Morphological and morphometric study of the subscapular arterial tree with its clinical implications

Background: Knowledge of anatomy of the subscapular arterial tree is very important as it is commonly used for microvascular grafting to substitute injured portions of the arteries of hands and forearm. So, the study aims to examine the subscapular arterial tree and its variants. Aim and Objectives: To observe the subscapular arterial tree and its variants in the origin and branching pattern as well as to locate the distance of origin of subscapular artery and circumflex scapular artery from various anatomical landmarks. Material and Methods: Twenty-six upper limbs were used in the study. The origin and branches of subscapular artery were noted down. Distance between the various anatomical landmarks and the origin of subscapular and circumflex scapular artery were noted down. Variants in the origin and branching pattern were noted. Results: In five limbs, the 2nd part of axillary artery gave rise to subscapular artery. On the right and left side, the mean distance between the origins of subscapular artery from pectoralis minor was 2.92 and 3.17 cm, respectively. In five limbs, subscapular artery was originating along with posterior circumflex humeral artery and in three limbs with lateral thoracic artery. In two limbs, the 3rd part of axillary artery was giving rise to circumflex scapular artery directly. The mean distance of the origin of circumflex scapular artery from pectoralis minor was 3.53 and 3.83 cm away on right and left side, respectively. On both right and left side, the mean distance of the origin of circumflex scapular artery was 2.18 cm away from the origin of subscapular artery. Conclusion: The measurements on the subscapular artery and its variations will help the surgeons in identifying subscapular artery easily for microvascular arterial grafting

Iterates of polynomials over \F_q(t) and their Galois groups

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ n}$ of $f$ is irreducible and the Galois group of the splitting field of $f^{\circ n}$ is isomorphic to $[S_d]^{n}$, the $n$ folded iterated wreath product of the symmetric group $S_{d}$. We prove an analogue this conjecture over \F_q(t), the field of rational functions in $t$ over a finite field \F_q of characteristic $p>0$. We present some examples and see that most polynomials in \F_q[t][x] satisfy these conditions.Comment: 10 page

Odoni's conjecture and Simultaneous prime specialization over large finite fields

Let \F_q be a finite field of odd cardinality $q$. Given a monic, irreducible, separable polynomial F(t,x) \in \F_q[t][x] of degree $n\geq 2$ in $x$ and a generic monic polynomial $\Phi(\mathbf{a},t)$ of positive degree $d$ in $t$, with coefficient vector $(\mathbf a)$ consisting of linearly independent variables over \F_q, we find that, for $n,d$ of fixed degree and $q\rightarrow \infty$, the Galois group of $F(t,\Phi(\mathbf{a},t))$ over \F_q(\mathbf a) is $S_{nd}$. Under this setting, we show that, the Galois group of the $n$-th iterate of $F(t,\Phi(\mathbf{a},t))$ over \F_q(\mathbf a) is $[S_{nd}]^n$, the $n$-fold iterated wreath product of the symmetric group $S_{nd}$. This is the function field analogue of Odoni's conjecture over \F_q(\mathbf a), which was originally stated over a Hilbertian field \F of characteristic $0$ and, for polynomials of degree $d\geq 2$. We further see that, F(t,\Phi(\mathbf{a},t))\in \F_q(\mathbf a)[t] is a stable polynomial. This indicates, function field analogue of the Odoni's conjecture implies the function field analogue of Schinzel Hypothesis H. Thus, a natural connection between the Odoni's conjecture in \F_q(\mathbf a)[t] and the number of prime polynomial tuples of the given form in \F_q[t] is established via an explicit form of the Chebotarev Density theorem over function fields.Comment: 21 page