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Word Matrix Rewriting Systems

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Bisht R. K. , Nishida T. Y. , Yamamoto K.

Fundamenta Informaticae

2019

tom Vol. 166, nr 3

199--226

We introduce word matrices and word matrix rewriting systems. A word matrix over an alphabet ∑ of k letters is a k × n rectangular matrix with entries of nonnegative integers. A word matrix represents a set of words. The i-th row corresponds to the i-th letter in ∑ (thus the letters in ∑ are ordered as a1, . . . , ak). Each column vector represents a set of words which have the vector as the Parikh vector. The word matrix represents the concatenation of all sets represented by column vectors. A word matrix rewriting system consists of a set of rewriting rules which rewrite word matrices and an initial word matrix (the axiom). A word matrix rewriting system generates a set of word matrices by iterated applications of rules to the axiom and generates a language which is the union of all sets represented by the matrices. A word matrix rewriting system is a kind of (restricted) parallel rewriting system with scattered context dependency and hence can generate non-context-free languages, e.g., {anbncn | 1 ≤ n}. We define variants of word matrix rewriting systems and show an infinite hierarchy among them. Some language theoretical problems, including relations among Chomsky hierarchy, closure properties, and decision properties, are considered.

Quantitative analysis of Gamma Knife stereotactic radiosurgery for uveal melanoma

100%

Bisht R. K. , Natanasabapathi G. , Kale S. S.

Polish Journal of Medical Physics and Engineering

2020

tom Vol. 26, Iss. 2

111--117

The purpose of the study was to analyze single fraction Gamma Knife stereotactic radiosurgery (SRS) for uveal melanoma (UM). In the treatment of UMs, the dose distribution exhibited by an irregular eye surface has more calculation uncertainty. A tissue-equivalent bolus was placed on the left eye surface of a human head-shaped phantom. It was assumed that the treated eye is fixed using retrobulbar anesthesia and suture on extraocular muscles for phantom study. Leksell stereotactic frame was fixed around phantom’s head and the stereotactic computed tomography (CT) was performed. Two sets of scans were acquired (a) without bolus and (b) with a bolus of 1.0 cm thickness. These scans were transferred into a treatment planning system (TPS). The skull contouring was performed using stereotactic CT images. The target, visual pathways, and eye lens were delineated in stereotactic CT space created on TPS. A clinical relevant plan was designed on the CT study set “a” to deliver a radiation dose of 30Gy at tumor margin. The plan superimposed over CT study set “b” and compiled for convincing treatment strategy. The tumor coverage was 95% at 50% prescription isodose line. The conformity index, selectivity and the gradient index were 1.27, 0.80 and 3.28 respectively. The left optic nerve and eye lens received a maximum dose of 11.1 Gy and 11.0 Gy respectively. The treatment plan overlay showed similar planning indices and critical organ doses. The plan comparison showed: an irradiated volume received the radiation dose > 15 Gy varies < 1.0% whereas the volume received < 15 Gy were larger (> 1.0%) in the study set “b”. The distant lateral points from the target volume which describe the phantom’s eyelid showed a radiation dose of 3.2 Gy - 2.5 Gy. The doses to these points were misled and ignored in the CT study set “a”. The eye bolus provides better dosimetric information in the estimation of low dose areas which is commonly misled on TPS in SRS planning for UMs.