Let G be a non-abelian group and let l2(G) be a finite dimensional Hilbert space of all complex valued functions for which the elements of G form the (standard) orthonormal basis. In our paper we prove results concern...Let G be a non-abelian group and let l2(G) be a finite dimensional Hilbert space of all complex valued functions for which the elements of G form the (standard) orthonormal basis. In our paper we prove results concerning G-decorrelated decompositions of functions in l2(G). These G-decorrelated decompositions are obtained using the G-convolution either by the irreducible characters of the group G or by an orthogonal projection onto the matrix entries of the irreducible representations of the group G. Applications of these G-decorrelated decompositions are given to crossover designs in clinical trials, in particular the William’s 6×3?design with 3 treatments. In our example, the underlying group is the symmetric group S3.展开更多
Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for 60 years until they were proved by O. Kharlampovich and A. Myasnikov and indep...Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for 60 years until they were proved by O. Kharlampovich and A. Myasnikov and independently by Z. Sela. The proofs, by both sets of authors, were monumental and involved the development of several new areas of infinite group theory. In this paper we explain precisely the Tarski problems and what has been actually proved. We then discuss the history of the solution as well as the components of the proof. We then provide the basic strategy for the proof. We finish this paper with a brief discussion of elementary free groups.展开更多
Let L be an abelian extension of the rationals Q whose Galois group Gal(L) is an abelian (q-group q is any prime number). The explicit law of prime decomposition in L for any prime number p, the inertia group, residue...Let L be an abelian extension of the rationals Q whose Galois group Gal(L) is an abelian (q-group q is any prime number). The explicit law of prime decomposition in L for any prime number p, the inertia group, residue class degree, and discriminant of L are given here; such fields L are classified into 4 or 8 classes according as q is odd or even with clear description of their structures. Then relative extension L/K is studied. L/K is proved to have a relative integral basis under certain simple conditions; relative discriminant D(L/K) is given explicitly; and necessary and sufficient conditions are obtained for D(L/K) to be generated by a rational square (and by a rational). In particular, it is proved that L/K has a relative integral basis and that D(L/K) is generated by a rational square if [L: K]≥x~* or x~*+1 (according as q is odd or even), where x~* is the exponent of Gal(L). These results contain many related results on similar fields in literature.展开更多
文摘Let G be a non-abelian group and let l2(G) be a finite dimensional Hilbert space of all complex valued functions for which the elements of G form the (standard) orthonormal basis. In our paper we prove results concerning G-decorrelated decompositions of functions in l2(G). These G-decorrelated decompositions are obtained using the G-convolution either by the irreducible characters of the group G or by an orthogonal projection onto the matrix entries of the irreducible representations of the group G. Applications of these G-decorrelated decompositions are given to crossover designs in clinical trials, in particular the William’s 6×3?design with 3 treatments. In our example, the underlying group is the symmetric group S3.
文摘Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for 60 years until they were proved by O. Kharlampovich and A. Myasnikov and independently by Z. Sela. The proofs, by both sets of authors, were monumental and involved the development of several new areas of infinite group theory. In this paper we explain precisely the Tarski problems and what has been actually proved. We then discuss the history of the solution as well as the components of the proof. We then provide the basic strategy for the proof. We finish this paper with a brief discussion of elementary free groups.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19771052).
文摘Let L be an abelian extension of the rationals Q whose Galois group Gal(L) is an abelian (q-group q is any prime number). The explicit law of prime decomposition in L for any prime number p, the inertia group, residue class degree, and discriminant of L are given here; such fields L are classified into 4 or 8 classes according as q is odd or even with clear description of their structures. Then relative extension L/K is studied. L/K is proved to have a relative integral basis under certain simple conditions; relative discriminant D(L/K) is given explicitly; and necessary and sufficient conditions are obtained for D(L/K) to be generated by a rational square (and by a rational). In particular, it is proved that L/K has a relative integral basis and that D(L/K) is generated by a rational square if [L: K]≥x~* or x~*+1 (according as q is odd or even), where x~* is the exponent of Gal(L). These results contain many related results on similar fields in literature.
