Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A₁,...,A_k are sets in F, not necessarily distinct, and A₁ ? ? ? A_k = ?, then A₁ ? ? ? A_k = S. We prove here that the maximum size of such a family is 2^|S|?1 + 1. If we require that the sets A₁,...,A_k be distinct, then the maximum size of F is again 2^|S|?1 + 1, provided that |S| ≥ log₂(K?2)+3.     

SEMIGROUP GRADINGS OF FULL MATRIX RINGS

Let F be a field, and let A = M_n (F) be the algebra of all n × n matrices over F. For each finite semigroup S, we describe all gradings A = ⊕ _s∈S A _S of A by S such that all standard matrix units e_ij of A are homogeneous elements of A.     

On fair parametric rational cubic curves

First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we numerically determine the distribution of inflection points and singularities on a segment which gives conditions that aC ² parametric rational cubic curve interpolating to dataS = {(x _i ^(k) ,y _i ^(k) ), 0 i n} is free of inflection points and singularities. When the parametric rational cubic curve reduces to the well-known parametric cubic one, we obtain a theorem on the distribution of the inflection points and singularities on the cubic curve segment which has been widely used for finding aC ¹ fair parametric cubic curve interpolating toS.     

Some results on the cartan matrix of a frobenius algebra

Let F be a field and A a Frobenius algebra over F. The Jacobson radical of A is denoted by J = J(A) and the kth socle of A by S _k (A). Let [Abar]=A/J ^k or A/S _k (A). This article gives new interesting relations between the Cartan matrix of A and that of [Abar]. From these results we prove that the Cartan matrix of A is diagonal if A/Soc(A) is a symmetric algebra. Let G be a finite group. If A is a block of F|G] with the above condition, then the Cartan matrix of A is (n), where n is the order of the defect group of A and the least integer such that Jⁿ (A)=0.     

A note on matrix subalgebras containing a full Jordan block

Let A be a subalgebra of the full matrix algebra M_n(F), and suppose J∈A, where J is the Jordan block corresponding to xⁿ. Let $\text{S}$ be a set of generators of A. It is shown that the graph of $\text{S}$ determines whether A is the full matrix algebra M_n(F).     

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP ²'s and for rational and integral homologyCP ²'s are given in terms of the typesA _k,D _k, orE _k of singularities allowed by the construction. Supported in part by National Science Foundation grant no. MCS 77-03540.     

The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

Disjoint matchings of graphs

In this paper, we prove a generalization of the familiar marriage theorem. One way of stating the marriage theorem is: Let G be a bipartite graph, with parts S₁ and S₂. If A ? S₁ and F(A) ? S₂ is the set of neighbors of points in A, then a matching of G exists if and only if Σ_x∈S2 min(1, | F^?1(x) ∩ A |) ≥ | A | for each A ? S₁. Our theorem is that k disjoint matchings of G exist if and only Σ_x∈S2 min (k, | F^?1(x) ∩ A |) ≥ k | A | for each A ? S₁.     

On some questions concerning permanents of (1,−1)-matrices

Let Ω _n denote the set of alln×n-(1,?1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ω _n such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2 ^k ?1,k≧2, even for singularA∈Ω _n . Moreover, we prove that perA≠0 ifA∈Ω _n ,n=2 ^k ?1, and we derive new results concerning the divisibility of the permanent in Ω _n by powers of 2.     

Factoriality of complete intersections in ℙ<Superscript>5</Superscript>

Let X be a complete intersection of two hypersurfaces F _n and F _k in ℙ⁵ of degree n and k, respectively, with n ≥ k, such that the singularities of X are nodal and F _k is smooth. We prove that if the threefold X has at most (n + k − 2)(n − 1) − 1 singular points, then it is factorial.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

A Principle for Critical Point Under Generalized Regular Constraint and Ill-Posed Lagrange Multipliers Under Nonregular Constraints

In this article, a kind of nonregular constraint and a principle for seeking critical point under the constraint are presented, where no Lagrange multiplier is involved. Let E, F be two Banach spaces, g: E → F a c ¹ map defined on an open set U in E, and the constraint S = the preimage g ^?1(y ₀), y ₀ ∈ F. A main deference between the nonregular constraint and regular constraint is that g′(x) at any x ∈ S is not surjective. Recently, the critical point theory under the nonregular constraint is a concerned focus in optimization theory. The principle also suits the case of regular constraint. Coordinately, the generalized regular constraint is introduced and the critical point principle on generalized regular constraint is established. Let f: U → ? be a nonlinear functional. While the Lagrange multiplier L in classical critical point principle is considered, its expression is given by using generalized inverse g′⁺(x) of g′(x) as follows: if x ∈ S is a critical point of f| _S , then L = f′(x) ○ g′⁺(x) ∈ F*. Moreover, it is proved that if S is a regular constraint, then the Lagrange multiplier L is unique; otherwise, L is ill-posed. Hence, in case of the nonregular constraint, it is very difficult to solve Euler equations; however, it is often the case in optimization theory. So the principle here seems to be new and applicable. By the way, the following theorem is proved: if A ∈ B(E, F) is double split, then the set of all generalized inverses of A, GI(A) is smooth diffeomorphic to certain Banach space. This is a new and interesting result in generalized inverse analysis.     

A determinantal version of the frobenius-könig theorem

Let F be a field and let {d ₁,…,d_k } be a set of independent indeterminates over F. Let A(d ₁,…,d_k ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d ₁,…,d_k }. We assume that no d ¹ appears twice in A(d ₁,…,d_k ). We show that if det A(d ₁,…,d_k ) = 0 then A(d ₁,…,d_k ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ? p ? 1: for some positive integer p.     

POLYNOMIALS RELATED TO HALL NUMBERS

Let Λ be a finite dimensional algebra of finite representation type over a finite field k. For any modules A, B and Pin mod Λ with P projective, we prove that there exists a polynomial ? ^B _(P)Λ over Z whose evaluation at |E| for any conservative finite field extension E of Λ is the sum of Hall numbers F ^{B E} _{C ^E A ^E} where C ^E runs through isoclasses in mod Λ ^E and P ^E is the projective cover of C ^E . As a consequence of this result and its dual version, Hall polynomials ? ^E _CA exist when C or A is semisimple. As applications of the main result, we obtain the existence of Hall polynomials for Nakayama algebras and some selfinjective algebras.

     

Free idempotent generated semigroups: subsemigroups,retracts and maximal subgroups

Let S be a subsemigroup of a semigroup T and let IG(E) and IG(F) be the free idempotent generated semigroups over the biordered sets of idempotents of E of S and F of T, respectively. We examine the relationship between IG(E) and IG(F), including the case where S is a retract of T. We give su?cient conditions satisfied by T and S such that for any e∈E, the maximal subgroup of IG(E) with identity e is isomorphic to the corresponding maximal subgroup of IG(F). We then apply this result to some special cases and, in particular, to that of the partial endomorphism monoid PEnd A and the endomorphism monoid EndA of an independence algebra A of finite rank. As a corollary, we obtain Dolinka’s reduction result for the case where A is a finite set.     

A Generalization of Engel Conditions with Derivations in Rings

Let R be a prime ring of characteristic different from 2 with Z the center of R and d a nonzero derivation of R. Let k, m, n be fixed positive integers. If ([d(x ^k ), x ^k ] _n ) ^m  ∈ Z for all x ∈ R, then R satisfies S ₄, the standard identity in 4 variables.     

Polynomial Lie Algebras

We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x ₁,...,x _n]/(f ₁,...,f _n) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A,B,C, D, and E ₆.     

On isometries of matrix spaces

Let A _kbe the group of isometries of the space of n-by-n matrices over reals (resp. complexes, quaternions) with respect to the Ky Fan k-norm (see the Introduction for the definitions). Let Γ⁰ be the group of transformations of this space consisting of all products of left and right multiplications by the elements of SO(n)(resp. U(n), Sp(n)). It is shown that, except for three particular casesA_k coincides with the normalizer of Γ in Δ group of isometries of the above matrix space with respect to the standard inner product. We also give an alternative treatment of the case D = R n = 4k = 2 which was studied in detail by Johnson, Laffey, and Li [4].