Algebraic functions of normal operators

The solutions ofp(T)=S, whereS is normal andp a polynomial, are described. This work was supported, in part, by Grant GP-3628 of the National Science Foundation.     

The inflation class operator

We consider the inflation class operator, denoted by F, where for any class K of algebras, F(K) is the class of all inflations of algebras in K. We study the interaction of this operator with the usual algebraic operators H, S andP, and describe the partially-ordered monoid generated by H, S, P andF (with the isomorphism operator I as an identity). Received February 3, 2004; accepted in final form January 3, 2006.     

On factorizing meromorphic functions

Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr) ^m+k,q=r ^m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t ^m) is a doubly-periodic function.     

Representability of Hom implies flatness

LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf ε onX, consider the set-valued contravariant functor Hom(ε,F)S-schemes, defined by Hom(ε,F) (T)= Hom(ε _T ,F _T) where ε _T andF _T are the pull-backs of ε andF toX _T =X x _S T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then Kom_ε,F) is representable for all ε. We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom(L ^-n ,F) is representable for infinitely many positive integersn, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorT ↦H°(T, F _t) on the category ofS-schemes is representable if and only ifF is locally free onS. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author’s earlier result (see [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.     

Amitsur cohomology of cubic extensions of algebraic integers

LetK be the rational fieldQ or a complex quadratic number field other than . LetL be a normal three-dimensional field extension onK. IfR andS are the rings of algebraic integers ofK andL respectively, then the Amitsur cohomology groupH ² (S/R, U) is trivial. Inflation and class numbers give information about cohomology arising from certain nonnormal cubic extensions. This work was supported in part by NSF Grant GP-28409.     

On a problem of Erdős concerning property B

A family ℱ of sets has propertyB if there exists a setS such thatS∩F≠0 andS⊄F for everyF∈ℱ. ℱ has propertyB(s) if there exists a setS such that 0<|F∩S|<s for everyF∈ℱ. Denote bym(n) (respectivelym(n, s)) the size of a smallest family ofn-element sets not having propertyB (respectivelyB(s)). P. Erdős has asked whetherm(n, s)≧m (s) for alln≧s. We show that, in general, this inequality does not hold.     

Linear problems (with extended range) have linear optimal algorithms

LetF ₁ andF ₂ be normed linear spaces andS:F ₀ F ₂ a linear operator on a balanced subsetF ₀ ofF ₁. IfN denotes a finite dimensional linear information operator onF ₀, it is known that there need not be alinear algorithm:N(F ₄) F ₂ which is optimal in the sense that (N(f)) –S(f is minimized. We show that the linear problem defined byS andN can be regarded as having a linear optimal algorithm if we allow the range of to be extended in a natural way. The result depends upon imbeddingF ₂ isometrically in the space of continuous functions on a compact Hausdorff spaceX. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.     

Integer programming and convex analysis: Intersection cuts from outer polars

Recently a new, geometrically motivated approach was proposed [1] for integer programming, based on generating intersection cuts from some convex setS whose interior contains the linear programming optimum but no feasible integer point. Larger sets tend to produce stronger cuts, and in this paper convex analysis is used to construct sets as large as possible within the above requirements. Information is generated from all problem constraints within a unit cubeK containing The key concept is that of outer polars, viewed as maximal convex extensions of the ballB circumscribingK, relative to the problem constraints. The outer polarF ^* of the feasible setF overB is shown to be a convex set whose boundary contains all feasible vertices ofK, and whose interior contains no feasible integer point. The existence of a dual correspondence betweenF andF ^*, and the fact that polarity is inclusion-reversing, leads to a dualization of operations onF. As one possible procedure based on this approach, we construct a generalized intersection cut, that can be strengthened whenever some vertex ofF is cut off. This makes it possible to fruitfully combine intersection cuts with implicit enumeration or branch-and-bound. While valid for arbitrary integer programs, the theory developed here is relevant primarily to (pure or mixed-integer) 0–1 problems. Other topics discussed include: generalized polars, intersection cuts from related problems, connections with asymptotic theory.This paper was presented at the 7th Mathematical Programming Symposium, 1970, The Hague, The Netherlands.The research underlying this paper was partially supported by the National Science Foundation under grant GP-31699 and by the Office of Naval Research under contract N00014-67-A-0314-0007 NR 047-048.     

The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space

LetE andF be Hilbert spaces with unit spheresS ₁(E) andS ₁(F). Suppose thatV ₀ S₁(E) →S ₁(F) is a Lipschitz mapping with Lipschitz constantk=1 such that −V ₀[S ₁(E)] ⊂V ₀[S ₁(E)]. Then V₀ can be extended to a real linear isometric mappingV fromE intoF. In particular, every isometric mapping fromS ₁(E) ontoS ₁(F) can be extended to a real linear isometric mapping fromE ontoF.     

A local-global principle for finiteness properties ofS-arithmetic groups over number fields

In the preceding paper [AT] compactness propertiesC _n andCP _n for locally compact groups were introduced. They generalize the finiteness propertiesF _n andFP _n for discrete groups. In this paper a local-global principle forS-arithmetic groups over number fields is proved. TheS-arithmetic group is of typeF _n , resp.FP _n , if and only if for allp inS thep-adic completionG _p of the corresponding algebraic groupG is of typeC _n resp.CP _n . As a corollary we obtain an easy proof of a theorem of Borel and Serre: AnS-arithmetic subgroup of a semisimple group has all the finiteness propertiesF _n .     

Shuffles of permutations and the Kronecker product

The Kronecker product of two homogeneous symmetric polynomialsP ₁,P ₂ is defined by means of the Frobenius map by the formulaP ₁oP ₂=F(F ⁻¹ P ₁)(F ⁻¹ P ₂). WhenP ₁ andP ₂ are the Schur functionsS _I ,S _J then the resulting productS _I oS _J is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofS _I oS _J with a third Schur functionsS _K gives the so called Kronecker coefficientc _I,J,K =<S _I oS _J ,S _K >. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thec _I,J,K in terms of some classes of permutations. In Lascoux's workI andJ are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of <S _I oS _J ,S _K > forS _I a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result. Work supported by NSF grant at the University of California, San Diego.     

On orbit equivalence of borel automorphisms

LetE andF be two Borel sets of the countable productZ of the two point space {0,1}. Assume thatE andF are invariant sets for the odometer transformationR and thatE andF are of measure zero with respect to the unique finiteR-invariant measure onZ. We show thatE andF areR-orbit equivalent in a strict sense.     

Some remarks on weakly compactly generated banach spaces

A simple example is given of a non WCG space whose dual is a WCG space with an unconditional basis. It is proved that ifX* is WCG andX is a subspace of a WCG space thenX itself is WCG. The first named author was supported in part by NSF GP-33578. An erratum to this article is available at .     

An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

Automorphism groups of certain simple 2-(q,3,λ) designs constructed from finite fields

Let F be a finite field of characteristic not 2, and SF a subset with three elements. Consider the collection

S={S·a+b | a,bF, a≠0}.

Then (F,S) is a simple 2-design and the parameter λ of (F,S) is 1, 2, 3 or 6. We find in this paper the full automorphism group of (F,S). Namely, if we put U={r | {0,1,r}S} and K the subfield of F generated by U, then the automorphisms of (F,S) are the maps of the form xg(α(x))+b, xF, where bF, α : F→F is a field automorphism fixing U, and g is a linear transformation of F considered as a vector space over K.     

Additive Maps Preserving Moore–Penrose Inverses of Matrices on Symmetric Matrix Spaces

Suppose F is a field of characteristic not 2. Let M_n F and S_n F be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from S_n F to S_n F (respectively, M_n F) preserving Moore–Penrose inverses of matrices are characterized. We first characterize all additive Moore–Penrose inverse preserving maps from S_n F to M_n F, and thereby, all additive Moore–Penrose inverse preserving maps from S_n F to itself are characterized by restricting the range of these additive maps into the symmetric matrix space.     

Doubly stochastic matrix equations

It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω _n of alln x n doubly stochastic matrices. The regular matrices in Ω _n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω _n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω _n . The paper is concluded with extensions of some of these results to the convex setS _n of alln x n nonnegative matrices having row and column sums at most one. His research was supported by the N. S. F. Grant GP-15943.     

A note on the valence of certain means

Given two functionsf(z),g(z) in the (usual) classS, we can form the new functions (arithmetric and geometric mean functions) F(itz)=∝(itf)(itz)+β(itg)(itz) and G(itz)=(itz)(f(itz)/(itz))(su∝)(g(itz)/(itz))(suβ), whereα, β ∈ (0, 1) andα+β=1. This paper determines the maximum valence of the functionsF andG.     

Normality and shared values

LetF be a family of meromorphic functions on the unit disc Δ and leta andb be distinct values. If for everyf∈F,f andf′ sharea andb on Δ, thenF is normal on Δ. The first author was supported by NNSF of China approved no. 19771038 and by the Research Institute for Mathematical Sciences, Bar-Ilan University.     

Testing linear operators

The goal of testing is to determine whether an implementation linear operatorA conforms to a specification linear operatorS within a given error bound for all elements from an input setF. Suppose that an upper boundK on the norm of the difference ofS andA is given a priori. Then it is shown that in general any finite number of tests is inconclusive both in the worst case and on the average. However, the testing problem is still decidable in the limit for an arbitraryK; there is an algorithm of an infinite sequence of test-and-guess such that all but finitely many guesses are correct. On the other hand, if the error bound is relaxed for weak conformance then finite tests suffice even in the worst case and tight lower and upper bounds on the number of tests are derived. The test set is universal; it only depends on the set of valid inputsF. Furthermore, the test elements are on the boundary ofF. Two examples are used to illustrate the approaches and the paper is concluded with comments on two related problems: computation and verification.This work was done while consulting at AT&T Bell Laboratories, and is partially supported by the National Science Foundation grant IRI-92-12597 and the Air Force Office of Scientific Research 91-0347.