Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Legitimate colorings of projective planes

For a projective plane _n of ordern, let( _n ) denote the minimum numberk, so that there is a coloring of the points of _n ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 ( _n ) 8 for every projective plane _n of ordern. Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation     

Degrees of freedom versus dimension for containment orders

Given a family of sets L, where the sets in L admit k degrees of freedom, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)nk log n where k is fixed and n is large.Research supported in part by Allon Fellowship and by a grant from Bat Sheva de Rothschild Foundation.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

Solving systems of polynomial inequalities over a real closed field in subexponential time

Let _m= (₁,..., _m) denote an ordered field, where _i+1>0 is infinitesimal relative to the elements of _i, 0 < –i < m (by definition, ₀= ). Given a system of inequalities f₁ > 0, ..., f_s > 0, f_s+1 0, ..., f_k 0, where f_{j m} [X₁,..., X_n] are polynomials such that, and the absolute value of any integer occurring in the coefficients of the f_js is at most 2^M. An algorithm is constructed which tests the above system of inequalities for solvability over the real closure of _m in polynomial time with respect to M, ((d)ⁿd₀)^n+m. In the case _m=, the algorithm explicitly constructs a family of real solutions of the system (provided the latter is consistent). Previously known algorithms for this problem had complexity of the order ofM(d d ₀ ^{m 2U(n)} .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 3–36, 1988.     

Proper 2-Covers of PG(3,q), q Even

A k-cover of =PG(3q) is a set S of lines of such that every point is on exactly k lines of S. S is proper if it contains no spread. The existence of proper k-covers of is necessary for the existence of maximal partial packings of q ²+q+1–k spreads of . Here we give the first construction of proper 2-packings of PG(3,q) with q even; for q odd these have been constructed by Ebert.     

Factorization intok-dimensional linear bijections

LetV be a vector space,k withkdimV andS _k{GL(V)|dimV(–1)=k}. ThenS _k generates GL _f (V){GL(V)|V(-1) is finite-dimensional} (with the exception that dimV=2=k and the field is GF2). We study the length problem in GL _f (V) withS _k as set of generators.     

Monotonic subsequences in permutations of n natural numbers

Let S_n be the set of all permutations of the numbers 1, 2,..., n, and letl _n() be the number of terms in the maximal monotonic subsequence contained in S_n. If M[l _n()] is the mean value ofl _n () on S_n, then, for all except a finite number of n, the bound M[l _n()] e n is valid.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 511–514, April, 1973.The author wishes to thank E. M. Nikishin for having posed the problem and for his constant interest in the work.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Computing on-line the lattice of maximal antichains of posets

We consider the on-line computation of the lattice of maximal antichains of a finite poset . This on-line computation satisfies what we call the linear extension hypothesis: the new incoming vertex is always maximal in the current subposet of . In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of , where |P| is the number of elements of the poset, , |MA(P)| is the number of maximal antichains of and (P) is the width of . This is more efficient than the best off-line algorithms known so far.     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

A survey of pseudo (v,k, λ)-designs

This work is an attempt to give a complete survey of all known results about pseudo (v, k, )-designs. In doing this, the author hopes to bring more attention to his conjecture given in Section 6; an affirmative answer to this conjecture would settle completely the existence and construction problem for a pseudo (v, k, )-design in terms of the existence of an appropriate (v, k, )-design.     

On blocking sets in affine planes

Denote by _q an affine plane of order q. In the desarguesian case _q=AG(2,q), q 5(q= p^h, p prime), we prove that the smallest cardinality of a blocking set is 2q–1. In any arbitrary affine plane _q (desarguesian or not) with q5, for any integer k with 2q–1 k(q–1)², we construct a blocking set S with ¦S¦=k. For an irreducible blocking set S of _q we determine the upper bound S [qq]+1. We prove that if _q contains a blocking set S which is irreducible with its complementary blocking set, then necessarily _q=AG(2, 4) and S is uniquely determined. Finally we introduce techniques to obtain blocking sets in AG(2, q) and in PG(2, q).Research partially supported by G.N.S.A.G.A. (CNR)     

On the Projections of Multivalued Maps

This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Indecomposable Sylow 2-Subgroups of Simple Groups

Let S be a Sylow 2-subgroup of a finite simple group and let S=S₁×S₂××S_k be the direct product and each component S_i, i=1,2,...,k is indecomposable. In this article, we prove that each S_i is also a Sylow 2-subgroup of some simple group. Mathematics Subject Classifications (2000) 20E32, 20D20.     

Steepest Descent with Curvature Dynamical System

Let H be a real Hilbert space and let <..,.> denote the corresponding scalar product. Given a function that is bounded from below, we consider the following dynamical system:

Circles and spheres in pseudo-Riemannian geometry

Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For ₀ = +1 or –1, ₁ = +1, –1 or 0 (2–2 ₀+ ₁2n–2–2) and a positive constantk, every circlec inM withg(c, c) = ₀ andg( _c c, _c c) = ₁ k ² is a circle in iffM is an extrinsic sphere. For ₀ = +1 or –1 (–₀n–), every geodesicc inM withg(c, c) = ₀ is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.     

m-Systems and Partial m-Systemsof Polar Spaces

LetP be a finite classical polar space of rankr, withr 2. A partialm-systemM ofP, with 0 m r - 1, is any set (₁), ₂,..., _k ofk ( 0) totally singularm-spaces ofP such that no maximal totally singular space containing _i has a point in common with (_{1 2} ... _k) — _i,i = 1, 2,...,k. In a previous paper an upper bound for ¦M¦ was obtained (Theorem 1). If ¦M¦ = , thenM is called anm-system ofP. Form = 0 them-systems are the ovoids ofP; form =r - 1 them-systems are the spreads ofP. In this paper we improve in many cases the upper bound for the number of elements of a partialm-system, thus proving the nonexistence of several classes ofm-systems.Dedicated to Hanfried Lenz on the occasion of his 80th birthday