( ( For ( regular ( cross-sectional shapes ) ) ) ( , ( ( ( the dimensions ) ( of ( the kern ) ) ) ( ( can ( be ( determined analytically ) ) ) . ) ) ) )
(ROOT (S (PP (IN For) (NP (JJ regular) (JJ cross-sectional) (NNS shapes))) (, ,) (NP (NP (DT the) (NNS dimensions)) (PP (IN of) (NP (DT the) (NN kern)))) (VP (MD can) (VP (VB be) (VP (VBN determined) (ADVP (RB analytically))))) (. .)))
For regular cross-sectional shapes, the dimensions of the kern can be determined analytically.
(ROOT (S (NP (NP (DT The) (NN volume)) (PP (IN of) (NP (DT a) (ADJP (ADVP (RB regularly)) (JJ shaped)) (JJ solid)))) (VP (MD can) (VP (VB be) (VP (VBN calculated) (PP (IN from) (NP (PRP$ its) (NNS dimensions)))))) (. .)))