Ith Row Research Articles

Semimagic Squares and Non-Semisimple Algebras

Now, define A 1 to be the matrix with 1 in the first row and column, -1 in the first row and jth column, -1 in the ith row and first column, 1 in the ith row and jth column, and 0 elsewhere for i, j = 2, , n. Then, as he remarked, the algebra (n has the dimension (n 1)2, and the matrices A q form a basis for .n These results are all he mentioned in his paper. But, in the following, it will be shown that (n iS isomorphic to the total matric algebra of degree n -1 when the characteristic p of the field Q of elements of matrices does not divide the order n, and that when n is divisible by p, the decomposition (1) does not hold, 2In is not semisimple, and there is the radical 91 such that 9 = 0. Moreover, by taking half of the condition, either E1 aJ==o(A) or Z=i aX1=o(A), for allj, we may obtain larger subalgebras of 9)1l, because the conditions (a), (b), (c) also hold. These algebras will be shown to be non-semisimple, having the radical 9 such that 92 = 0. The investigation of these algebras offers us a suitable illustration of the general theory of non-semisimple algebras [2]. 2. We start from the half of the condition for semimagic squares.

The asymptotic behavior of the stable initial manifolds of a system of nonlinear differential equations

Here we shall show that those solutions of (1.1) whose initial y vector lies on a certain initial manifold, which depends on E and the initial x vector, are very well approximated for small t and E by the corresponding solutions of a boundary layer equation, in which the initial x vector enters as a parameter, associated with (1.1). Moreover, it will also be shown that as E->O+ the stable initial manifolds associated with (1.1) tend to the stable initial manifold associated with the boundary layer equation. Problems of this nature, i.e., problems in which there is a system of differential equations possessing the property that the setting of a parameter equal to zero reduces the order of the system, have been treated under various hypotheses by several authors, e.g., [2; 3; 4; 5; 6 ] and [7 ]. The same hypotheses as in [5 ] will be assumed here. We now state these hypotheses as well as the principal theorem of [5]; this will then serve to introduce the problem and the results of the present paper. Let f=f,(t, x, y, E) be the matrix with ofil/xj in the ith row and jth column and let f,(t) be the matrix f,(t, p(t), q(t), 0). The matrices fy, gx, gy as well as fy(t), gx(t), gy(t) are similarly defined. HI:f, g, fx, fy, gx, gy are continuous in (t, x, y, E) for 0? t< T if I x-p(t)| +|y-q(t)| +E is sufficiently small. H2: There exists a real nonsingular matrix P(t) CC'(O <t ? T) such that

Cycles in algebraic systems

2. Cycles and their terminology. Let L be an n Xn Latin square or, alternatively, let Q be a quasigroup of order n. Let M be the set of n2 ordered triplets ijk, where k is the entry in the ith row and jth column of L. If S is the set of n2 ordered pairs ij, and if rir: M->S is the projection parallel to the ith coordinate (for example, ir2(ijk) =ik), then for each i, 1_i_3, 7ri is onto S (or equivalently is one-one). There is clearly a one-one correspondence between Latin squares L of order n and sets M of n2 ordered triplets for which the iri are all onto S. Let T: S-S be the involution defined by T(j) = (ji) and Pi: M-M be 7r 'Thrs. For each i, 1 <i<n, let MiCM be the set of triplets which contain i. Each tE Mk, with leading element i, generates what we shall call a cycle on i in the following manner. If 0: Mi-*Mi denotes P2P1P3, the cycle beginning with t shall consist of the triplets

On the existence problem of linear programming

1. Notation and introduction. We shall consistently use Latin capital letters to represent real rectangular matrices, lower-case Latin letters for real column vectors, and lower-case Greek letters for real scalars. The appearance of a subscript on a symbol lowers its classification one step; thus: A will represent a matrix, Aj will represent the /th column of A (a column vector), A(j will be the entry in the ith row and the /th column of A; b( will be the ith component of the column vector b; and so on. The dimensions of the matrices and vectors will not always be mentioned, but it is of course implied that they are consistent in the sense that all indicated operations are meaningful; for instance, the appearance of the product Ax implies that the number of columns in the matrix A is equal to the number of components of x. Vector and matrix inequalities are based upon the following notations (where Γ represents either a matrix or a

The Hook Graphs of the Symmetric Group

Each irreducible representation [λ] of the symmetric group Sn may be identified by a partition [λ] of n into non-negative integral parts λ1 ≥ λ2 ≥ … λn ≥ 0, of which the first λ'j parts are ≥j, or by a right (Young) diagram also called [λ], that contains λi nodes in its ith row and λ'j nodes in its jth column.

On the characteristic polynomial of the product of two matrices

The validity of Lemma I is obvious. The rank of D does not exceed unity, hence every two of its rows (columns) are proportional and D=(risj) =RTS, where R=(ri, r2, * * , rn) and S=(5si, S2, Sn) and ri and si, i= 1, 2, * *. n, are in F. To prove Lemma II, let D = RTS, where R and S are matrices established by Lemma I. The determinant of M+D may be expressed as the sum of 2n determinants. Of these I MI is one; n others are I Mil, i= 1, 2, ... n, where the matrix Mi is obtained from M by replacing its ith row by Sri of D; and the remaining 2n -1 n are zeros for their matrices are obtained by replacing two or more of the

On the Irreducible Representations of the Symmetric Group

Let T be a Young diagram of n nodes:(1)ai being the length of its ith row. With respect to a prime p, we denote by T0 the p-core of T. If T0 consists of m nodes, then(2) m=n-lp,where l is the number of successive p-hooks [3] removable from T to yield its p-core T0. We have stated in [4] the following theorem :