Solutions of the matrix equation (1):X m+A 1Xm−1+…+A m=0

SupposeX and the coefficientsA ₁, …,A _m aren×n matrices. LetB be anmn×mn matrix as in (7). LetJ be the Jordan canonical matrix ofB andB=PJP ⁻. LetE _i denote thei×i unit matrix.V is defined bydV/dt=JV andV(t=0)=E _mn. ThenZ=PV satisfiesdZ/dt=BZ.P ^* is a matrix which consists of the firstn rows ofP. The author proves: There is a solution of (1) ↔ there are anmn×n matrixC, ann×n matrixQ and ann×n function matrixN such thatP ^*VC=QN, where detQ≠0 andN is defined byN(t=0)=E _n anddN/dt=RN, in whichR is ann×n Jordan canonical matrix. There are three cases regarding the solutions of (1): No solution, finitek solutions, 1<k<C _n ^m , and infinite solutions which containj parameters, 1<-j<-mn ².     

A direct method for the general solution of a system of linear equations

A computationally stable method for the general solution of a system of linear equations is given. The system isA ^Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA ^T and the augmented matrix [A ^T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx _m and a symmetricn ×n matrixH _m of rankn–m, so thatx=x _m+H _my satisfies the system for ally, ann-vector. Whenm=n, the matrixH _m reduces to zero andx _m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158.     

Semidifferentiable functions and necessary optimality conditions

We consider Lyapunov's equationPA+A ^T P+Q=0, whereQ is symmetric positive definite andA is in controllable companion form. We prove that a necessary and sufficient condition thatA be stable is that the first rowP ₁ of theP-matrix be a stablen–1 coefficient vector. This result is related to the minimum phase property of linear systems and is useful in designing robust controllers.     

两类惯量惟一的对称符号模式

§ 1　 IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .　　 For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…     

On the sequence of powers of a stochastic matrix with large exponent

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n²−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λⁿ−αλ^n−j−(1−α)λ^n−k. In this paper, we prove that for any mn, if α1/2, then A^m+k−A^m_∞A^m−1w^T_∞, where 1 is the all-ones vector and w^T is the left-Perron vector for A, normalized so that w^T1=1. We also prove that if jn/2, n31 and , then A^m+j−A^m_∞A^m−1w^T_∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence A^m.     

Monomorphism Operator and Perpendicular Operator

For a quiver Q, a k-algebra A, and an additive full subcategory 𝒳 of A-mod, the monomorphism category Mon(Q, 𝒳) is introduced. The main result says that if T is an A-module such that there is an exact sequence 0 → T _m  → … → T ₀ → D(A _A ) → 0 with each T _i  ∈ add(T), then Mon(Q, ^⊥ T) =^⊥(kQ ? _k T); and if T is cotilting, then kQ ? _k T is a unique cotilting Λ-module, up to multiplicities of indecomposable direct summands, such that Mon(Q, ^⊥ T) =^⊥(kQ ? _k T).

As applications, the category of the Gorenstein-projective (kQ ? _k A)-modules is characterized as Mon(Q, 𝒢𝒫(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, 𝒳) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.     

Convergence of Krylov methods for sums of two operators

We discuss the convergence of Krylov subspace methods for equationsx =Tx +f whereT is a sum of two operators,T =B +K, whereB is bounded andK is nuclear. Bounds are given for inf Q _k (B+K) and for inf p _k (B+K), where the infimum is over all polynomials of degreek, such thatQ _k is monic andp _k is normalized:p _k (1) = 1.     

Derived equivalence of algebras

The derived equivalence and stable equivalence of algebrasR _A ^m and R _B ^m are studied. It is proved, using the tilting complex, thatR _A ^m andR _B ^m are derived-equivalent whenever algebrasA andB are derived-equivalent.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Robustness of positional scoring over subsets of alternatives

Positional score vectorsw=(w ₁,,w _m ) for anm-element setA, andv=(v ₁,,v _k ) for ak-element proper subsetB ofA, agree at a profiles of linear orders onA when the restriction toB of the ranking overA produced byw operating ons equals the ranking overB produced byv operating on the restriction ofs toB. Givenw ₁>w _mandv ₁>v _k , this paper examines the extent to which pairs of nonincreasing score vectors agree over sets of profiles. It focuses on agreement ratios as the number of terms in the profiles becomes infinite. The limiting agreement ratios that are considered for (m, k) in {(3,2),(4,2),(4,3)} are uniquely maximized by pairs of Borda (linear, equally-spaced) score vectors and are minimized when (w,v) is either ((1,0,,0),(1,,1,0)) or ((1,,,1,0),(1,0,,0)).This research was supported by the National Science Foundation, Grants SOC 75-00941 and SOC 77-22941.     

Conservative loads on shells

Generalized load vectorsp and edge load vectorsF are denned in terms of the body force and surface on a shell. Necessary and sufficient conditions are derived forp andF, and therefore the body force and the surface force, to be conservative. It is shown for example thatp must satisfyp _i=P _ijk q _j,1 q _k,2+Q _ij ¹ q _j,2–Q _ij ² q _j,1+R _i whereq is the generalized position vector andP _ijk, Q_i,j ¹ and Q_ij ² are skew tensors.The case of hydrostatic pressure is examined in detail.This work was supported in part by NSF grant MSM 8618657.     

Perturbation bounds for the Cholesky andQR factorizations

LetA, A+E be Hermitian positive definite matrices. Suppose thatA=LL ^H andA+E=(L+G)(L+G)^H are the Cholesky factorizations ofA andA+E, respectively. In this paper lower bounds and upper bounds on |G|/|L| in terms of |E|/|A| are given. Moreover, perturbation bounds are given for the QR factorization of a complexm ×n matrixA of rankn.This research was supported by the National Science Foundation of China and the Department of Mathematics of Linköping University in Sweden.     

Cyclic homology and the Macdonald conjectures

Summary LetA+(k) denote the ring [t]/t ^k+1 and letG be a reductive complex Lie algebra with exponentsm ₁, ...,m n. This paper concerns the Lie algebra cohomology ofGA ₊(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA ₊(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m _s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m _s +1, one has weight 0 and the others have weights (k+1)m _s +t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A ₊(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA _n ,B _n ,C _n , orD _n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship     

Bounds on some van der Waerden numbers

For positive integers s and k₁,k₂,…,ks, the van der Waerden number w(k₁,k₂,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided.     

On the evaluation of the Eigenvalues of a banded toeplitz block matrix

Let A = (a_ij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, a_ij = a_j−i, i, J = 1,… ,n, a_i = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if a_i are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m³k(log² k + log n) + m²k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B.     

Optimal pairs of score vectors for positional scoring rules

Letw=(w ₁,,w _m ) andv=(v ₁,,v _m-1 ) be nonincreasing real vectors withw ₁>w _m andv ₁>v _m-1 . With respect to a lista ₁,,a _n of linear orders on a setA ofm3 elements, thew-score ofaA is the sum overi from 1 tom ofw _i times the number of orders in the list that ranka inith place; thev-score ofaA{b} is defined in a similar manner after a designated elementb is removed from everya _j .We are concerned with pairs (w, v) which maximize the probability that anaA with the greatestw-score also has the greatestv-score inA{b} whenb is randomly selected fromA{a}. Our model assumes that linear ordersa _j onA are independently selected according to the uniform distribution over them linear orders onA. It considers the limit probabilityP _m (w, v) forn that the element inA with the greatestw-score also has the greatestv-score inA{b}.It is shown thatP _m (m,v) takes on its maximum value if and only if bothw andv are linear, so thatw _i –w _i+1=w _i+1–w _i+2 forim–2, andv _i –v _i+1 =v _i+1 –v _i+2 forim–3. This general result for allm3 supplements related results for linear score vectors obtained previously form{3,4}.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024     

On two set-systems with restricted cross-intersections

If A₁, …, A_m; B₁, …, B_m are finite sets such that for l t 0 and any r, s, we have |A_i| r, |B_i| s and |_i ∩ B_i| t for 1 i m and |A_i ∩ B_i| > l for 1 i < j m, what is the maximum value that m can attain? In this paper we answer this question of Füredi and extend an inequality of Bollobás.im]0658.TIF     

Convex interpolating splines of arbitrary degree II

For given data {(x _i ,y _i )} _i=0 ⁿ , (x ₀<x ₁<...<x _n ) we consider the possibility of finding a spline functions of arbitrary degreek+1 (k 1) with preassigned smoothnessl, where 1 l [(k+1)/2]. The splines should be such thats(x _i )=y _i ,i=0, 1,...,n ands is increasing and convex on [x ₀,x _n ]. Sufficient conditions which guarantee the existence ofs and an explicit formula for this function are derived.