the sense that μ = ∑j=1mθjI and θj ∈ θj* ± Δ, j = 1, ..., m, for some small Δ ∈ . Take Km to be the set of all such densities of Bin. Then μ ∈  everywhere, since θj ∈ θj* ± Δ and θj* ∈  for all j.The approximation property of the piecewise constants implies that, for any true p0 and p be the densities corresponding to μ0 and μ, respectively. Then supp∈Kmd12 = supp∈Km∫01dx(μ0 − μ)2/(μ(1 − μ))≤supp∈Km∫01dx(|μ0 − μ* | +|μ − μ* | )2/{minx(μ(1 − μ))}≤(D/m + Δ)2/((δ − Δ)(1 − δ + Δ)) due to the triangle inequality. The prior probability over  is πm(2Δ)m. Therefore, the “penalized divergence” πm ∝ eK(m−1)lnn− for some large enough constant K > 0 and apply d2  and the resulting convergence rate d−1/22 are both of order O(n−2/3lnn), which is within a lnn factor to the minimax optimal result. It is noted that the model averaging automatically achieves this near optimal rate.We can take m0-piecewise constant, where we do not know the value m0. We call this a sparse case since we only need an m-piecewise constant model μ* to approximate the true mean function μ0 perfectly, where m = m0 can be much smaller than the choice of m ~ n1/3 in Consider a second setup, where we assume that the true model is a d12 by taking m = m0 and Δ = 1/n and obtain (using |μ* − μ0 | = 0) the following: d−1/22 are, therefore, both close to the parametric rate O(m0/n), as if we knew m0 beforehand.Then modifying the above reasoning, we can bound the infimum in the “penalized divergence” π is adaptive in the sense that in either the dense or the sparse case, the resulting posterior distribution works nearly optimally, even if we do not really know whether the true model is dense or sparse.In summary, the prior π(·∣m) with a bounded support. In this subsection, we will consider a parametrization in the log-odds scale, with an unbounded prior support, for illustrating how to calculate the norm complexity described in Remarks Z∣X ~ Bin, X ∈ Unif, where the true μ, denoted as μ0, is bounded away from 0 and 1. We consider an m-piecewise constant working model for the mean function μ = ∑j=1meθj/(1 + eθj)I. Suppose the prior is iid for each “log-odds” parameter θj supported on ℜ; then In the example above, we have considered a uniform prior for Z, X): p1,2 = μ1,2zz1−, with parameters θj1's and θj2's, respectively. Then one can easily derive a following relationship for the L1 distance: ∫ | p1 − p2 | = 2∫ | μ1 − μ2 | ≤(1/2)supj=1m | θj1 − θj2|. For covering all the densities in this working model with L1 balls Bmj with radius nu−1/ ), we can use the densities with parameters in ℓ∞ balls  : s = 1,…, m], with radius h/2 = 2nu−1/.Consider two densities of (Nu(π(·∣m))≤[∑j1=−∞∞⋯∑jm=−∞∞πh],…, θm ∈ ∣m)u]u1/ = [∏s=1m∑js=−∞∞πh]∣m)u]u1/ = [∑j=−∞∞πh]∣m)u]m/u, since the priors of θ1,…, θm∣m are iid.Then θj (given model m) is continuous, symmetric, and decreasing from the origin π(θj∣m) = fm(|θj|) for some decreasing functions fm ∈ Lu  priors or double-exponential densities).Now assume that the prior density for each j=−∞∞πh]∣m)u = 2∑j=0∞πh]∣m)u ≤ 2∑j=0∞fm(jh)uhu−1≤[2hfm(0)u + ∫∞−∞fm(|w | )udw]hu−1, where we used the decreasingness of fm in the last two steps. The integration exists since fm ∈ Lu.Then ∑Nu(π(·∣m)) ≤ O(hm(u−1)/u) = O(nm(1−u)/u2), which is finite despite the unbounded priors support. Then according to Nu(π) = [∑m(πmNu(π(·∣m)))u]u1/ will be O(nu)/u2(1−) if πm ∝ eK(m−1)lnn− for some large enough constant K > 0. So the norm complexity term in O(lnn/n), which, when compared with the last formula in O(1) by model averaging. Therefore, the norm complexity term does not affect the convergence rate significantly due to model averaging, and the convergence rate is mainly determined by the penalized divergence d12. The bounding of the penalized divergence is similar to the example discussed in the previous subsection and we omit the details. The resulting convergence rates are essentially the same as when the uniform priors (with bounded supports) are used, despite the fact that we now allow priors with unbounded supports .Then we have P0Π(A) ≤ P0ϕ + P0Π(A)(1 − ϕ) for any “test” ϕ as a function of D valued in . Consider P0 represents the expectation under the true density p0.P0∫f∈A(f(D)/p0(D))(1 − ϕ)π(df) = ∫f∈APf(D)(1 − ϕ)π(df) due to Fubini's Theorem, where Pf represents the expectation under density f.t > 0, we have Using Markov's theorem, for any All these combine to (∗) Bj, there exist ϕj such that for any α, β > 0, u ∈ , supf∈Bj(αP0ϕj + βPf(1 − ϕj)) = supf∈Bj∫min{(αp0), (βf)} ≤ supf∈Bj∫{(αp0)u1−(βf)u} = αu1−βusupf∈BjP0(p0/f)u−.Now apply a result that is a straightforward extension of Ghosal et al.  so that α/β = λj in the above statement.Given any j∈NBj to be a convex cover of A and define a combined test ϕ = maxj∈Nϕj and plug it into (∗), we have If we take ∪Therefore, we have λj = (1 − u)−1uaπ(Bj)−1 and Take c = (t/(t + u))t/(t+u)−(u/(t + u))u/(t+u)−(uu(1 − u)u1−)t/(u+t)− ≤ 4. Then notice that P0(p0/f)t = 1 + tdt2; we obtain the proof of p0 and f, are the densities for the entire data set p0D and fD, resp., and we do not assume iid.)We get t ∈ ∪, Use the fact that, for any This leads to the proof by applying πm focuses on only one model.This is a special case of 𝒫 = ∪m∈M𝒫m, where 𝒫m is the support of π(·∣m).Repeat the proofs of Propositions A∩𝒫 is doubly indexed as ∪m∈M∪j∈𝒩mBmj, where 𝒩m has cardinality at most j∈𝒩mBmj is a convex cover of the support A∩𝒫m. Then the result in Suppose the convex cover of A = [f : du−2 > ϵ].In Bmj are such that infj,minff∈Bmjdu−2 > δ. Then we have (†) Bmj needed to cover A∩𝒫m.Suppose all the convex sets Bmj in more detail. They are used to cover A, so without generality, each Bmj contains a point in A, say p1, which is not close to p0 since du−2 > ϵ. If Bmj is small so that any two points are close together, then any point p2 in Bmj  can be made to be also not close to p0, so that du−2 > δ for some δ > 0 related to ϵ. This would be easy to establish by a triangular inequality, were it not for the difficulty that the divergence du− is not a true distance for u ≠ 1/2. So we would not be able to say, for example, that Bmj should be a small du−-ball.Now we try to define the convex sets u = 1/k for any k = 2,3, 4,…, we have |p1u − p2u|u1/ = |p1k1/ − p2k1/ | |p1k1/ − p2k1/|k−1 ≤ |p1k1/ − p2k1/ | ∑i=0k−1 | p1i/kp2k−1−i)/k( | 2k−1 = |(p1k1/)k − (p2k1/)k | 2k−1, so that To resolve this difficulty, we derive the following inequalities: Bmj is an L1 ball with small enough radius λ, then any point p2 in Bmj has a not too small divergence du−2 ≥ ϵ − |du−2 − du−2|≥ϵ − u−12(2λ)u ≥ ϵ − u−14λu = δ. We can take λ = [(u/4)(ϵ − δ)]u1/. The L1 balls are also convex as required. We will take ϵ − δ = 4/(nu). Then the radius λ = (1/n)u1/. Therefore, we conclude that we will take Bmj's to be small L1 balls with radius nu−1/.Therefore, if A ∪ 𝒫m. We can use this upper bound L1 balls of radius (1/n)u1/ needed to cover the entire prior-support 𝒫m under model m.Now we try to find ϵ > 0, (‡), dt2(p0π) ≡ infK∈𝒫[supq∈Kdt2 − n−1lnπ(K)] and 𝒫 is the support of the prior.Then (†) implies, for any eu/t)−2/(1+n)2 under a choice of ϵ = ϵn for the right-hand side of (‡). Then the event [du−2 ≤ ϵn] will happen for all large enough n, almost surely, due to the Borel-Cantelli lemma.Let the probability in (‡) be bounded by 4/ = infK∈𝒫[supq∈Kdt2 − n−1lnπ(K)], under a mixture prior π(K) = ∑s∈Mπsπ(K∣s), can be bounded by taking K = Km ⊂ 𝒫m, for any m ∈ M, as infK[supq∈Kdt2 − n−1lnπ(K) ≤ infminfKm{supq∈Kmdt2 − n−1ln[πmπ(Km∣m)]}≡infm[dt2) − n−1lnπm].The quantity


The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 3 G = (V(G), E(G)) denote a graph with no multiple edges and no loops and with vertex set V(G) = {v1, v2,…, vn} and edge set E(G). The degree ks of a vertex vs is the number of edges attached to it. A k-regular graph is a graph with the property that each of its vertices has the same degree k. The adjacency matrix A(G) of G is the n × n matrix with elements A(G)sj = 1 if vs and vj are connected by an edge and zero otherwise. The Laplacian matrix Q(G) is the n × n matrix with the element Q(G)sj = ksδsj − A(G)sj, where δsj is the Kronecker delta, equal to 1 if s = j, and zero otherwise. Denote by t(G) the number of spanning trees of a graph G. Enumeration of spanning trees on the graph is a problem of fundamental interest in mathematics and physics. This number can be calculated in several ways. A basic result is “the Matrix-Tree Theorem.”Let G be a graph with vertex set {v1, v2,…, vn} and let Q(G) be its Laplacian matrix. Then,Q(G)s}{ is the submatrix of Q(G) by deleting the sth row and the sth column from Q(G) for 1 ≤ s ≤ n.Let Q(G) is always zero. We can express t(G) that can be expressed by the nonzero eigenvalue of Q(G) as follows.Note that one of the eigenvalues of μ1 ≤ μ2 ≤ ⋯≤μn−1 be the Laplacian eigenvalues of a connected graph G with n vertices. Then, t(G) = μ1μ2 … μn−1/n.Let 0 < G is said to be n-rotational symmetric if the cyclic group of order n is a subgroup of the automorphism group of G. Yan and Zhang [n-rotational symmetric graph. As applications, they got explicit expressions for the numbers of spanning trees and the asymptotic tree number entropy for some lattices with cylindrical boundary condition.By two methods, Ciucu et al.  obtainednd Zhang  also obtt(G) has asymptotically exponential growth; one defines the quantity z(G) byLattices are of special interest for their structures. In particular, the number of spanning trees in a lattice was studied extensively. It turns out that t(G) have been obtained for many lattices. Wu [d dimensions under free, periodic, and a combination of free and periodic boundary conditions and a quartic lattice embedded on a Möbius strip and the Klein bottle. Shrock and Wu [d-dimensional body-centred cubic lattice and thermodynamical limit. They also gave an exact integral expression for thermodynamical limit on the face-centred cubic lattice and 4.8.8 lattice. Chang and Wang [Closed-form expressions for ices. Wu  evaluateices. Wu  obtainedk and Wu  got a gek and Wu  got closand Wang  considerand Wang , 11.3 · 42 lattice. The number of spanning trees of 4.8.8 lattice is gotten in 3 · 42 lattice in Sections In this paper, we present an exact closed-form result for the asymptotic growth constant for spanning trees of lattices embedded on Klein bottle, exactly for 4.8.8 lattice, hexagonal lattice, and 3B−1 and BT be the inverse and the transpose of a matrix B. And let Im denote the m × m identity matrix. SetU be an m × m matrices with entriesm × m matrices U−1 arem × m matrices U−1K1U, U−1(K1T)U and U−1K2U areθt = 2tπ/m for t, j = 1,2,…, m.Introduce some notation firstly. Let L4.8.8 is shown in bj, bj*), for 1 ≤ j ≤ n in L4.8.8, we obtain a graph with cylindrical boundary condition, denoted by L4.8.8c. Adding edges , for 1 ≤ s ≤ m in L4.8.8c, a 4.8.8 lattice with toroidal boundary condition, denoted by L4.8.8t, can be gotten.The 4.8.8 lattice L4.8.8c:a = 1 − cos(2jπ/m), b = 10 − 2cos(2jπ/m), and c = 14 − 6cos(2jπ/m).Yan and Zhang  got the L4.8.8t can be expressed asθ1 = 2sπ/n and θ2 = 2jπ/m. Chang and Shrock [L4.8.8t by an exact closed-form evaluation of the integral given in [Shrock and Wu  showed tas, as*), for 1 ≤ s ≤ m in L4.8.8c, 4.8.8 lattice L4.8.8K with Klein bottle boundary condition can be gotten. By a suitable labelling of vertices of L4.8.8K, the adjacency matrix X of it can be written in terms of a linear combination of direct products of smaller ones:By adding edges , the Laplacian characteristic polynomial of L4.8.8K can be expressed asj = 1,…, m/2 − 1, ϕj(x) = det(xIn4 − Aj′ − Cj′) and for j = m/2, m. Note thatμ1, μ2,…, μmn−14 are the nonzero Laplacian eigenvalues of L4.8.8K.For an even value of dIn4 − A − B − BT − C also is a Laplacian matrix of a graph, denoted by L4.8.80 (see ϕm(0) = det(−Am′ − Cm′) = det(−dIn4 + A + B + BT + C) = 0 and ϕm′(0) = (−1)n−144nt(L4.8.80). So, we haveL4.8.80 by Lh0 . Let Y be a subset of the row/column index set of P. For convenience, let PY denote the determinant of the matrix obtained from P by deleting all rows and columns whose indices are in Y. For j = 1,…, m/2 − 1, noticing that θj = −θm−j and D1(θj) = D1(−θj), expanding the determinant In the following, we turn to calculate ϕj(0){1}, ϕj(0){1,2,3,4}, ϕj(0){1,2,3,4,5}, and ϕj(0)n}{1,8.Now, we turn to calculate Fn2 = ϕj(0){1}, Ln−12 = ϕj(0){1,2,3,4}, Fn2′ = ϕj(0)n}{1,8, and Ln−12′ = ϕj(0)n}{1,2,3,4,8. Also set Γj−1 = Γj{1,2,3,4}, j = 2,…, n, Γn ∈ {Fn, Ln, Fn′, Ln′}.Let By the Laplace expansion theorem, we obtain several expansions. First, an expansion by rows 1,2, and 3:The recursion relations  and 21)21) giveΓn=(28−12By combining  and 24)24), we oExpanding the determinant along the first row and then expanding the resulting determinants along the first column, we haveThe number of spanning trees of 4.8.8 lattice can be expressed asLh is shown in a1 and b1*, am* and bn2*, and bs and bs* for f or s = 2,…, 2n − 1 in Lh, we obtain a graph with cylindrical boundary condition, denoted by Lhc. Adding edges  for 1 ≤ s ≤ m, in Lhc, a hexagonal lattice with toroidal boundary condition, denoted by Lht, can be gotten.The hexagonal lattice Lhc:a = 1 − cos(2jπ/m), b = 7 − cos(2jπ/m), and c = 3 − cos(2jπ/m).Yan and Zhang  got the Lht can be expressed asθ1 = 2sπ/n and θ2 = 2jπ/m.Shrock and Wu  showed tas, as*) for 1 ≤ s ≤ m, in Lhc, a hexagonal lattice LhK with Klein bottle boundary condition can be gotten. For the number of spanning trees of LhK, we have the following result.By adding edges n×n, in which asj1 = ajs1 = 1 if s is odd and j = s + 1; else, asj1 = 0; A2 = (asj2)n×n, in which asj2 = ajs2 = 1, if s is even, and j = s + 1; else, asj2 = 0; C1 = (csj)n×n, in which cn1 = cn1 = 1, otherwise 0.By suitable labelling of vertices of A3(x) = (asj3)n×2n2, in which asj3 = aj+2,s+23 = ei/2x(−1), if s is even, and j = s + 1, an−1,2n23 = 1 + eix−, and an,2n−123 = 1 + eix; else, asj3 = 0; C3(x) = (csj3)n×2n2, in which cn1,23 = −1 − eix, cn,123 = −1 − eix−, otherwise 0. Expanding the determinant T along the first row and then expanding the resulting determinants along the first column, we haveFn−14 = T{1}, Ln−1)22 + c, b = 3 − cos(2jπ/m) − c and The recursion relations  and 37)37) giveΓn=(6−2coombining  and 40)(38)Γn= for 1 ≤ s ≤ m, in L3·423c, a 33 · 42 lattice with toroidal boundary condition, denoted by L3·423t, can be gotten.The 3L3·423c:a = 7 − 9cos(2jπ/m) + 2cos2(2jπ/m), b = 17 − 13cos(2jπ/m) + 2cos2(2jπ/m), and c = 11 − 11cos(2jπ/m) + 2cos2(2jπ/m).Yan and Zhang  got the L3·423t can be expressed asθ1 = 2sπ/m and θ2 = 2jπ/n.Chang and Wang  showed tas, as*) for 1 ≤ s ≤ m, in L3·423c, a 33 · 42 lattice L3·423K with Klein bottle boundary condition can be gotten. For the number of spanning trees of L3·423K, we have the following theorem.By adding edges n×2n2, where bsj = 1, if s is odd, and j = s + 1; else, bsj = δsj, C = (csj)n×2n2, in which cn1,2 = cn,12 = 1, otherwise 0. Using the same notations as By a suitable lebelling of vertices of Fn2 = ϕj(0){1}, Ln−12 = ϕj(0){1,2}, Fn−12′ = ϕj(0)n}{1,2, and Ln−12′ = ϕj(0)n}{1,2,2. Also set Γj−1 = Γj{1,2}, j = 2,…, n, Γn ∈ {Fn, Ln, Fn′, Ln′}.Let By the Laplace expansion theorem, we obtain several expansions. First, an expansion by rows 1 and 2 is as follows:c = (11 − 11cos(2jπ/m) + 2cos(2jπ/m))2 − 2 − 2cos(2jπ/m). By combining (The recursion relations  and 50)50) giveΓn=(22−22ombining  and 53)(51)Γn=/|V(Gn)|) = h. If {Gn′} is a sequence of connected subgraph of {Gn}, such that limn→∞(|{v ∈ V(Gn′); dGn(v) = dGn′(v)}|/|V(Gn)|) = 1, then limn→∞(logt(Gn′)/|V(Gn′)|) = h.Let {3 · 42 lattice have the same property.By


On page 1, Importance section, line 5, “trichloroacetic acid cycle” should be “tricarboxylic acid cycle.” This error was due to a copyediting mistake.Volume 1, no. 2, doi:


Cℵ-fibration property. We give and prove the relationship between the Cℵ-fibration property and an approximate fibration property. Furthermore, we study the pullback maps for Cℵ-fibrations.We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the  Sℵ-fibrations as extension of Hurewicz fibrations. In  into Xτ with the compact-open topology. Recall  be two given S-maps withy ∈ Y, s ∈ I, where ρc is a compact-open topology on P(O) which is induced by ρ. Define an S-homotopy H :  → P byy ∈ Y, s, t ∈ I. That is, s ∈ I. Hence Cℵ-fibration.Since c-lifting function λf is called regular if for every x ∈ Xτ2, Xτ2 ∈(X × O)τ2,t ∈ I.In The following theorem is an analogue of results of Fadell in Hurewicz fibration theory .f12 : → be a regular Cℵ-fibration and let S-map defined by Mi(α) = (α(0), fi∘α) for all α ∈ P(Xτi) where i = 1,2. Then (1)M1∘λf = X × id|f2)Δ S-homotopy S-maps λf∘M2 and f1[(H(α)(s))(t)] = f2(α(t)) for all t, s ∈ I, α ∈ P(Xτ2).Let x, α) ∈ Δ(f2),M1∘λf = X × id|f2)Δ and s ∈ I, define a path βs1− ∈ P(Oρ) byλf, we can define an S-homotopy H : P → P bys ∈ I, α ∈ P(Xτ2). Thenα ∈ P(Xτ2), t ∈ I. That is, s, t ∈ I, α ∈ P(Xτ2). Hence For the second part, for f : Xτ → Oρ of compact metrizable spaces Xτ and Oρ is called an approximate fibration if, for every space Yω and for given ϵ > 0, there exists δ > 0 such that whenever g : Yω → Xτ and H : Yω × I → Oρ are maps with d < δ, then there is homotopy G : Yω × I → Xτ such that G0 = g andCoram and Duvall  introducCNπ-fibration property in inducing an approximate fibration property.The following theorem shows the role of the  S-map f : → with metrizable spaces Xτ and Oρ, by dτ and dρ, we mean the metric functions on X and O, respectively; by X × O we mean the product metrizable space of Xτ and Oρ with a metric functionGf we mean the graph of f ) : x ∈ X}) which is an S-subspace of ((X × O)τ×ρ, π); for a positive integer n > 0, by Gn(f), we mean the (1/n)-neighborhood of Gf in a metrizable space X × O which is also S-subspace of ((X × O)τ×ρ, π).For anf : Xτ → Oρ be a map with compact metrizable spaces Xτ and Oρ. Then f is an approximate fibration if and only if, for every positive integer n > 0, there exists a positive integer m ≥ n such that the S-map fn : (Gn(f), π)→ has the CNπ-fibration property by the inclusion S-map Imn : (Gm(f), π)→(Gn(f), π), where fn = b for all  ∈ Gn(f).Let n be any positive integer. For ϵ = 1/n > 0, let δ be given in the definition of approximate fibration. Since δ/2 > 0 and f is a continuous function, then let δ′ be chosen such that if x, x′ ∈ X and dτ < δ′, then dρ(f(x), f(x′)) < ϵ/2. Choose a positive integer m⩾n, such that 1/m ≤ δ′, δ/2.Let Yω, π) ∈ Nπ and let g : →(Gm(f), π) and G :  → P be two given S-maps with G0 = fn∘(Imn∘g). Define a map g′ : Yω → Xτ by g′(y) = J1[g(y)] and a homotopy G′ : Yω × I → Oρ by G′ = G(y)(t) for all y ∈ Y and t ∈ I. We get that g(y) = (g′(y), G′) for all y ∈ Y. Since g(y) ∈ Gm(f), then there exists x ∈ X such thaty ∈ Y. This impliesy ∈ Y. Hence, since f is an approximate fibration, there exists a homotopy H′ : Yω × I → Xτ such that H0′ = g′ andy ∈ Y, t ∈ I. Define an S-homotopy H :  → P(Gn(f), π) byy ∈ Y and fn∘Ht = Gt for all t ∈ I. Hence fn has the CNπ-fibration property by Imn.Now let  < δ′, then dρ(f(x), f(x′)) < ϵ/2. Choose a positive integer n > 0 such that 1/n ≤ δ′, ϵ/2. By hypothesis, there exists a positive integer m ≥ n such that fn has the CNπ-fibration property by Imn.Conversely, let δ = 1/m. Let Yω be any space and let g : Yω → Xτ and G : Yω × I → Oρ be two given maps withy ∈ Y. Define an S-map g′ : →(Gm(f), π) by g′(y) = (g(y), G) and an S-homotopy G′ :  → P by G′(y)(t) = G for all y ∈ Y and t ∈ I. Since G0′ = fn∘(Imn∘g′), then there exists an S-homotopy F :  → P(Gn(f), π) such that F0 = Imn∘g′ and fn∘Ft = Gt′ for all t ∈ I. By the last part, we can define a homotopy H : Yω × I → Xτ byF(y)(t) = , G). Since F(y)(t) ∈ Gn(f), then there exists x ∈ X such thaty ∈ Y, t ∈ I. Hence f is an approximate fibration.Take


A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shownto verify the validity of the results. A discussion is presented for further study. Cancer is a well-known killer of humans worldwide, and its treatments are varied and sporadically successful. There are four main types of cancer treatments, which are surgery, chemotherapy, radiotherapy, and immunotherapy. In this paper, we only consider cancer treatment by radiotherapy.Radiotherapy, as a primary treatment strategy, has been proven to be an effective tool in combating with cancer , 2. RadiIt is an important and effective way to deeply understand the real-world problems by establishing mathematical models and analyzing their dynamical behaviors  be the concentration of healthy cells, and let x2(t) be the concentration of cancer cells; then our model takes the formD(t) is the strategy of the radiotherapy. We assume that D(t) ≡ γ > 0 when t ∈ /[(α1 − K2β1) + ɛ(α2 − K1β2)]. Furthermore, if [α2(α1 − K2β1) + α1(α2 − K1β2)]/[(α1 − K2β1) + ɛ(α2 − K1β2)] < min{α1/ɛ, α2}, thenα2(α1 − K2β1) + α1(α2 − K1β2)]/[(α1 − K2β1) + ɛ(α2 − K1β2)]⩾min{α1/ɛ, α2}, γ is inexistent.(c) Consider  for any positive initial values. However, if we treat it all the time, by condition . Hence, if the treatment is periodic, the solution of system  during the treatment stage, and it will tend to the cancer win state  during the no treatment stage. This will lead to the appearance of a periodic solution.Under conditions of ondition  we know ondition  has a glondition  and Lemmondition  has a unf system  will tenω-periodic solution is guaranteed by The existence of a positive x2(t) ≡ 0:t ∈ } → 0 as n → ∞. Together with (V(t) → 0 as t → ∞. Hence, x1(t) → x1*(t) and x2(t) → 0 as t → ∞. The cancer eradication periodic solution is unique and global stable. This completes the proof.By a simple calculation and from the last inequality of condition , it can her with , we finaγL < α2ω holds. Further, ifσ2 = mint∈{x2*(t)}. Then system , x2(t)) be any solution of system  + ɛ(α2 − K1β2) = 0.007625 > 0 and μ2 = α2(α1 − K2β1) + α1(α2 − K1β2) = 0.03075 > 0. The range of γ is determined by case (a) in γ < 0.45. We choose γ = 0.35 here. It is easy to verity that all the conditions of Theorems L = 15 hours , 0) exists by L = 8 hours) and L = 9 hours). From σ1 = mint∈{x1*(t)}⩾0.44, and, σ1⩾0.44. It is easy to verify that condition (L = 8 hours) and Figures L = 9 hours).To show the existence and global stability of the cancer eradication periodic solution, we illustrate ondition  is satisIt can be seen from Figures γ to the cancer eradication periodic solution with ω = 10 hours and L = 8 hours. Here, we also choose ɛ = 0.3. For the sake of getting a cancer eradication periodic solution, we need the condition ɛγL < α1ω. Then we have γ < α1ω/(ɛL) = 0.83. However, the true γ(ω − L)+(−η)ω < 0. From the fact that −η = max{α1 − ɛγ − α1σ1/K1, α2 − γ}, then we have γ(ω − L)+(α2 − γ)ω < γ(ω − L)+(−η)ω < 0, which implies γ > 0.625. Hence, here we choose γ = 0.65,0.75 and 0.8 separately to investigate the influence of the variance of the γ to the cancer eradication periodic solution. The dynamical behavior of the system with γ = 0.65 can be seen in Figures As we all know, larger dosage radiation can kill cancer cells more effectively, but it also may increase the rate to the healthy cells from the radiation. We now investigate effects of the variance of the parameter γ = 0.75, it is easy to verify that ɛγL < α1ω and system }⩾0.43; then condition }⩾0.42, then it is easy to verify that condition (γ = 0.65), Figures γ = 0.75), and Figures γ = 0.8) that under the circumstance that the system has a cancer eradication periodic solution and all the other conditions are not changed, the increase of the radiation dosage will quicken the extinction of the cancer cells but will also decrease the concentration of the healthy cells at the same time. One can refer to the three figures; in γ = 0.65, 0.1 ⩽ x1, in γ = 0.75, 0.05 ⩽ x1 ⩽ 0.1, and in γ = 0.8, x1 ⩽ 0.05.If we take ondition  is satisondition  is satisα1 = 0.2, α2 = 0.5, β1 = 0.48, β2 = 0.05, K1 = 0.65, and K2 = 1. It is easy to verify that condition ) by L = 2 hours. From σ2 = mint∈{x2*(t)}⩾0.55. It is easy to verify that condition ) by L = 1 hour. From σ2 = mint∈{x2*(t)}⩾0.6. It is easy to verify that condition  = 0.5 ⩽ K1 and x2(0) = 0.8 ⩽ K2. Also note that the choice of the values in the theorems are based on the range determined by [Throughout Figures mined by , but theγx2 to the cancer cells and ɛγx1 to the healthy cells. However, during the recovery stage, the model has taken the most basic Lotka-Volterra competition type. We gave the range of radiation dosage γ under the following three results: the healthy cells and cancer cells are coexist, the cancer eradication periodic solution is globally stable, and the cancer win periodic solution is globally stable. Note that, in the discussion on the global stability of the cancer eradication periodic solution, we showed the relationship between the radiation dosage γ and the treatment time L; that is, ɛγL < α1ω and γ(ω − L) < ηω, which are also given during the discussion on the cancer win periodic solution (γL < α2ω). However, during the discussion on the coexistence of the healthy cells and cancer cells, we only give the range of the radiation dosage γ, which shows that it fits all the treatment time L ∈ . This is a flaw, which is caused by the analysis methods in this paper. We will improve our analysis techniques and make a further study in the future.In this paper, we employed a pair of ordinary differential equations to model the dynamics between the healthy cells and cancer cells for the cancer treatment by radiotherapy. We separated the treatment into two stages: treatment stage and recovery stage (no treatment stage). During the treatment stage, the radiation harvesting amount is The cancer treatment model discussed in this paper is only based on one treatment measure. It may be more effective for the cancer treatment if we add a medication during the recovery stage, which is still an open problem and we will carry out the research in the further work.


Apocrita has a special structure that its first abdominal segment has been incorporated into the thorax as the propodeum. The remaining abdomen, metasoma, is connected to this hybrid region via a narrow propodeal-metasomal articulation forming a “wasp waist”, which serves an important function of providing maneuverability, flexibility and posture for oviposition. However, the origin and transformation of the propodeal-metasomal articulation are still vague. Ephialtitidae, as the basal group of Apocrita from the Early Jurassic to the Early Cretaceous, have shown various types of propodeal-metasomal articulations.Acephialtitia colossa gen. et sp. nov., Proephialtitia acanthi gen. et sp. nov. and P. tenuata sp. nov., collected respectively from the Early Cretaceous Yixian Formation at Liutiaogou and the Middle Jurassic Jiulongshan Formation at Daohugou, both in Inner Mongolia, China. These genera are assigned to the Ephialtitidae based on their complete wing venation, e.g. 2r-rs, 2r-m, 3r-m and 2 m-cu always present in the forewings and Rs, M and Cu in the hind wings. These new fossil ephialtitids have well-preserved propodeal-metasomal articulations indicating metasoma is broadly attached to propodeum.This study describes and illustrates two new genera with three new species, The broad articulation between the propodeum and metasoma in basal Ephialtitidae, likely passed on from a still more basal family Karatavitidae, suggests three separate pathways of the transformation of the “wasp waist” in three different derived lineages leading from Ephialtitidae to: (i) Kuafuidae and further to the remaining Apocrita, (ii) Stephanidae, and (iii) Evanioidea. In addition, the demise of ephialtitid wasps lagging behind the flourishing of angiosperms suggests that ephialtitid extinction might have been mainly driven by competition with numerous new taxa (eg. the abundant Cretaceous xylophilous Baissinae and Ichneumonoidea) appeared just before or/and soon after the J/K boundary.The online version of this article (doi:10.1186/s12862-015-0317-1) contains supplementary material, which is available to authorized users. The extinct family Ephialtitidae was considered as the second most stem group in the suborder Vespina of Hymenoptera. Together with the most stem group of Karatavitidae, they jointly constituted the superfamily Ephialtitoidea . In receAcephialtitia colossa gen. et sp. nov., Proephialtitia acantha gen. et sp. nov. and Proephialtitia tenuata sp. nov., based on three well-preserved, nearly complete female specimens. These specimens were collected respectively from the Early Cretaceous Yixian Formation at Liutiaogou and the late Middle Jurassic Jiulongshan Formation at Daohugou, both are in Inner Mongolia, China. According to the accurate Ar–Ar and SHRIMP U–Pb dating, the age of the Yixian Formation is considered as the Early Cretaceous  - Cell 3rm much longer than 1mcu……………………………………………………..……..8Cells 2-3rm short, 3rm shorter than 1mcu……………… Stephanogaster Rasnitsyn, 1975 [distinctly bent…………………………………..……….- Forewing 4 mm or shorter. Hind wing when known with r cell closed and 1-M shorter andAsiephialtites Rasnitsyn, 1975 [straight (or nearly so)…………………….……..…………Forewing 5 mm or longer. Hind wing when known with r cell closed and 1-M longer and- 3r-m present………………………………………………………………………………11(2). 3r-m and 1a-2a lost…………………………………………………………………….10…………………………….…………………………………………Liadobracona Zessin, 1981 [in, 1981 Sessiliventer Rasnitsyn, 1975 [- 2 m-cu present, 2-RS almost straight with no sign of 1r-rs ……yn, 1975 2 m-cu lost, 2-RS angular at junction with rudimentary 1r-rs………………………………Montsecephialtites Rasnitsyn & Martínez-Delclòs, 2000 [- 2r-m present ……………………………………………………………………………….12(9). 2r-m lost, 1a-2a lost…….. òs, 2000 - 2r-m pr- 2r-m subvertical, not sinuate, often distant from 3r-m for much more than its length…….152r-m oblique, sinuate, distant from 3r-m for about its own length or less………………..13Tuphephialtites Zhang et al. 2002 [- 2r-rs and 2r-m clearly distant, 1a-2a lost …………………………………….……………142r-rs and 2r-m practically coincide, 1a-2a present…..al. 2002 - 2r-rs aCratephialtites Rasnitsyn, 1999 [Cretephialtites Rasnitsyn & Ansorge, 2000 [- Cell 3rm receiving 2cu-a ……………………ge, 2000 Cell 2rm receiving 2cu-a…………………………..……Ephialtites Meunier, 1903 [- If (rarely) cell 2rm as above, hind wing with 1-RS oblique. Ovipositor never bent upward…...16(12). Cell 2rm long, surpassing level of pterostigmal base basally. Hind wing with 1-RS vertical to both R and 2-RS. Ovipositor evenly bent upward…......er, 1903 - If ……..…………………………….Antenna 12-segmented, 1a-2a present, metasoma widest near midlength…………………..………………………………..……………. Brigittepterus Rasnitsyn, Ansorge & Zessin, 2003 [in, 2003 - 1-RS reclivous, 1r-rs rarely present…………………………………………………………22(1). 1-RS proclivous, short 1r-rs present……………………………………………………………………………………………………………………….……Symphyogaster Rasnitsyn, 1975 [Cephenopsis might run here but differs in having complete 1r-rs reaching- 3r-m lost, female metasoma either longer or not subcylindrical…………………….…….233r-m lost, female metasoma subcylindrical, as long as head and mesosoma combined……..- 1a-2a lost……………………………………………………………………………….…261a-2a present……………………………..………………………………………………24………………..Micrephialtites Rasnitsyn, 1975 [distinctly angled with 2nd in side view…- Cell 3rm usually shorter than 2rm and/or 1mcu. 1st metasomal segment narrowing basal…………………………………………………………………………………………….25Cells 2-3rm and 1mcu of about same length. 1st metasomal segment big, cylindrical,Karataus Rasnitsyn, 1977 [Hind femur very thick…………………………………..…… yn, 1977 - Cells 2rm and 3rm not simultaneously as long as above. Hind femur ordinary……………Symphytopterus Rasnitsyn, 1975 […………………………………………………………Cell 2rm along M much longer than cell 1mcu, and 3rm along M about as long as 1mcu.……………………………………..………Karataviola Rasnitsyn, 1975 [Body more elongate- 2r-m distant from 2r-rs for about its length. Antenna filiform .Trigonalopterus Rasnitsyn, 1975 [Body more robust………………………………….….(23). 2r-m oblique, almost reaching 2r-rs. Antenna setaceous (narrowing towards apex).Symphyogaster, Pracproapocritus, Acephialtitia and Karataviola form a branch Ephialtitidae, which as the sister group of the remaining big clade (Kuafuidae + (Orussoidae + ((Stephanidae + Evanioidae) + (Ceraphronmorpha + (Proctotrupomorpha + (Ichneumonomorpha + Vespomorpha))))). This first big clade supported by the following characters: 1-Rs short, proclined or subvertical, reclined , 1r-rs  longer than 2r-rs, or lost traceless , hind wing m-cu lost , hind wing jugal lobe  not delimited , first abdominal segment/propodeum fused with metapleuron . In the second big clade, Kuafuidae as the sister group to the remain groups is supported by 2r-rs entirely lost . In the third big clade, Stephanidae is sister group to Evanioidea, and together they are sister group to the remaining (Ceraphronomorpha + (Proctotrupomorpha + (Ichneumonomorpha + Vespomorpha))), which supported hind wing cell r open or very small .An analyses using NONA resulted in forty-five most parsimonious trees, each consisting of 29 steps, consistency index = 0.82; retention index = 0.83. . The family Ephialtitidae was interpreted as a stem group of Apocrita demonstrating the origin and transformation of the wasp waist characters of the Apocrita [A key character of Ephialtitidae, as well as of more primitive taxa , is that the first abdominal segment (propodeum) is more or less convex (bent) transversally but even  longitudinally Figure A. The fiThe second pathway of transformation is the pattern Figure A basicalSome other Ephialtitidae Figure D demonstAfter studying the wasp waists of described fossil species of Ephialtitidae, we identified three typical but different propodeal-metasomal articulations: the shape of the first metasomal segment varies from narrow and greatly elongate with distal end broader than the proximal end , to sli& Zhang, ), then, Apocrita ,5,28. OuApocrita ,30. ThisThe above hypothesis suggests three separate pathways of the wasp waist transformation. Concerning our considerations about various ovipositing postures of Apocrita, this poses limitation only on the most sophisticated “=”-posture for both ephialtitid-stephanid and evanioid lineages, as well as on the “L”-posture for only the most basal versions of the ephialtitid mechanism Figure A. The ovBased on the information of Table Based on the information and data summarized in Table Archaefructus liaoningensis, Archaefructus sinensis and Archaefructus eoflora from the Early Cretaceous Yixian Formation, are now widely accepted as important fossil angiosperm plants [Sinocarpus decussates, from the same Formation. These fossil angiosperms provided important information about early angiosperms which co-existed with many gymnosperm and other plants in the same ecosystems [The above considerations suggest addressing to profound changes in vegetation occurred in the Cretaceous while seeking for a possible cause of demise of the Ephialtitidae. It is well known that the non-marine Earth transformed from the gymnosperm- to angiosperm-forested that time. Early angiosperms, vascular flowering plants with seeds enclosed in an ovary, have many primitive features which are considerably different from extant angiosperms . Primitim plants -37 Leng m plants  describeosystems ,40. Duriosystems , and theosystems ,32 . At the same time, Ephialtitidae appeared as a rare group since the very beginning of Cretaceous and lost the fossil record after the Aptian Table . This paA thorough review of the various types of the propodeal-metasomal articulation of Apocrita suggests that the wide articulation between the propodeum and metasoma in basal Ephialtitidae was passed on from a still more basal family of Karatavitidae and provided three separate pathways of transformation of the wasp waist. In addition, the demise of ephialtitid wasps lagging behind the flourishing of angiosperms suggests that ephialtitid extinction driven by competition with numerous new taxa (e.g. the abundant Cretaceous xylophilous Baissinae and Ichneumonoidea) appeared just before or/and soon after the J/K boundary, rather than the transformed from the gymnosperm to angiosperm-forested led to shortage of food sources for hosts of the larvae of ephialtitids.The authors declare that the study makes no uses of human, clinical tools and procedures, vertebrate and regulated invertebrate animal subjects and/or tissue, and plants.


R0 algebras and R0-algebras to the noncommutative forms is given. We show that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras and that pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak-R0 algebras and R0-algebras, are investigated. The simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras are obtained.A positive answer to the open problem of Iorgulescu on extending weak- It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. Due to strict and complete logical foundation , making inferences about certain information can be done with high confidence levels. Thus, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. It is evident that this kind of logic cannot be two-valued logic itself but might form a certain extension of two-valued logic. Various kinds of nonclassical logic systems have therefore been extensively researched in order to construct natural and efficient inference systems to deal with uncertainty.In recent years, motivated by both theory and application, the study of t-norm-based logic systems and the corresponding pseudo-logic systems has become of greater focus in the field of logic. Here, t-norm-based logical investigations preceded the corresponding algebraic investigations, and in the case of pseudo-logic systems, algebraic development preceded the corresponding logical development.A noncommutative generalization of reasoning can be found, for example, in psychological processes. In clinical medicine on behalf of transplantation of human organs, an experiment was performed in which the same two questions have been posed to two groups of interviewed people as follows. (1) Do you agree to donate your organs for medical transplantation after your death? (2) Do you agree to accept organs of a donor if you need them? When the order of questions was changed in the second group, the number of positive answers here was much higher than in the first group.The following reviews some situation concerning some important logic algebras and the corresponding pseudo-logic algebras. BCK- and BCI-algebras were introduced by Imai and Iseki  and haveR0-algebras were introduced by Wang R0-algebProblem. Recall that the IMTL-algebras, introduced in 2001 by Esteva and Godo, are categorically isomorphic to weak-R0 algebras, introduced in 1997 by Wang, and that NM-algebras are categorically isomorphic to R0-algebras, introduced also in 1997 by Wang. Extend weak-R0 algebras and R0-algebras to the noncommutative case.R0 algebras and R0-algebras to the noncommutative forms, called pseudo-weak-R0 algebras and pseudo-R0 algebras. We show that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL-algebras and that pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak-R0 algebras and R0-algebras, are investigated. Furthermore, we discuss the simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras.In this paper, we extend weak-We recall some definitions and results which will be used in the sequel.A, ∨, ∧, ⊙, →, 0,1) of type  such that for all x, y, z ∈ A(B1)A, ∨, ∧, 0,1) is a bounded lattice,((B2)A, ⊙, 1) is a monoid,((B3)x⊙y ≤ z if and only if x ≤ y → z,(B4)x → y)∨(y → x) = 1. is a structure ((B5)x−− = x,where x− = x → 0.An IMTL- (involutive MTL-) algebra is an MTL-algebra satisfying the following additional condition:(B6)x⊙y)−∨((x∧y)→(x⊙y)) = 1.(A WNM- (weak nilpotent minimum-) algebra is an MTL-algebra satisfying the following additional condition:An NM- (nilpotent minimum-) algebra is an IMTL-algebra satisfying condition (B6).A, ∨, ∧, ⊙, →, ⇝, 0,1) of type  such that for all x, y, z ∈ A(pB1)A, ∨, ∧, 0,1) is a bounded lattice,((pB2)A, ⊙, 1) is a monoid,((pB3)x⊙y ≤ z if and only if x ≤ y → z if and only if y ≤ x⇝z,(pB4)x → y)∨(y → x) = (x⇝y)∨(y⇝x) = 1. is a structure ((pB5)x~− = x−~ = x,where x− = x → 0 and x~ = x⇝0,A pseudo-IMTL (pseudo-involutive MTL) algebra is a pseudo-MTL algebra satisfying the following additional condition:(pB6)x⊙y)−∨((x∧y)→(x⊙y)) = (x⊙y)~∨((x∧y)⇝(x⊙y)) = 1,(where x− = x → 0 and x~ = x⇝0.A pseudo-WNM (pseudo-weak nilpotent minimum) algebra is a pseudo-MTL algebra satisfying the following additional condition:A pseudo-NM (pseudo-nilpotent minimum) algebra is a pseudo-IMTL algebra satisfying condition (pB6).M be a -type algebra, where ¬ is a unary operation and ∧, ∨, and → are binary operations. If there is a partial ordering ≤ on M, such that  is a bounded distributive lattice, ∧, ∨ are infimum and supremum operations with respect to ≤, ¬ is an order-reversing involution with respect to ≤, and the following conditions hold for any a, b, c ∈ M:(R1)a → ¬b = b → a,¬(R2)a = a, a → a = 1,1 → (R3)b → c ≤ (a → b)→(a → c),(R4)a → (b → c) = b → (a → c),(R5)a → (b∨c) = (a → b)∨(a → c), a → (b∧c) = (a → b)∧(a → c),where 1 is the largest element of M, then one calls M a weak-R0 algebra.Let R0 algebra M is a weak-R0 algebra satisfying the following additional condition:(R6)a → b)∨((a → b)→(¬a∨b)) = 1. such that  is a bounded distributive lattice, − and ~ are order-reversing pseudo-involution , and the following axioms hold for any x, y, z ∈ A: x → y = y−⇝x−,  x⇝y = y~ → x~,(pR1)  x = 1⇝x = x; x → x = x⇝x = 1,(pR2*) 1 →  x → y ≤ (z → x)→(z → y), x⇝y ≤ (z⇝x)⇝(z⇝y),(pR3)  x → (y⇝z) = y⇝(x → z),(pR4)  x → (y∨z) = (x → y)∨(x → z), x⇝(y∨z) = (x⇝y)∨(x⇝z);  x → (y∧z) = (x → y)∧(x → z), x⇝(y∧z) = (x⇝y)∧(x⇝z).(pR5*) A pseudo-weak-R0 algebra A is a pseudo-weak-R0 algebra satisfying the following additional axiom: x → y)∨((x → y)⇝(x−∨y)) = (x⇝y)∨((x⇝y)→(x~∨y)) = 1.(pR6)  Comparing − and ~ in (ii) The operations A, ∧, ∨, 0,1),the distributivity of bounded lattice  = (x → y)∧(x → z), x⇝(y∧z) = (x⇝y)∧(x⇝z).Hence, we have the following simplified definition. In what follows, we will use the following definition.(iii) As we will see in R0 algebra is a structure  satisfying the following:(pL1)A, ∧, ∨, 0,1) is a bounded lattice,((pL2)x ≤ y, then y− ≤ x− and y~ ≤ x~,if (pL3)x~− = x−~ = x,(pR1)x → y = y−⇝x−, x⇝y = y~ → x~,(pR2)x = 1⇝x = x,1 → (pR3)x → y ≤ (z → x)→(z → y), x⇝y ≤ (z⇝x)⇝(z⇝y),(pR4)x → (y⇝z) = y⇝(x → z),(pR5)x → (y∨z) = (x → y)∨(x → z), x⇝(y∨z) = (x⇝y)∨(x⇝z).A pseudo-weak-R0 algebra A is a pseudo-weak-R0 algebra satisfying the following additional axiom:(pR6)x → y)∨((x → y)⇝(x−∨y)) = (x⇝y)∨((x⇝y)→(x~∨y)) = 1. is a bounded lattice satisfying x−~ = x~− = x for any x ∈ A. Define operations → and ⇝ as pseudo-Lukasiewicz implication on A:A, ∧, ∨, →, ⇝, −, ~, 0,1) is a pseudo-weak-R0 algebra.Let R0 algebra, the following properties hold:~ = 0− = 1, 1~ = 1− = 0,0x− = x → 0, x~ = x⇝0,x⇝x = x → x = 1,x ≤ y if and only if x⇝y = 1 if and only if x → y = 1,i∈Ixi)~ = ⋁i∈Ixi~, (⋀i∈Ixi)− = ⋁i∈Ixi−, whenever the arbitrary meets and unions exist,(⋀i∈Ixi)~ = ⋀i∈Ixi~, (⋁i∈Ixi)− = ⋀i∈Ixi−, whenever the arbitrary meets and unions exist,⇝z = (x⇝z)∨(y⇝z), (x∧y) → z = (x → z)∨(y → z),(x⇝(y∧z) = (x⇝y)∧(x⇝z), x → (y∧z) = (x → y)∧(x → z),x∨y)⇝z = (x⇝z)∧(y⇝z), (x∨y) → z = (x → z)∧(y → z),→(x⇝z), x → y ≤ (y → z)⇝(x → z),A, ∧, ∨, 0,1) is a bounded distributive lattice. and (L2), 1 = 1~− ≤ 0− and 1 = 1−~ ≤ 0~, and hence 0~ = 0− = 1. 1~ = 1− = 0 follows similarly.(1) Since 0 ≤ 1x− = 1⇝x− = x−~ → 1~ = x → 0, x~ = 1 → x~ = x~−⇝1− = x⇝0.(2) By (1), (pR1), and (pR2), ~ → 0~ = 1 → 1 = 1. By (pR3) and (L3), 0⇝0 ≤ (x−⇝0)⇝(x−⇝0); that is, 1 ≤ x⇝x, and so x⇝x = 1. The second equality follows similarly.(3) By (pR1) and (pR2), 0⇝0 = 0x → y = 1, by (pR1), (pR3), and (pR2), x = 1 → x = x−⇝0≤(y−⇝x−)⇝(y−⇝0) = (x → y)⇝(1 → y) = 1⇝y = y. If x ≤ y, by (pR5) and (3), x → y = x → (x∨y) = (x → x)∨(x → y) = 1∨(x → y) = 1. Similarly, we have x ≤ y if and only if x⇝y = 1.(4) If i∈Ixi ≤ xi for each i ∈ I, by (L2), xi~ ≤ (⋀i∈Ixi)~ for each i ∈ I, and so ⋁i∈Ixi~ ≤ (⋀i∈Ixi)~. Since xi~ ≤ ⋁i∈Ixi~ for each i ∈ I, by (L2) and (L3), (⋁i∈Ixi~)− ≤ xi for each i ∈ I, and so (⋁i∈Ixi~)− ≤ ⋀i∈Ixi, (⋀i∈Ixi)~ ≤ ⋁i∈Ixi~. The second equality follows similarly.(5) Since ⋀(6) Similarly.x ≤ y, by (pR5), z⇝y = z⇝(x∨y) = (z⇝x)∨(z⇝y), and so z⇝x ≤ z⇝y. Similarly, z → x ≤ z → y.(7) If x ≤ y, then y− ≤ x−. By (pR1), y → z = z−⇝y− ≤ z−⇝x− = x → z. Similarly, y⇝z ≤ x⇝z.(8) If x∧y) → z = z−⇝(x∧y)− = z−⇝(x−∨y−) = (z−⇝x−)∨(z−⇝y−) = (x → z)∨(y → z). Similarly, (x∧y)⇝z = (x⇝z)∨(y⇝z).(9) By (pR1), (5), and (pR5),  if and only if u → (x → (y∧z)) = 1 if and only if u → ((y∧z)−⇝x−) = 1 if and only if (y∧z)−⇝(u → x−) = 1 if and only if (u → x−)~ → (y∧z) = 1 if and only if (u → x−)~ ≤ y∧z if and only if (u → x−)~ ≤ y and (u → x−)~ ≤ z if and only if (u → x−)~ → y = 1 and (u → x−)~ → z = 1 if and only if u ≤ x → y and u ≤ x → z if and only if u ≤ (x → y)∧(x → z). The second equality follows similarly.(10) By (4), (pR1), and (pR4), for any x∨y) → z = z−⇝(x∨y)− = z−⇝(x−∧y−) = (z−⇝x−)∧(z−⇝y−) = (x → z)∧(y → z). Similarly, (x∨y)⇝z = (x⇝z)∧(y⇝z).(11) By (10),  Since x → y)→((y → z)⇝(x → z)) = (y → z)⇝((x → y)→(x → z)) = 1. Hence, x → y ≤ (y → z)⇝(x → z). Similarly, we have the second inequality.(13) By (pR4) and (pR3), (x∧y)∨(x∧z) ≤ x∧(y∨z); by (pR5), (10), and (4),x∧(y∨z))≤(x∧y)∨(x∧z).(14) Obviously, x⇝y ≤ x∨z⇝y∨z, x → y ≤ x∨z → y∨z,(16)x⇝y ≤ x∧z⇝y∧z, x → y ≤ x∧z → y∧z,(17)x⇝y)≤(x⇝z)∨(z⇝y), (x → y)≤(x → z)∨(z → y),((18)x⇝y)∨(y⇝x) = (x → y)∨(y → x) = 1,((19)x ≤ (x → y)⇝y, x ≤ (x⇝y) → y,(20)x → y = ((x → y)⇝y) → y, x⇝y = ((x⇝y) → y)⇝y,(21)x → (y → x) = x⇝(y⇝x) = x⇝(y → x) = x → (y⇝x) = 1,(22)x− → (x → y) = x~⇝(x⇝y) = x−⇝(x → y) = x~ → (x⇝y) = 1,(23)y ≤ (x⇝y)∧(x → y),(24)x∨y = ((x⇝y) → y)∧((y⇝x) → x) = ((x → y)⇝y)∧((y → x)⇝x),(25)x∨y) → x = y → x, (x∨y)⇝x = y⇝x,((26)x → (x∧y) = x → y, x⇝(x∧y) = x⇝y,(27)x ≤ y− if and only if y ≤ x~,(28)x → y~ = y⇝x−, x⇝y− = y → x~,(29)x → y−)~ = (y⇝x~)−.(In a pseudo-weak-z⇝(y∨z) = 1, x⇝y ≤ x⇝(y∨z) = (x⇝(y∨z))∧(z⇝(y∨z)) = (x∨z)⇝(y∨z). The second inequality follows similarly.(15) Since x∧z)⇝z = 1, x⇝y ≤ (x∧z)⇝y = ((x∧z)⇝y)∧((x∧z)⇝z) = (x∧z)⇝(y∧z). The second inequality follows similarly.(16) Since (x⇝y ≤ (x∨z)⇝(y∨z) = ((x∨z)⇝y)∨((x∨z)⇝z) = ((x⇝y)∧(z⇝y))∨ ((x⇝z)∧(z⇝z))≤(x⇝z)∨(z⇝y). The second inequality follows similarly.(17) Consider x∨y)⇝(x∨y) = ((x∨y)⇝x)∨((x∨y)⇝y) = ((x⇝x)∧(y⇝x))∨((x⇝y)∧(y⇝y)) = (x⇝y)∨(y⇝x). The second equality follows similarly.(18) Consider 1 = (x → ((x → y)⇝y) = (x → y)⇝(x → y) = 1 and x⇝((x⇝y) → y) = (x⇝y)→(x⇝y) = 1.(19) Consider x → y)⇝y) → y)⇝(x → y) ≥ x → ((x → y)⇝y) = (x → y)⇝(x → y) = 1 and (x → y)⇝(((x → y)⇝y) → y) = ((x → y)⇝y)→((x → y)⇝y) = 1, then ((x → y)⇝y) → y = (x → y). Similarly, we have ((x⇝y) → y)⇝y = x⇝y.(20) Since (((x → (y → x) = 1 if and only if x⇝(y → x) = 1 if and only if y → (x⇝x) = 1. The rest follow similarly.(21) x− → (x → y) ≥ 0 → y = 1. The rest follow similarly.(22) Consider y = 1⇝y ≤ x⇝y and y = 1 → y ≤ x → y.(23) Consider x⇝((x⇝y) → y) = (x⇝y)→(x⇝y) = 1 and y⇝((x⇝y) → y) = (x⇝y)→(y⇝y) = 1, then x, y ≤ (x⇝y) → y, and so x∨y ≤ (x⇝y) → y. Similarly, x∨y ≤ (y⇝x) → x. Hence, x∨y ≤ ((x⇝y) → y)∧((y⇝x) → x). On the other hand, by (9), (pR5), (20), and (18),(24) Since x∨y) → x = (x → x)∧(y → x) = y → x, (x∨y)⇝x = (x⇝x)∧(y⇝x) = y⇝x.(25) Consider (x → (x∧y) = (x → x)∧(x → y) = x → y, x⇝(x∧y) = (x⇝x)∧(x⇝y) = x⇝y.(26) Consider  x ≤ y− if and only if x⇝y− if and only if y → x~ = 1 if and only if y ≤ x~.(27) x → y~ = y~−⇝x− = y⇝x−, x⇝y− = y−~ → x~ = y → x~.(28) Consideru, (x → y−)~ ≤ u if and only if u− ≤ x → y− if and only if u−⇝(x → y−) = 1 if and only if x → (u−⇝y−) = 1 if and only if x → (y → u) = 1 if and only if x⇝(y → u) = 1 if and only if y → (x⇝u) = 1 if and only if y⇝(x⇝u) = 1 if and only if y⇝(u~ → x~) = 1 if and only if u~ → (y⇝x~) = 1 if and only if u~ ≤ y⇝x~ if and only if (y⇝x~)− ≤ u.(29) For each R0 algebra  A, we define a binary operation ⊙ as follows, for any x, y ∈ A:In a pseudo-weak-x⊙y = (x→y−)~ = (y⇝x~)−.(30) R0 algebra, the following properties hold:(31)x → y = (x⊙y~)−, x⇝y = (y−⊙x)~,(32)x⊙y)⊙z = x⊙(y⊙z),((33)x = x⊙1 = x,1⊙(34)x⊙y ≤ z if and only if x ≤ y → z if and only if y ≤ x⇝z,(35)x⊙(x⇝y) ≤ y ≤ x⇝(x⊙y), (x → y)⊙x ≤ y ≤ x → (y⊙x),(36)x⊙(x⇝y) ≤ x ≤ y⇝(y⊙x), (x → y)⊙x ≤ x ≤ y → (x⊙y),(37) if x ≤ y, then x⊙z ≤ y⊙z and z⊙x ≤ z⊙y,(38)x⊙(x⇝y) ≤ x∧y, (x → y)⊙x ≤ x∧y,(39)x⊙0 = 0⊙x = 0,(40)x⊙(⋁i∈Ixi) = ⋁i∈I(x⊙xi), (⋁i∈Ixi)⊙x = ⋁i∈I(xi⊙x), whenever the arbitrary unions exist,(41)x⊙y) → z = x → (y → z), (y⊙x)⇝z = x⇝(y⇝z),((42)y⇝(⋀i∈Ixi) = ⋀i∈I(y⇝xi), y → (⋀i∈Ixi) = ⋀i∈I(y → xi), whenever the arbitrary meets exist,(43)i∈Ixi)⇝y = ⋀i∈I(xi⇝y), (⋁i∈Ixi) → y = ⋀i∈I(xi → y), whenever the arbitrary unions and meets exist,(⋁(44)x⊙x~ = x−⊙x = 0,(45)x⊙y ≤ x∧y ≤ x, y,(46)x∨(y⊙z)≥(x∨y)⊙(x∨z),(47)x → y ≤ (x⊙z)→(y⊙z), x⇝y ≤ (z⊙x)⇝(z⊙y),(48)x⊙(y → z) ≤ y → (x⊙z), (y⇝z)⊙x ≤ y⇝(z⊙x).In a pseudo-weak-(31) Straightforward.x⊙y)⊙z = (x → y−)~⊙z = ((x → y−)~ → z−)~ = (z−−⇝(x → y−))~ = (x → (z−−⇝y−))~ = (x → (y → z−))~ and x⊙(y⊙z) = x⊙(y → z−)~ = (x → (y → z−))~.(32) By (pR1) and (pR4), (x = (1 → x−)~ = x−~ = x, x⊙1 = (x → 1−)~ = (x → 0)~ = x−~ = x.(33) Consider1⊙x⊙y ≤ z⇔(x → y−)~ ≤ z⇔z− ≤ x → y−⇔z−⇝(x → y−) = 1⇔x → (z−⇝y−) = 1⇔x → (y → z) = 1⇔x ≤ y → z. And(34) Considerx⊙y ≤ z⇔(y⇝x~)− ≤ z⇔z~ ≤ y⇝x~⇔z~ → (y⇝x~) = 1⇔y⇝(z~ → x~) = 1⇔y⇝(x⇝z) = 1⇔y ≤ x⇝z.Consider x⇝y ≤ x⇝y, by (34), x⊙(x⇝y) ≤ y. Since x⊙y ≤ x⊙y, by (34), y ≤ x⇝(x⊙y). The second inequality follows similarly.(35) Since x → y ≤ 1 = x → x, we have (x → y)⊙x ≤ x. Since x⊙y ≤ x⊙y, we have x ≤ y → (x⊙y). The second inequality follows similarly.(36) Since x ≤ y⇒y → z− ≤ x → z−⇒(x → z−)~ ≤ (y → z−)~; that is, x⊙z ≤ y⊙z. Similarly, we have that x ≤ y implies that z⊙x ≤ z⊙y.(37) x⊙(x⇝y) ≤ x. By (35), x⊙(x⇝y) ≤ y. So, x⊙(x⇝y) ≤ x∧y. The second inequality follows similarly.(38) By (36), x⊙0 = (x → 0−)~ = (x → 1)~ = 1~ = 0,0⊙x = (0 → x−)~ = 1~ = 0.(39) Consider u, x⊙(⋁i∈Ixi) ≤ u⇔⋁i∈Ixi ≤ x⇝u⇔xi ≤ x⇝u for each i ∈ I⇔x⊙xi ≤ u for each i ∈ I⇔⋁i∈I(x⊙xi) ≤ u for each i ∈ I. The second equality follows similarly.(40) For any x⊙y) → z = (x → y−)~ → z = z−⇝(x → y−) = x → (z−⇝y−) = x → (y → z). The second equality follows similarly.(41) Consider ⇔y⊙z ≤ ⋀i∈Ixi⇔y⊙z ≤ xi for each i ∈ I⇔z ≤ y⇝xi for each i ∈ I⇔z ≤ ⋀i∈I(y⇝xi). The second equality follows similarly.(42) For any (43) Similarly.x⊙x~ = (x → x~−)~ = 1~ = 0, x−⊙x = (x− → x−)~ = 0.(44) Consider x⊙y⇝x = y⇝(x⇝x) = 1 and x⊙y → y = x → (y → y) = 1. Hence, x⊙y ≤ x∧y.(45) By (41), x∨y)⊙(x∨z) = (x⊙x)∨(x⊙z)∨(y⊙x)∨(y⊙z) ≤ x∨x∨x∨(y⊙z) = x∨(y⊙z).(46) By (40), (x⊙z)→(y⊙z) = (x → z−)~ → (y → z−)~ = (y → z−)⇝(x → z−) ≥ x → y. Similarly, we have the other one.(47) By (13), (y⊙(y⇝z) ≤ z. By (37), (y⊙(y⇝z))⊙x ≤ z⊙x; that is, y⊙((y⇝z)⊙x) ≤ z⊙x, and so (y⇝z)⊙x ≤ y⇝(z⊙x). The second inequality follows similarly.(48) By (35), R0 algebra, the following property holds:In a pseudo-x⊙y)−∨((x∧y)→(x⊙y)) = (y⊙x)~∨((x∧y)⇝(y⊙x)) = 1.(49)  (By (pR1), (5), and (pR6),R0 algebra. If A is a chain, then the operations → and ⇝ can be written by the following:Let A be a pseudo-x ≤ y, by (4), x → y = x⇝y = 1. Now, we show that if x≰y, then x → y = x−∨y and x⇝y = x~∨y. By (12), we have x−∨y ≤ x → y and x~∨y ≤ x⇝y. On the other hand, by (pR6), we havex≰y, x → y ≠ 1 and x⇝y ≠ 1. By A being a chain, we have (x → y)⇝(x−∨y) = 1 and (x⇝y)→(x~∨y) = 1, and so x → y ≤ x−∨y and x⇝y ≤ x~∨y.If R0 algebras coincide with pseudo-IMTL algebras and that pseudo-R0 algebras coincide with pseudo-NM algebras. Hence, as the corollaries of commutative cases, weak-R0 algebras coincide with IMTL-algebras and R0-algebras coincide with NM-algebras, which are obtained in [We show that pseudo-weak-ained in .𝒜 =  be a pseudo-weak-R0 algebra. Define Φ(𝒜) =  byxR− = x→R0, xR~ = x⇝R0. Then, Φ(𝒜) is a pseudo-IMTL algebra.(i) Let 𝒜 =  be a pseudo-IMTL algebra. Define Ψ(𝒜) =  by𝒜) is a pseudo-weak-R0 algebra.(ii) Conversely, let (iii) The above defined maps Φ and Ψ are mutually inverse.(i) (pB1) is verified by (pL1).(pB2) is verified by (32) and (33).(pB3) is verified by (34).(pB4) is verified by (18).(pB5) is verified by (pL3).𝒜) is a pseudo-IMTL algebra by Thus, Φ(R” and “I” of the operations.(ii) For convenience, in the following proof, we omit the indices “(pL1): By (pB1).x ≤ y, by (pB3), x ≤ y ≤ z → (y⊙z) and x ≤ y ≤ z⇝(z⊙y), and so(pL2): If x ≤ y, by y → z)⊙x ≤ (y → z)⊙y and x⊙(y⇝z) ≤ y⊙(y⇝z) ≤ z, and soIf x ≤ y, by y → 0 ≤ x → 0 and y⇝0 ≤ x⇝0; that is, y− ≤ x− and y~ ≤ x~, which means that (pL2) holds.Thus, if (pL3): By (pB5).x⊙((x⇝y)⊙(y⇝z)) = (x⊙(x⇝y))⊙(y⇝z) ≤ y⊙(y⇝z) ≤ z and ((y → z)⊙(x → y))⊙x = (y → z)⊙((x → y)⊙x)≤(y → z)⊙y ≤ z, and so(pR1): By (pB2) and (pB3), x → y ≤ (y → 0)⇝(x → 0) = y−⇝x− ≤ x−~ → y−~ = x → y and x⇝y ≤ (y⇝0)→(x⇝0) = y~ → x~ ≤ x~−⇝y~− = x⇝y, which means that x → y = y−⇝x− and x⇝y = y~ → x~.Thus, by x⊙1 = x ≤ x and 1⊙x = x ≤ x, and, by (pB3), x ≤ 1 → x and x ≤ 1⇝x. On the other hand, 1 → x = (1 → x)⊙1 ≤ x and 1⇝x = 1⊙(1⇝x) ≤ x. Hence, 1 → x = x and 1⇝x = x.(pR2): By (pB2), z⊙(z⇝x))⊙(x⇝y) ≤ x⊙(x⇝y) ≤ y; that is, z⊙((z⇝x)⊙(x⇝y)) ≤ y, so (z⇝x)⊙(x⇝y) ≤ z⇝y; hence, x⇝y ≤ (z⇝x)⇝(z⇝y). Similarly, (x → y)⊙((z → x)⊙z)≤(x → y)⊙x ≤ y; that is, ((x → y)⊙(z → x))⊙z ≤ y, so (x → y)⊙(z → x) ≤ z → y; hence, x → y ≤ (z → x)→(z → y).(pR3): By (pB2), (pB3), and y⊙(x → (y⇝z)))⊙x = y⊙((x → (y⇝z))⊙x) ≤ y⊙(y⇝z) ≤ z, and so y⊙(x → (y⇝z)) ≤ x → z; it follows that x → (y⇝z) ≤ y⇝(x → z). Similarly, we have y⇝(x → z) ≤ x → (y⇝z). Hence, y⇝(x → z) = x → (y⇝z).(pR4): By (pB2), (pB3), and x ≤ y, by (pB3), z⊙(z⇝x) ≤ x ≤ y and (z → x)⊙z ≤ x ≤ y, and so(pR5): If y → z ≤ y → z and (y → z)⊙z ≤ z, by (pB3), y ≤ (y → z)⇝z and z ≤ (y → z)⇝z, and so y∨z ≤ (y → z)⇝z. Hence, by x → (y∨z) ≤ x → ((y → z)⇝z) = (y → z)⇝(x → z), so (y → z)⊙(x → (y∨z)) ≤ x → z. Similarly, (x⇝(y∨z))⊙(y⇝z) ≤ x⇝z. Namely,Since By (pB3) and the same proof of (40), we havex → (y∨z) = 1⊙(x → (y∨z)) = ((y → z)∨(z → y))⊙(x → (y∨z)) = ((y → z)⊙(x → (y∨z))∨((z → y))⊙(x → (y∨z))≤(x → z)∨(x → y). On the other hand, by x → (y∨z) ≥ x → y and x → (y∨z) ≥ x → z, and so x → (y∨z)≥(x → y)∨(x → z). Hence, x → (y∨z) = (x → y)∨(x → z). Similarly, we have x⇝(y∨z) = (x⇝y)∨(x⇝z). Namely, (pR5) holds.By (pB2), (pB4), 𝒜) is a pseudo-weak-R0 algebra by Thus, Ψ((iii) We put the index “c” to the operations of the structure obtained by composition of Φ and Ψ.𝒜 =  be a pseudo-weak-R0 algebra. We prove that Ψ(Φ(𝒜)) = 𝒜. Indeed,x∨cy = x∨Iy = x∨Ry, x∧cy = x∧Iy = x∧Ry, xc− = x→I0 = x→R0, xc~ = x⇝I0 = x⇝R0, x→cy = x→Iy = x→Ry, and x⇝cy = x⇝Iy = x⇝Ry.Let 𝒜 =  be a pseudo-IMTL algebra. We prove that Φ(Ψ(𝒜)) = 𝒜. Indeed, x∨cy = x∨Ry = x∨Iy, x∧cy = x∧Ry = x∧Iy, x⊙cy = (x→RyR−)R~ = (x→IyI−)I~ = x⊙Iy, x→cy = x→Ry = x→Iy, andx⇝cy = x⇝Ry = x⇝Iy.Let 𝒜 =  be a pseudo-R0 algebra. Define Φ(𝒜) =  byxR− = x→R0, xR~ = x⇝R0. Then, Φ(𝒜) is a pseudo-NM algebra.(i) Let 𝒜 =  be a pseudo-NM algebra. Define Ψ(𝒜) =  by𝒜) is a pseudo-R0 algebra.(ii) Conversely, let (iii) The above defined maps Φ and Ψ are mutually inverse.𝒜). (pB6) is verified by (49).(i) By 𝒜). Now, we prove that (pR6) also holds in Ψ(𝒜). For convenience, in the following proof, we omit the indices “R" and “I” of the operations.(ii) By u ∈ A, u ≤ x → (y → z)⇔u⊙x ≤ y → z⇔u⊙(x⊙y) = (u⊙x)⊙y ≤ z⇔u ≤ (x⊙y) → z and u ≤ x⇝(y⇝z)⇔x⊙u ≤ y⇝z⇔(y⊙x)⊙u = y⊙(x⊙u) ≤ z⇔u ≤ (y⊙x)⇝z. Hence,By (pB2) and (pB3), for any x⊙y)~ = (x⊙y)⇝0 = y⇝(x⇝0) = y⇝(1 → x~) = y⇝x~. Similarly, (x⊙y)− = (x⊙y) → 0 = x → (y → 0) = x → (1⇝y−) = x → y−. Hence,By By (pL2), (pL3), and the same proof of (5) and (6), we haveThus, by (iii) By R = ⇝R, R−=R~, and →I = ⇝I, then we have the main results in [In Theorems sults in .R0 algebras coincide with IMTL-algebras.Weak-R0-algebras coincide with NM-algebras.R0 algebras and R0-algebras to the noncommutative forms, called pseudo-weak-R0 algebras and pseudo-R0 algebras. The properties of pseudo-weak-R0 algebras and pseudo-R0 algebras were investigated, and the simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras were discussed. Finally, we showed that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras and that pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Based on these results, we will study filter theory of pseudo-weak-R0 algebras and pseudo-R0 algebras and investigate relations between various kinds of filters of pseudo-logic algebras. We may also study fuzzy type of filters of pseudo-weak-R0 algebras and pseudo-R0 algebras.We gave a positive answer to Iorgulescu's open problem. We extended weak-


The present study investigated the effects of ε‐viniferin and resveratrol on epithelial secretory and barrier functions in isolated rat small and large intestinal mucosa. Mucosa–submucosa tissue preparations of various segments of the rat large and small intestines were mounted on Ussing chambers, and short‐circuit current (Isc) and tissue conductance (Gt) were continuously measured. The mucosal addition of ε‐viniferin (>10−5 mol/L) and resveratrol (>10−4 mol/L) to the cecal mucosa, which was the most sensitive region, induced an increase in Isc and a rapid phase decrease (P‐1) followed by rapid (P‐2) and broad (P‐3) peak increases in Gt in concentration‐dependent manners. Mucosal ε‐viniferin (10−4 mol/L), but not resveratrol (10−4 mol/L), increased the permeability of FITC‐conjugated dextran (4 kDa). The mucosal ε‐viniferin–evoked changes in Isc (Cl− secretion), but not in Gt, were attenuated by a selective cyclooxygenase (COX)‐1 inhibitor and a selective EP4 prostaglandin receptor. The mucosal ε‐viniferin–evoked increase in Isc was partially attenuated, and P‐2, but not P‐1 or P‐3, change in Gt was abolished by a transient receptor potential cation channel, subfamily A, member 1 (TRPA1) inhibitor. Moreover, the mucosal ε‐viniferin concentration‐dependently attenuated the mucosal propionate (1 mmol/L)‐evoked increases in Isc and Gt. Immunohistochemical studies revealed COX‐1–immunoreactive epithelial cells in the cecal crypt. The present study showed that mucosal ε‐viniferin modulated transepithelial ion transport and permeability, possibly by activating sensory epithelial cells expressing COX‐1 and TRPA1. Moreover, mucosal ε‐viniferin decreased mucosal sensitivity to other luminal molecules such as short‐chain fatty acids. In conclusion, these results suggest that ε‐viniferin modifies intestinal mucosal transport and barrier functions. Resveratrol has a variety of beneficial effects, including antioxidant  in a colonic epithelium‐like monolayer‐cultured cell line, T84 cells, and in the mouse jejunum  Cl− secretion  were purchased from Japan SLC, Inc. , and were allowed food and water ad libitum before all experiments. The animals were handled and euthanized in accordance with the Guidelines for the Care and Use of Laboratory Animals of the University of Shizuoka, and the study was approved by the University of Shizuoka Animal Usage Ethics Committee. SD rats were anesthetized with isoflurane and decapitated with a guillotine. The terminal ileum, cecum, proximal colon, middle colon, distal colon, and rectum were removed and placed in ice‐cold Krebs–Ringer solution saturated with O2 and CO2  rats  maintained at 37°C and gassed with 95% O2/5% CO2. The transepithelial potential difference  was measured using paired Ag–AgCl electrodes  through Krebs–agar bridges, and was clamped at 0 mV by applying a short‐circuit current (Isc) with another pair of Ag–AgCl electrodes  connected to a voltage‐clamp apparatus . The lumen negative electrogenic current represents a positive Isc per unit area of tissue (μA/cm2). To record tissue conductance (Gt [mS/cm2]), voltage command pulses  were applied at 1 min intervals, and Gt was calculated using Ohm's law as the current needed to alter the clamped voltage. The current output was continuously recorded on a data acquisition and analog‐to‐digital conversion system . Before the experiments, the tissues were stabilized for 1 h and tissue viability was checked by electrical field stimulation  using a pair of aluminum foil ribbon electrodes.The cross‐sectional area of the mucosa–submucosa preparations in the Ussing flux chamber was 0.64 cm−4 mol/L) was added to the mucosal bathing solution 30 min before the addition of EtOH , ε‐viniferin (10−4 mol/L), and resveratrol (10−4 mol/L). Samples (100 μL) were taken from the serosal bathing solution, and fresh Krebs–Ringer (100 μL) was added at −15, 0 15, 30, 45, and 60 min (6 samples). The concentrations of FD4 were determined by measuring the fluorescence intensity of samples and standard solutions using a microplate reader  at emission and excitation wavelengths of 520 nm and 490 nm, respectively. Based on a volume of the serosal bathing solution (Vm) of 10 mL and sample volume (Vs) of 100 μL, the total FD4 mass that crossed the tissue at each time (Mn) was:C1, C2, … Cn are the sample concentrations at sampling times −15 … 60 min. The FD4 flux (Jm→s) per unit area (0.64 cm2) was calculated as:Jnm→s=Mn+1−Mn15min×10.64cm2where J1m→s, J2m→s, …Jn‐1m→s are the FD4 fluxes in each period . Because the decrease in the mucosal concentration of FD4 during experiments is considered to be negligible, the permeability (Pnm→s) of FD4 in each period was calculated by Fick's law as follows:Pnm→s=Jnm→sCm−Cs≅Jnm→sCm=Jnm→s10−4mol/Lwhere Cm and Cs (Cm ≫ Cs) are the concentrations of FD4 in the mucosal and serosal bathing solutions, respectively. This method was built in reference to a chapter in the textbook, Molecular Biopharmaceutics ‐conjugated dextran  in the mucosa–submucosal tissue preparation was measured under short‐circuit conditions in the Ussing chamber. In this experiment, FD4  117 NaCl, 4.7 KCl, 1.2 MgClε‐Viniferin, resveratrol, and sodium propionate were purchased from Wako Pure Chemical Industries, Ltd. . Carbachol (CCh), FD4, bumetanide, 5‐nitro‐2‐(3‐phenylpropylamino) benzoic acid (NPPB), atropine, hexamethonium, SC‐560, and NS‐398 were purchased from Sigma . TTX and HC030031 were purchased from Tocris . Piroxicam was purchased from Biomol Research Laboratories . 4,4′‐Diisothiocyanatostilbene‐2,2′‐disulfonic acid (DIDS) was purchased from ANA SPEC . AH‐6809 was purchased from Cayman Chemical . ONO‐8713, ONO‐AE3‐208, and ONO‐AE3‐240 were kind gifts from ONO Pharmaceutical Co., Ltd. .ε‐Viniferin and resveratrol were dissolved in ethanol (EtOH) just before use. TTX was dissolved in citrate buffer (pH 4.8) and stored at −20°C until use. CCh, FD4, propionate, atropine, and HEX were dissolved in distilled water, and stored at 4°C until use. All of the other chemicals were dissolved in DMSO and stored at −20°C until use.μm‐thick sections on a cryostat . The sections were placed on glass slides, fixed in 100% methanol at −20°C for 10 min, and air‐dried under a cold blower for 30 min. The sections were incubated with 10% normal donkey serum and 1% Triton X‐100 in phosphate‐buffered saline (PBS) at room temperature for 1 h to block nonspecific antibody binding. The sections were incubated with goat anticyclooxygenase (COX)‐1 antibody  in PBS containing 0.3% Triton X‐100 at 4°C overnight. After washing in PBS (3 × 10 min), the sections were incubated with donkey anti‐goat IgG conjugated with Alexa594  at room temperature for 1 h. The sections were washed in PBS (3 × 10 min), and coverslips were mounted on the glass slides with mounting medium . Immunoreactivity for COX‐1 was visualized using a fluorescence microscope , and the images were captured using a cooled charge‐coupled device digital camera and digital imaging software .The isolated cecal tissues were immediately frozen with Tissue‐Tek optimal cutting temperature compound  in a −75°C acetone bath . The tissues were cut into 4‐t tests were used to determine the statistical significance between two groups because some tissue preparations from the same animals were used in both experimental groups. In all analyses, Ps < 0.05 were considered statistically significant.All physiologic data in the present study are expressed as the mean ± SEM. Multiple comparisons were made using Dunnett's test and Tukey's test for each group relative to the control group and among all combinations of groups, respectively. Paired ε‐viniferin  evoked a monophasic increase in Isc  and triphasic changes in Gt . In contrast, mucosal addition of 10−4 mol/L resveratrol evoked very small changes in Isc and Gt  and Gt, likely via ε‐viniferin–evoked P‐1  and P‐3  , ε‐viniferin evoked a small, but long‐lasting decrease in Gt, as shown in Figure μL), did not affect Isc or Gt.In the rat cecal mucosa–submucosal preparations, mucosal, but not serosal, addition of ε‐viniferin (3 × 10−6 to 3 × 10−4 mol/L) and resveratrol (10−5 to 3 × 10−4 mol/L) were added to the mucosal bathing solution, and the mean changes in Isc  of the mucosal ε‐viniferin–evoked ∆Isc was 4.81 × 10−5 mol/L, the maximum effect (Emax) was 87.19 μA/cm2, and the Hill coefficient value (nH) was 2.37 (coefficient of determination [R2] = 1.000). At 10−5 mol/L or lower concentrations, mucosal ε‐viniferin only evoked a decrease in Gt, which was long‐lasting. The EC50 of the mucosal ε‐viniferin–evoked P‐1 ∆Gt was 2.53 × 10−6 mol/L, Emax was −0.69 mS/cm2, and nH was 1.22 (R2 = 0.887). Although it is unclear whether the P‐2 ∆Gt evoked by 3 × 10−4 mol/L ε‐viniferin was at the maximum effective concentration, the EC50 of ∆Gt was calculated to be 9.44 × 10−5 mol/L, when we assumed that the Emax was the mean value for ∆Gt at 3 × 10−4 mol/L ε‐viniferin (6.92 mS/cm2) and that the nH was same as the that for P‐3 described below (R2 = 1.000). Therefore, the EC50 for mucosal ε‐viniferin–evoked P‐3 ∆Gt was calculated to be 2.35 × 10−5 mol/L, the Emax of mucosal ε‐viniferin–evoked P‐3 ∆Gt was 4.49 mS/cm2, and nH was 4.53 (R2 = 0.999).Various concentrations of Isc did not seem to reach the maximum concentrations, but the EC50 of P‐2 ∆Isc was 2.52 × 10−4 mol/L (R2 = 0.975) when we assumed that Emax and nH were the same as those for ε‐viniferin–evoked ∆Isc. Likewise, the resveratrol‐evoked increase in Gt did seem to reach the maximum concentration, but the EC50 was 1.82 × 10−4 mol/L (R2 = 0.912) when we assumed the Emax and nH were the same as those for ε‐viniferin–evoked P‐3 ∆Gt.The resveratrol‐evoked P‐1 and P‐2 ∆ε‐viniferin may evoke more potent responses at concentrations of >3 × 10−4 mol/L, ε‐viniferin at these concentrations evoked unlimited continuous increases in Gt. This indicates that ε‐viniferin at concentrations >3 × 10−4 mol/L may have cytotoxic effects because the epithelial barrier function was thought to have failed. Thus, ε‐viniferin was used at 10−4 mol/L in the following experiments.Although ε‐viniferin (10−4 mol/L) on Isc and Gt were measured and compared among the rat terminal ileum, cecum, proximal colon, middle colon, and distal colon. Mucosal addition of ε‐viniferin (10−4 mol/L) had weak effects on Isc in the intestinal tissues, except for the cecum –evoked increase was most potent in the cecum and the magnitude of the increase gradually decreased from the proximal to distal segments of the large intestine  were measured in terms of Gt. Basal Isc was 18.07 ± 1.62 μA/cm2, basal Gt was 11.15 ± 0.41 mS/cm2, and basal FD4 Pm→s was 1.26 ± 0.20 × 10−7 cm/s (n = 20).To investigate the effects of mucosal ε‐viniferin evoked an increase in Isc, and the mean Isc reached its maximum at 7 min . The Gt first decreased at 3 min , the P‐2 peak occurred at 8 min , and the P‐3 peak occurred at 26 min . FD4 Pm→s increased, and reached a maximum  at 30–45 min after mucosal addition of ε‐viniferin. This value was significantly greater (P < 0.05 by Dunnett's test) than the FD4 Pm→s after the addition of the vehicle control  at 30–45 min.Mucosal addition of −4 mol/L), Isc decreased initially , and then slowly increased . Gt slowly increased . However, FD4 permeability did not significantly change.After the mucosal addition of resveratrol . All preparations were bathed with normal solution at first. After the bathing solutions of the serosal, mucosal, or both sides were replaced with Cl−‐free solution, and the compensations of fluid resistance were reset after changing the solutions. Basal Isc significantly increased in the serosal Cl−‐free solution, but decreased in the mucosal Cl−‐free solution –evoked increases in Isc were almost completely abolished for Cl−‐free solution on the serosal and both sides, but not the mucosal side . These results indicate that the mucosal ε‐viniferin–evoked increase in Isc is due to electrogenic Cl− secretion. Moreover, with serosal Cl−‐free solution, mucosal ε‐viniferin evoked a decrease in Isc . Serosal and both side Cl−‐free solutions led to unlimited increases in Gt, which made it difficult or impossible to measure the mucosal ε‐viniferin–evoked changes in Gt.The ε‐viniferin evoked Cl− secretion, a Na+‐K+‐2Cl− (NKCC) transporter inhibitor, bumetanide (10−4 mol/L), was added to the serosal bathing solution. Alternatively, a Ca2+‐dependent channel blocker, DIDS (10−4 mol/L), or a cAMP‐dependent Cl− channel  blocker, NPPB (10−5 to 10−4 mol/L), were added to the mucosal bathing solution before the mucosal addition of ε‐viniferin. Bumetanide (10−4 mol/L) and NPPB (≥5 × 10−5 mol/L) significantly attenuated the ε‐viniferin–evoked increase in Isc, but not DIDS . Although bumetanide attenuated the ε‐viniferin–evoked increase in Isc, it did not attenuate the Gt responses . In contrast, NPPB (5 × 10−5 mol/L) enhanced the ε‐viniferin–evoked P‐1 Gt reduction and nearly abolished the P‐3 Gt change . The ε‐viniferin–evoked changes in Gt in the presence of 10−4 mol/L NPPB were not measured because Gt gradually increased after the addition of 10−4 mol/L NPPB.To confirm that mucosal –4 mol/L resveratrol evoked only a weak decrease in P‐1 ΔIsc (−5.66 ± 1.61 μA/cm2) and an increase in ΔGt (0.63 ± 0.30 mS/cm2) , resveratrol at 3 × 10−4 mol/L elicited Isc and Gt responses, which were similar to those induced by 10−4 mol/L ε‐viniferin  was added to the mucosal bathing solution 30 min after mucosal addition of resveratrol (at 0 to 3 × 10−4 mol/L), and the changes in Isc and Gt were measured. In this experiment, resveratrol significantly attenuated the ε‐viniferin–evoked changes in Isc and Gt in concentration‐dependent manners . Mucosal resveratrol at 10−4 mol/L significantly attenuated the mucosal ε‐viniferin (10−4 mol/L)–evoked increase in Isc to 26.07 ± 4.12% of the control value  and P‐3 ∆Gt to 29.31 ± 8.20% of the control value . When the mucosal bathing solution contained 3 × 10−4 mol/L resveratrol, the administration of ε‐viniferin (10−4 mol/L) decreased Isc  and abolished the P‐2 response. However, there were no statistically significant differences because the P‐2 response in the control conditions was weak and varied considerably between tests. Moreover, in the presence of 3 × 10−4 mol/L resveratrol, ε‐viniferin (10−4 mol/L) caused a continuous increase in Gt, as shown in Figure 6C, so P‐3 ∆Gt could not be determined.Although mucosal addition of 10ε‐viniferin–evoked changes in Isc and Gt were mediated via TRPA1 channels, like those evoked by AITC. In this setting, a TRPA1 inhibitor, HC030031 (10−4 mol/L), significantly reduced the ε‐viniferin–evoked increase in Isc from 60.95 ± 10.70 μA/cm2 to 38.04 ± 8.02 μA/cm2 , and the peak time after the addition of ε‐viniferin was significantly delayed from 6.71 ± 0.93 min to 14.68 ± 2.05 min . For Gt, the ε‐viniferin–evoked P‐1 and P‐3 responses were not affected, but the P‐2 response was almost completely abolished by HC030031, as shown in Figure 7A and B .In our previous study, we reported that mucosal administration of a pungent principle, allyl isothiocyanate (AITC), evoked anion secretion by activating TRPA1 channels , HEX , or atropine (a muscarinic ACh receptor antagonist) to the serosal bathing solution 30 min before the addition of ε‐viniferin (10−4 mol/L). However, TTX, HEX, and atropine did not affect the ε‐viniferin–evoked changes in Isc and Gt .The neural reflex pathways in the submucosal plexus in the gut wall play important roles in luminal stimuli‐evoked secretory responses in the intestinal mucosa Cooke . Thus, wn‐propyl‐2‐thiouracil) , a selective COX‐1 inhibitor , a selective COX‐2 inhibitor , or a combination of SC‐560 and NS‐398 were added to the serosal bathing solution 30 min before the mucosal addition of ε‐viniferin (10−4 mol/L). Piroxicam, SC‐560, and the combination of SC‐560 and NS‐398, but not NS‐398 alone, significantly attenuated the ε‐viniferin–evoked increase in Isc from the control value of 59.40 ± 10.53 μA/cm2 (n = 6) to 13.33 ± 2.79 μA/cm2 with piroxicam , to 23.10 ± 6.36 μA/cm2 with SC‐560 , and 22.92 ± 7.70 μA/cm2 with the combination of SC‐560 and NS‐398 . These results indicate that the ε‐viniferin–evoked increase in Isc is mediated by COX‐1 production of prostaglandin (P)G. In contrast, piroxicam, SC‐560, and NS‐398 did not significantly affect the ε‐viniferin–evoked changes in Gt .In our previous studies, some mucosal stimulants including a bitter tastant , an EP1 and EP2 receptor antagonist , a selective EP3 receptor antagonist , or a selective EP4 receptor antagonist  were added to the serosal bathing solution 30 min before the mucosal addition of ε‐viniferin (10−4 mol/L). In this experiment, neither ONO‐8713 nor AH‐6809 affected the ε‐viniferin–evoked increase in Isc, but ONO‐AE3‐208 at concentrations ≥10−6 mol/L significantly attenuated the ε‐viniferin–evoked increase in Isc . These results indicate that the ε‐viniferin–evoked increase in Isc is mediated by EP4 receptors.To determine which PG receptor subtype mediates the ε‐viniferin would attenuate the mucosal propionate‐evoked secretory responses. To confirm this hypothesis, mucosal propionate (10−3 mol/L)‐evoked Isc responses were measured 30 min after the addition of ε‐viniferin at a variety of concentrations, as described earlier.Short‐chain fatty acids, including acetate (two carbons), propionate (three carbons), and butyrate (four carbons), are the predominant anions in the large intestine and exist at concentrations of ≥100 mmol/L. They are produced by bacterial fermentation of indigestible dietary fibers and oligosaccharides. Mucosal SCFAs (with a potency order of propionate ≥ butyrate ≫ acetate) are known to stimulate the large intestinal mucosa to secrete transepithelial anions Yajima . In our ε‐viniferin, propionate (10−3 mol/L) transiently increased Isc  and Gt  (n = 6). However, the mucosal propionate (10−3 mol/L)‐evoked increases in Isc and Gt were attenuated by mucosal ε‐viniferin in a concentration‐dependent manner at ε‐viniferin concentrations ranging from 3 × 10−6 to 3 × 10−4 mol/L . Pretreatment with 3 × 10−4 mol/L ε‐viniferin nearly abolished the propionate response. The concentration–response curves of ε‐viniferin for inhibiting the propionate‐evoked Isc and Gt responses were drawn by fitting the data to the inhibitory Hill equation, as described in Figure 11. In this analysis, the half‐maximal inhibitory concentration (IC50) of ε‐viniferin for ΔIsc was 4.12 × 10−5 mol/L and the nH was 2.11 (R2 = 0.998). For Gt, the IC50 was 2.68 × 10−5 mol/L and nH was 1.68 (R2 = 0.985), where 119.4%, which was the mean value of the propionate‐evoked Gt in 3 × 10−6 mol/L ε‐viniferin, was used as the value for the maximal effect.In the absence of ε‐viniferin might stimulate some epithelial sensory cells expressing COX‐1, because luminal ε‐viniferin evoked Isc and Gt responses, but serosal ε‐viniferin did not. An earlier study revealed that COX‐1–expressing epithelial cells are scattered throughout the crypts in the rat large intestine  induce transepithelial electrogenic Cl− secretion by activating EP4 receptors via PGs produced by COX‐1; (2) elicit a rapid decrease and a sustained increase in transepithelial ion permeability; and (3) increase transepithelial permeability of a nonionic macromolecule, FD4. In addition, mucosal ε‐viniferin inhibited mucosal propionate‐evoked Cl− secretion. These results indicate that ε‐viniferin stimulates the intestinal epithelium from the luminal side and effects mucosal barrier functions.The results of the present study show that ε‐viniferin at concentrations ≥10−5 mol/L elicited a concentration‐dependent monophasic positive change in Isc and triphasic changes in Gt, which comprised an abrupt decrease (P‐1) followed by fast (P‐2) and sustained (P‐3) increases in the rat cecal mucosa, whereas serosal administration of 10−4 mol/L ε‐viniferin did not  and a sustained positive ∆Isc (P‐2), and a single sustained increase in Gt. In contrast, serosal administration of 3 × 10−4 mol/L resveratrol did not affect Isc and a small decrease (P‐1) and increase (P‐2) in Gt , ε‐viniferin evoked the long‐lasting decreases in P‐1 Gt, as shown in Figure −5 mol/L), the ε‐viniferin–evoked changes in Gt consisted of an abrupt and long‐lasting decrease in ion permeability and subsequent increases in ion permeability. At lower concentration, ε‐viniferin may enhance epithelial barrier function, and at higher concentrations ε‐viniferin may further enhance ionic and nonionic permeability (as mentioned below) to induce secretory and inflammatory functions as part of a host defense mechanism.At lower concentrations ≤100−5 mol/Lε‐viniferin in terms of the Isc and Gt responses occurred in the cecal mucosa , but not resveratrol (10−4 mol/L), elicited a transient (15–45 min after addition) increase in FD4 Pm→s  also increased FD4 Pm→s in the rat cecum. These findings suggest that luminal ε‐viniferin directly and/or indirectly affects epithelial tight junctions, and increases paracellular permeability of nonionic macromolecules. However, the mechanism underlying the ε‐viniferin–evoked increase in FD4 Pm→s needs to be examined in future studies.Mucosal m→s Fig. . In our ε‐viniferin–induced increase in Isc was due to transepithelial Cl− secretion, because these responses were attenuated by Cl−‐free solution on the serosal and both sides, but not on the mucosal side  blocker  and apical DIDS (10−4 mol/L), the apical administration of NPPB at concentrations ≥5 × 10−5 mol/L significantly enhanced P‐1 and significantly attenuated P‐3 ∆Gt  antagonist, significantly attenuated ε‐viniferin–evoked Cl− secretion ‐evoked increase in Isc and Gt in a concentration‐dependent manner  and/or FFA3 (GPR41) (Karaki et al. ε‐viniferin may allosterically bind to these receptors, with a possible stoichiometry of 2:1 because the nH was nearly 2. Nevertheless, further studies are necessary to confirm this hypothesis. Moreover, at the lower concentration of 3 × 10−6 mol/L, ε‐viniferin very weakly enhanced the propionate‐evoked increase in Gt (Fig. ε‐viniferin on transepithelial ion permeability need to be investigated in future studies.Bacterial fermentation in the lumen of the large intestine produces numerous metabolites. The predominant molecules are SCFAs, particularly acetate, propionate, and butyrate. Propionate and butyrate, but not acetate, were reported to induce anion secretion in the rat Yajima  and guinner Fig. . This inGt Fig. C. Therefε‐viniferin may activate COX‐expressing cecal crypt cells, increase the tissue PG level as part of a tissue alarm‐response system, and enhance the tissue's host defense functions.In the gastrointestinal lumen, especially in the large intestine, the luminal microbiota can synthesize a variety of compounds and some of these compounds may have cytotoxic effects. Thus, the intestinal mucosa has protective roles, including enhancing the integrity of the epithelial barrier and fluid secretion when the mucosa senses potentially cytotoxic chemicals. We hypothesized that the tissue concentrations of PGs may constitute an alarm‐response system, in which an increase in the PG concentration might enhance the protective functions of the mucosa in host defense (Karaki and Kuwahara ε‐viniferin to the mucosal side of the rat intestine modulates transepithelial ion transport, ion permeability, and the permeability of nonionic macromolecule as well as the effects of the other luminal molecules, such as SCFAs. These results also imply that ε‐viniferin has beneficial effects on intestinal functions by enhancing the mucosal host defense mechanism.The present study shows that administration of A. Kuwahara have received research fund from FANCL Corporation. I. Ishikawa is an employee of FANCL Corporation.