Our goal is to settle a fading problem, the Jacobian Conjecture $(JC_n)$~: If $f_1, cdots, f_n$ are elements in a polynomial ring $k[X_1, cdots, X_n]$ over a field $k$ of characteristic zero such that $ det(partial f_i/ partial X_j) $ is a nonzero
constant, then $k[f_1, cdots, f_n] = k[X_1, cdots, X_n]$. Practically, what we deal with is the generalized one, oindent The Generalized Jacobian Conjecture$(GJC)$ :{it Let $S hookrightarrow T$ be an unramified homomorphism of Noetherian domains. Assume that $S$ is a simply connected UFD ({sl i.e.,} ${rm Spec}(S)$ is simply connected and $S$ is a unique factorization domain) and that $T^times cap S = S^times$. Then $T = S$.} In addition, for consistency of the discussion, we raise some serious (or idiot) questions and some comments about the examples appeared in the papers published by the certain excellent mathematicians (though we are not willing to deal with them). However, the existence of such examples would be against our Main Result above, so that we have to dispute in Appendix B their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples which is shown explicitly.
We have proved the following Problem: Let $R$ be a $mathbb{C}$-affine domain, let $T$ be an element in $R setminus mathbb{C}$ and let $i : mathbb{C}[T] hookrightarrow R$ be the inclusion. Assume that $R/TR cong_{mathbb{C}} mathbb{C}^{[n-1]}$ and that
$R_T cong_{mathbb{C}[T]} mathbb{C}[T]_T^{[n-1]}$. Then $R cong_{mathbb{C}} mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Essen : Conjecture E : Let $A:=mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u), g(u)$ and $h(u)$ be the polynomials defined above. Let $D:= f(u)partial_x + g(u)partial_y + h(u)partial_z + tpartial_u$ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D otcong_{mathbb{C}} mathbb{C}^{[4]}$.} Consequently it gurantees that the conjectures : the Cancellation Problem for affine spaces, the Linearization Problem and the Embedding Problem and the affine $mathbb{A}^n$-Fibration Problem are still open.
We have proved the following Problem:{it Let $R$ be a $mathbb{C}$-affine domain, let $T$ be an element in $R setminus mathbb{C}$ and let $i : mathbb{C}[T] hookrightarrow R$ be the inclusion. Assume that $R/TR cong_{mathbb{C}} mathbb{C}^{[n-1]}$ and t
hat $R_T cong_{mathbb{C}[T]} mathbb{C}[T]_T^{[n-1]}$. Then $R cong_{mathbb{C}} mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : {it Let $A:=mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u):=u^3-3u, g(u):=u^4-4u^2$ and $h(u):=u^5-10u$ be the polynomials in $mathbb{C}[u]$. Let $D:= f(u)partial_x + g(u)partial_y + h(u)partial_z + tpartial_u$ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D otcong_{mathbb{C}} mathbb{C}^{[4]}$.} Consequently our result in this short paper guarantees that the conjectures : the Cancellation Problem for affine spaces, the Linearization Problem, the Embedding Problem and the affine $mathbb{A}^n$-Fibration Problem are still open.
The following is shown : Let $S={a_1,a_2,..,a_{2n}}$ be a subset of a totally ordered commutative semi-group $(G,*,leq)$ with $a_1leq a_2leq...leq a_{2n}$. Provided that a system of $n$ $a_{i_k} * a_{j_k} (a_{i_k}, a_{j_k} in G ; 1 leq k leq n)$, whe
re all $2n$ elements in $S$ must be used, are less than an element $N (in G)$, then $a_1*a_{2n}, a_2*a_{2n-1},..., a_n*a_{n+1}$ are all less than $N$. This may be called the Upper Bounding Case. Moreover in the same way, we shall treat also the Lower Bounding Case.
The B-physics program at the ATLAS experiment, which covers the mid-rapidity region, complements that at the dedicated LHCb experiment, which covers the forward rapidity region. At the early stage of the LHC operation, the program concentrated on und
erstanding of detector performance and measurements of quarkonia and D mesons. This article presents recent results of the B-physics program at ATLAS.
The J/psi is considered to be among the most important probes for the deconfined quark gluon plasma (QGP) created by relativistic heavy ion collisions. While the J/psi is thought to dissociate in the QGP by Debye color screening, there are competing
effects from cold nuclear matter (CNM), feed-downs from excited charmonia (chi_c and psi) and bottom quarks, and regeneration from uncorrelated charm quarks. Measurements that can provide information to disentangle these effects are presented in this paper.
