Some properties of Q-matrices

This paper deals with the class of Q-matrices, that is, the real n × n matrices M such that for every q ∈ $\text{R}$^n×1, the linear complementarity problem

$\text{Iw ? Mz = q}$

,

$\text{w ? 0, z ? 0, and w}{}^{\mathrm{T}}\text{z = 0}$

, has a solution. In general, the results are of two types. First, sufficient conditions are given on a matrix M so that M ∈ Q. Second, conditions are given so that M ? Q.     

On the simplex method and a class of linear complementary problems

We consider the linear complementarity problem of finding vectors w?Rⁿ, z?Rⁿ satisfying w ? Mz = q, w ? 0, z ? 0, w^Tz = 0. We show that if the off diagonal elements of M are nonpositive, then the above problem is solved by applying the simplex method to the problem Minimize z₀ subject to w ? Mz ? e_nz₀ = q, (z₀, w, z) ? 0, where e_n is a column vector of 1's. In fact the sequence of basic feasible solutions obtained by the simplex method and by Lemke's algorithm are the same. We also obtain necessary and sufficient conditions for the problem to have solutions for all q.     

Completely- matrices

A matrix is a real square matrixM such that for everyq the linear complementarity problem: Findw andz satisfyingw = q + Mz, w ≥ 0, z ≥ 0, w ^T z = 0, has a solution. We characterize the class of completely- matrices, defined here as the class of -matrices all of whose nonempty principal submatrices are also -matrices.     

Solving more linear complementarity problems with Murty's bard-type algorithm

The linear complementarity problem (M|q) is to findw andz inR ⁿ such thatw–Mz=q,w0,z0,w ^t z=0, givenM inR ^n×n andq in . Murty's Bard-type algorithm for solving LCP is modeled as a digraph.Murty's original convergence proof considered allq inR ⁿ andM inR ^n×n , aP-matrix. We show how to solve more LCP's by restricting the set ofq vectors and enlarging the class ofM matrices beyondP-matrices. The effect is that the graph contains an embedded graph of the type considered by Stickney and Watson wheneverM is a matrix containing a principal submatrix which is aP-matrix. Examples are presented which show what can happen when the hypotheses are further weakened.     

Q-matrices and boundedness of solutions to linear complementarity problems

This paper is concerned with the existence and boundedness of the solutions to the linear complementarity problemw=Mz+q,w0,z0,w ^T z=0, for eachq ⁿ . It has been previously established that, ifM is copositive plus, then the solution set is nonempty and bounded for eachq ⁿ iffM is aQ-matrix. This result is shown to be valid also forL ₂-matrices,P ₀-matrices, nonnegative matrices, andZ-matrices.     

A linear-algebra problem from algebraic coding theory

Given an m×n matrix M over E=GF(q^t) and an ordered basis A={z₁,…,z_t} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M¹ with entries from the field K. Let r=rank(M) and r¹=rank(M¹). We show that r?r¹?min{rt,n}, and we determine the number b(m,n,r,r¹,q,t) of m×n matrices M of rank r over GF(q^t) with associated mt×n matrix M¹ of rank r¹ over GF (q).     

The transcendental meromorphic solutions of a certain type of nonlinear differential equations

In this paper, we study the differential equations of the following form w²+R(z)²(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w²+P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w²+(z−z₀)P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w²+A(z−z₀)²(w^′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab²=1, a²=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B²=C and q^′(z)=±P(z).     

Atomic decomposition characterizations of weighted multiparameter Hardy spaces

Let w ?? A _??. In this paper, we introduce weighted-(p, q) atomic Hardy spaces H _w ^p,q (? ⁿ ×? ^m ) for 0 < p ? 1, q >q _w and show that the weighted Hardy space H _w ^p (? ⁿ × ? ^m ) defined via Littlewood-Paley square functions coincides with H _w ^p,q (? ⁿ × ? ^m ) for 0 < p ? 1, q > q _w . As applications, we get a general principle on the H _w ^p (? ⁿ × ? ^m ) to L _w ^p (? ⁿ ×? ^m ) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.     

Degeneracy in linear complementarity problems: a survey

Given a square matrixM of ordern and a vectorq ⁿ , the linear complementarity problem is the problem of either finding aw ⁿ and az ⁿ such thatw–Mz=q,w0,z0 andw ^T z=0 or showing that no such (w, z) exists. This problem is denoted asLCP(q, M). We say that a solution (w, z) toLCP(q, M) is degenerate if the number of positive coordinates in (w, z) is less thann. As in linear programming, degeneracy may cause cycling in an adjacent vertex following methods like Lemke's algorithm. Moreover, ifLCP(0,M) has a nontrivial solution, a condition related to degeneracy, then unless certain other conditions are satisfied, the algorithm may not be able to decide about the solvability of the givenLCP(q, M). In this paper we review the literature on the implications of degeneracy to the linear complementarity theory.     

Some classes of matrices in linear complementarity theory

The linear complementarity problem is the problem of finding solutionsw, z tow = q + Mz, w0,z0, andw ^T z=0, whereq is ann-dimensional constant column, andM is a given square matrix of dimensionn. In this paper, the author introduces a class of matrices such that for anyM in this class a solution to the above problem exists for all feasibleq, and such that Lemke's algorithm will yield a solution or demonstrate infeasibility. This class is a refinement of that introduced and characterized by Eaves. It is also shown that for someM in this class, there is an even number of solutions for all nondegenerateq, and that matrices for general quadratic programs and matrices for polymatrix games nicely relate to these matrices.Research partially supported by National Science Foundation Grant NSF-GP-15031.     

Eine Bemerkung über DOTU-Matrizen

It is proved that there exist real n × n-matrices of determinant unity, which are not DOTU-matrices.     

Einige weitere Resultate in Analogie zu einem Gleichverteilungssatz von Koksma

Continuing former investigations by the authors (see the references) the present paper contains metric results on the distribution modulo 1 of the powers of special kinds of real matricesA, namely of (2×2)- and (3×3)-triangle matrices, symmetric (2×2)-matrices and so-called “cosymmetric” (2×2)-matrices (i. e. matrices, symmetric with respect to the secondary diagonal). For almost all such matricesA (in the sense of the Lebesgue measure in ?³ resp. ?⁶) possessing no eigenvalue of modulus smaller than 1 the inequality $$D(N) \leqslant C(A, \varepsilon ) N^{ - 1/2} (\log N)^{d + 3/2 + \varepsilon } (\varepsilon > 0)$$ is proved as an estimate for the discrepancy of the sequence (A ^s(n) ) where (s(n)) _n ^=1/∞ is an arbitrary fixed strictly increasing sequence of positive integers andd is the dimension of the appropriate space ? ^d (d=3 or 6).     

On the radius of α-convexity of certain classes of starlike functions

LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functions? (z) =z + Σ _n=2/^∞ a _n z ⁿ in the unit discD = {z: |z | < 1} which satisfies inD the condition ?′ (z) ?(z)/z ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z f'(z)}}{{f (z)}} + a \left( {1 + \frac{{z f''(z)}}{{f' (z)}}} \right)} \right\} > 0. Let W (a) $$ denote the class of meromorphic α-convex functions. α real, that is the class of analytic functions ? (z) =z ^?1 + Σ _n=0/^∞ b _n z ⁿ inD* = {z: 0 < |z | < 1} which satisfies inD* the conditionsz?′(z)/?(z) ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z\phi ' (z)}}{{\phi (z)}} + a \left( {1 + \frac{{z\phi ''(z)}}{{\phi ' (z)}}} \right)} \right\}< 0. $$ In this paper we obtain the relation betweenM (a) and W(α). The radius of α-convexity for certain classes of starlike functions is also obtained.     

Multipliers of a wandering subspace for a shift invariant subspace

For a shift-invariant subspace M of the two variable Hardy space H², we consider the associated wandering subspace M₀=M?zM. Then there exists a nonconstant function ? in H^∞ such that ?M₀⊆M₀ if and only if M=qH² for some inner function q.     

Volume mean densities for the heat equation

It is shown that, for solid caps D of heat balls in ? ^{d + 1} with center z ₀ = (0, 0), there exist Borel measurable functions w on D such that inf w(D) > 0 and ∫ v(z)w(z) dz ≤ v(z ₀), for every supertemperature v on a neighborhood of D?. This disproves a conjecture by N. Suzuki and N.A. Watson. On the other hand, it turns out that there is no such volume mean density, if the bounded domain D in ? ^d × (?∞, 0) is only slightly wider at z ₀ than a heat ball.     

An existence theorem for the complementarity problem

LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE ⁿ ,C* its polar cone,M:C→E ⁿ a map, andq a vector inE ⁿ . The complementarity problem (q|M) overC is that of finding a solution to the system $$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q ₀ ∈ intC*, then it has a solution for allq ∈E ⁿ .     

Sign-nonsingular matrices and even cycles in directed graphs

A sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (a_ij) such that at least one of a_ij and a_ij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and a_ij=0 iff a_ji=0. We also show that for any n×n L-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects.     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? ⁿ⁺¹ there exist an affine connection ?, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=?h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ? and a nondegenerate metrich on a differentiable manifoldM ⁿ such that ?h is totally symmetric and satisfies the apolarity condition relative toh, canM ⁿ be locally immersed in ? ⁿ⁺¹ in such a way that (?,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ?. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (?,h) is that the conjugate connection of ? relative toh is projectively flat.     

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).     

Signsolvability revisited

For a real matrix A, Q(A) denotes the set of all matrices with the same sign pattern as A. A linear system Ax=b is signsolvable if solvability and Q(x) depend only on Q(A) and Q(b). The study of signsolvability can be decomposed into the study of L-matrices and of S-matrices, where A is an L-matrix [S-matrix] if the nullspace of each member of Q(A) is {0} [is a line intersecting the open positive orthant]. The problem of recognizing L-matrices is shown to be NP-complete, even in the [almost square] case. Recognition of square L-matrices was transformed into a graph-theoretic problem by Bassett, Maybee, and Quirk in 1968. The complexity of this problem remains open, but that of some related graph-theoretic problems is determined. The relation between S-matrices and L-matrices is studied, and it is shown that a certain recursive construction yields all S-matrices, thus proving a 1964 conjecture of Gorman.