A general quintic hypersurface in CP 4 is a compact Calabi-Yau threefold. Its rational Gromov-Witten invariants have been predicted by mirror symmetry discovered in string theory [1]. The prediction proved by Givental [7] is to be explained as follows. Let n d be the virtual number of degree d rational curves in the quintic threefold and let F(q) =. The Givental J-function J for the quintic hypersurface satisfies the so-called quantum differential equation Equivariant Mirrors 861 every b β depends on e ±t 0 . So, [∆, α 0 =k a α ∂ α ] has no order m + 1 part. Now we apply the induction hypothesis to α 0 =k a α ∂ α whose coefficients do not depend on t 0 and conclude that a α ∈ C[h] if α 0 = k. The conclusion is contradictory to the assumption that for all α, a α is not in C[h]. Theorem 2.2. The operators D i , i = 1, . . . , n + 1, commute. Proof. The commutativity of H and D i is proven in [8]. Since [H, [D i , D j ]] = 0, by Proposition 2.1, the highest order part of [D i , D j ] has coefficients in C[h]. Now it is enough to prove the following claim. For any multi-indices α and β and any a and b in the polynomial ring C[h, q 1 , . . . , q n ], if we let [a∂ α , b∂ β ] = c γ ∂ γ , then any c γ cannot be in C[h]

Quantitative structure-activity relationship (QSAR) regression models are mathematical ones which relate the structural properties of chemicals to the potencies of the biological activities of the chemicals. In QSAR models, the physical and chemical information of the molecules is encoded into quantitative numbers called descriptors. Recently, experimental test results (profiles) have been used as descriptors of chemicals. Profile QSAR 2.0 (pQSAR) model suggested by Martin et. al, is a multitask, two step machine learning prediction method with a combination of random forest regressions (RFRs) and partial least squares regression (PLSR). In pQSAR model, one fills the profile table's missing values with RFRs and then builds PLSR using the profile predictions. Note that in the second step of the pQSAR method, PLSR's predictor variables are profiles; so activity values, and the response variables are also activity values. Thus we can use the PLSRs to update the profile table and then repeat the second step. In this work, we propose an extended model of pQSAR generated by RFRs and PLSRs. Experiment of updating the given full initially predicted profile table by two kinds of prediction models, RFRs and PLSRs, has been conducted iteratively for the PKIS and ChEMBL data sets. Even though prediction performance of individual combination of RFRs and PLSRs varies, the average of the all possible predicted profile tables for given iteration shows better performance. This ensemble model has better prediction performance in sense of Pearson's R 2 compared to that of the pQSAR model.

Among the distance based algorithms in phylogenetic tree reconstruction, the neighbor-joining algorithm has been a widely used and effective method. We propose a new algorithm which counts the number of consistent quartets for cherry picking with tie breaking. We show that the success rate of the new algorithm is almost equal to that of neighbor-joining. This gives an explanation of the qualitative nature of neighborjoining and that of dissimilarity maps from DNA sequence data. Moreover, the new algorithm always reconstructs correct trees from quartet consistent dissimilarity maps.2000 Mathematics Subject Classification. 92D15, 68R10, 05C05, 68Q25.

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