Controllability and the linear equation Ax=b

The linear equation Ax=b, with An×n, is considered, and it is shown that the controllable subspace of (A, b) can intersect the solution set of this equation in at most one point. Some additional properties of the controllable subspace related to the solution set are presented, and the role of controllability in establishing nonsingularity of A is exhibited.     

On the block structure of BIB designs with parameters v = 22, b = 33, r = 12, k = 8, and λ = 4

It is unknown whether a BIB design with parameters (22, 33, 12, 8, 4) exists or not. In this paper, a necessary condition for the block structure of BIB designs is given. Using this necessary condition, it is shown that the intersection numbers {n_i} for any block in any BIB design with parameters (22, 33, 12, 8, 4) must be one of the four types shown in Table I if the BIB design exists. It is also shown that if there exists a BIB design with parameters (22, 33, 12, 8, 4), the BIB design must contain at least one block of Type 1 or 3 in Table I.     

Solutions of the equation zez = a(z + b)

The theory of complex variables is used to develop exact closed-form solutions for the, in general, complex zeros of the exponential polynomial F(z) = z exp z ? a(z + b), a complex and b real. The established zeros are related to canonical solutions of suitably posed Riemann problems and are expressed ultimately in terms of elementary quadratures.     

Laplacian integral graphs in S(a, b)

Let G_n,m be the family of graphs with n vertices and m edges, when n and m are previously given. It is well-known that there is a subset of G_n,m constituted by graphs G such that the vertex connectivity, the edge connectivity, and the minimum degree are all equal. In this paper, S(a, b)-classes of connected (a, b)-linear graphs with n vertices and m edges are described, where m is given as a function of a,b∈N/2. Some of them have extremal graphs for which the equalities above are extended to algebraic connectivity. These graphs are Laplacian integral although they are not threshold graphs. However, we do build threshold graphs in S(a, b).     

The matrix equations AX = C, XB = D

For the pair of matrix equations AX = C, XB = D this paper gives common solutions of minimum possible rank and also other feasible specified ranks.     

Block designs with v = 10, k = 5, λ = 4

It was shown by Singhi that there are 21 nonisomorphic block designs BD (10, 5; 18, 9, 4) which are residual designs of (19, 9, 4) Hadamard designs. In this paper we show that there are no other block designs with these parameters, i.e., each such design is embeddable in a symmetric design. We give a complete list of these designs and their automorphism groups.     

Ranked solutions of AXC=B and AX=B

Let GF(pⁿ) denote the finite field of pⁿ elements, p odd. Let A be an s×m matrix of rank ?, B be an s×t matrix of rank β, and C be an f×t matrix of rank v. This paper discusses the number of m×f matrices X of rank k over GF(pⁿ) which are solutions to the matric equations AXC=B or AX=B.     

Controllability,observability and the solution of AX - XB = C

A closed-form finite series representation of the unique solution X of the matrix equation AX ? XB=C is developed. Using this representation, the image, kernel, and rank of X are related to the controllable and unobservable subspaces of the (A, C) and (C, B) pairs respectively. Bounds on the rank of X are obtainedin terms of the dimensions of these subspaces. In the case that C has unitary rank, an exact calculation of rank X is made. The generic rank of X with A, B fixed and C generic is evaluated.     

Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices

The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX ? XB = C and X ? AXB = C can be transformed to equations whose coefficients are block companion matrices: $\text{C}\text{?}{}_{\mathrm{L}}\text{X?XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$ and $\text{X?}\text{C}\text{?}{}_{\mathrm{L}}\text{XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$, respectively, where ?_L and C_M stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λ^sI + Σ^s?1_j=0λ^jL_j and M(λ) = λ^tI + Σ^t7minus;1_j=0λ^jM_j. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ?, where the matrix S is r²l × r²l[l = max(s, t)] and enjoys some symmetry properties.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

On the Diophantine Equation x+q=y

In this paper it has been proved that if q is an odd prime, q?7 (mod 8), n is an odd integer ?5, n is not a multiple of 3 and (h,n)=1, where h is the class number of the filed Q(√−q), then the diophantine equation x²+q^2k+1=yⁿ has exactly two families of solutions (q,n,k,x,y).     

On the Diophantine equation ax + by = ck

In this paper we investigate the solvability and the representation of the solutions of the equation ax² +by² = ckⁿ. We extend and improve many known results. In particular, we completely solve the equation (a ± 1)x² + (3a ? 1) = 4aⁿ, 2 ? n.     

Solvability of the Matric Equation AX = B

Let A be an n×s matrix of rank r, B be an n×t matrix of rank ρ?r, and X be an s×t matrix. This paper discusses conditions on the matrices A and B so that the matric equation AX=B will have solutions for the matrix X.