Standard mark table and standard USCI-CF table of the point group D6h. Unification of candidates derived from different D6h-skeletons

Combined-permutation representations (CPRs) for characterizing -skeletons (a benzene skeleton, a Haworth-projected skeleton, a superphane skeleton, and a coronene skeleton) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to treat both achiral and chiral substituents under the GAP system. Thereby, the CPR of degree 8 (= 6 + 2) for the benzene skeleton, the CPR of degree 14 (= 12 + 2) for the Haworth-projected skeleton, the CPR of degree 14 (= 12 + 2) for the superphane skeleton, the CPR of degree 14 (= 12 + 2) for the coronene skeleton are generated to give primary mark tables (tables of marks) based on these CPRs. These primary mark tables generated by the GAP system are different in the sequence of subgroups from each other, although they stem from the same point group . They are unified into a single standard mark table by means of a newly-devised GAP function MarkTableforUSCI. Moreover, another newly-devised GAP function constructUSCITable is employed to construct a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table concordantly. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

Alternative Splicing of Neurofibromatosis Type 1 Exon 23a Modulates Ras/ERK Signaling and Learning Behaviors in Mice

Neurofibromin is one of the few Ras-GTP activating proteins (Ras-GAPs) expressed in the brain. Disruption of its expression leads to the detrimental disease neurofibromatosis type 1 (NF1). Many studies have revealed the crucial role of NF1 in developing and adult tissues. However, these studies have focused on the expression of the entire NF1 gene and largely ignored the role of an alternative splicing event that controls the Ras-GAP function of neurofibromin. The focus of this chapter is NF1 exon 23a. This exon is located in the GAP-related domain (GRD) of neurofibromin. Its expression level, indicated by the percentage of its inclusion in the NF1 mRNA transcripts, has a profound effect on the Ras-GAP function of neurofibromin. In this chapter, we review the expression pattern of exon 23a and the molecular mechanisms that regulate its expression. We then discuss the role of its expression in Ras/ERK signaling and learning behaviors in mice. Lastly, we propose a few directions for future studies.

S-derivative of perturbed mapping and solution mapping for parametric vector equilibrium problems

In this paper, we deal with the sensitivity analysis in vector equilibrium problems by using the S-derivative of a set-valued mapping. We first investigate the S-derivative on a kind of set-valued gap function for the vector equilibrium problems. Based on these results, S-derivative estimations on a perturbed mapping for the parametric vector equilibrium problem are given. Moreover, we provide some examples to illustrate the obtained results. Finally, we derive the S-derivative estimations of a solutions mapping of the parametric vector equilibrium problems via S-derivative estimations of a kind of the parametric variational system.

GAP FUNCTION WITH ERROR BOUND FOR RANDOM GENERALIZED VARIATIONAL-LIKE INEQUALITY PROBLEMS

In this paper, we used random generalized variational-like inequality problem and establish a new gap function for this problem in a fuzzy environment. In addition, we used this gap function, to evaluate an error bound for random generalized variational-like inequality problem in a fuzzy environment.

A Hybrid Newton Method for Stochastic Variational Inequality Problems and Application to Traffic Equilibrium

In this paper, we consider a class of stochastic variational inequality problems (SVIPs). Different from the classical variational inequality problems, the SVIP contains a mathematical expectation, which may not be evaluated in an explicit form in general. We combine a hybrid Newton method for deterministic cases with an unconstrained optimization reformulation based on the well-known D-gap function and sample average approximation (SAA) techniques to present an SAA-based hybrid Newton method for solving the SVIP. We show that the level sets of the approximation D-gap function are bounded. Furthermore, we prove that the sequence generated by the hybrid Newton method converges to a solution of the SVIP under appropriate conditions, and some numerical experiments are presented to prove the effectiveness and competitiveness of the hybrid Newton method. Finally, we apply this method to solve two specific traffic equilibrium problems.

Difference gap functions and global error bounds for random mixed equilibrium problems

The aim of this paper is to study the difference gap (in short, D-gap) function and error bounds for a class of the random mixed equilibrium problems in real Hilbert spaces. Firstly, we consider regularized gap functions of the Fukushima type and Moreau-Yosida type. Then difference gap functions are established by using these terms of regularized gap functions. Finally, the global error bounds for random mixed equilibrium problems are also developed. The results obtained in this paper are new and extend some corresponding known results in literatures. Some examples are given for the illustration of our results.