Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)

Let Г_r,n—r denote the infimum of all number Г > 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c₁; c₂,…, c_n, there exist integersx ₁,x₂,…,x_n satisfying 0 < Q(x₁+c₁,x₂ + c₂,…,x_n + c_n) ≤(Г|Z > |)^1/n. All the values of Г_r,n—r are known except for г_1,4. Earlier it was shown that 8 ≤Г_1,4 ≤16. Here we improve the upper bound to get Г_1,4 < 12.     

Some remarks about embeddings ofl 1 k in finite-dimensional spaces

The spanX _n of functionsx _i(t)=±1,i=1, …,n, on a setT in the supremum norm is considered. It is proved, for example, thatX _n contains an isometric copy ofl ₁ ^k fork≧cM _n ² /n logn whereM _n is the Rademacher average of {x _i} ₁ ⁿ . This generalizes a result of Pisier for characters. The proof uses a new combinatorial tool.     

The additive subgroup generated by a polynomial

SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x ₁, …,x _n) be a polynomial overC in noncommuting variablesx ₁, …,x _n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a ₁, …,a _n):a ₁, …,a _n ∈I}. Then eitherp(x ₁, …,x _n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=G _R with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {u _t} ofG and anyg∈G, the time spent inC by the {u _t}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g ⁻¹u_tg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].     

Completely monotone multisequences, symmetric probabilities and a normal limit theorem

Let G _n,k be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc _n(β₁,β₂,…, β _k ), β₁,β₂,…, β_k = 0,1,2,…, β₁+β₂ + … +β _k ≤n,c _n(0,0,…, 0) = 1 and whenever β₀ ≤n - (β₁ + β₂ + … + β _k ) where Δc _n(β₁,β₂,…, β _k ) =c _n(β₁ + 1, β₂,…, β _k )+c _n(β₁,β₂+1,…, β _k )+…+c _n (β₁,β₂,…, β _k +1) -c _n(β₁,β₂,…, β _k ). Further, let Π _n,k be the set of all symmetric probabilities on {0,1,2,…,k} ⁿ . We establish a one-to-one correspondence between the sets G _n,k and Π _n,k and use it to formulate and answer interesting questions about both. Assigning to G _n,k the uniform probability measure, we show that, asn→∞, any fixed section {it{c_n}(β₁,β₂,…, β _k ), 1 ≤ Σβ _i ≤m}, properly centered and normalized, is asymptotically multivariate normal. That is, converges weakly to MVN[0, Σ _m ]; the centering constantsc ₀(β₁, β₂,…, β _k ) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex inR ^k.     

Characterization of polynomials and divided difference

For distinct points x₁,x₂,…,x_n in ℛ (the reals), letϕ[x₁, x₂,…,x_n] denote the divided difference ofϕ. In this paper, we determine the general solutionϕ,g: ℛ → ℛ of the functional equationϕ[x₁,x₂,…,x_n] =g(x₁,+ x₂ + … + x_n) for distinct x₁,x₂,…, x_n in ℛ without any regularity assumptions on the unknown functions.     

Disjoint homotopic paths and trees in a planar graph

In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ², a subset {I ₁, …,I _p} of the faces ofG, and pathsC ₁, …,C _k inG, with endpoints on the boundary ofI ₁ ∪ … ∪I _p; find: pairwise disjoint simple pathsP ₁, …,P _k inG so that, for eachi=1, …,k, P _i is homotopic toC _i in the space ℝ²\(I ₁ ∪ … ∪I _p). Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ² and pairwise disjoint setsW ₁, …,W _k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT ₁, …,T _k ofG whereT _i (i=1, …, k).     

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.     

Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

A decomposition property of polyhedra

A polyhedronP = {x∈ ℝ ₊ ⁿ ∣d ⩽Ax ⩽e} is said to have the real decomposition property (RDP) if for any positiveT and any realx∈TP, there are positive coefficients λ₁,…, λ _r and integers ¹, …,s ^r ∈P withx = λ _l s ^l + … + λ _r s ^r andT=λ _l + ⋯ + λ _r . We give a constructive proof that this property holds for a polyhedronP iffP is integral. This construction is used to show that some classes of polyhedra have an integral decomposition property. Furthemore, the RDP provides a generalization of the theorem of Birkhoff-von Neumann. So RDP may be used in some scheduling problems on parallel processors with preemptions.     

Finitary reconstruction of a measure preserving transformation

This paper considers thefinitary reconstruction of an ergodic measure preserving transformationT of a complete separable metric spaceX from a single trajectoryx, Tx, …, or more generally, from a suitable reconstruction sequence x=x ₁,x ₂, … withx _i∈X. Ann-sample reconstruction is a functionT _n: X ⁿ⁺¹ →X; the map (·;x ₁, …,x _n)is treated as an estimate ofT(·) based on then initial elements of x. Given a reference probability measureμ ₀ and constantM>1, functionsT ₁,T ₂, … are defined, and it is shown that for everyμ with 1/M≤dμ/dμ ₀≤M, everyμ-preserving transformationT, and every reconstruction sequence x forT, the estimates (·;x ₁, …,x _nconverge toT in the weak topology. For the family of interval exchange transformations of [0, 1] a simple family of estimates is described and shown to be consistent both pointwise and in the strong topology. However, it is also shown that no finitary estimation scheme is consistent in the strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval, even if x is assumed to be a generic trajectory ofT. Supported in part by NSF Grant DMS-9501926.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

Enriched homology and cohomology modules of simiplicial complexes

For a simplicial complex Δ on {1, 2,…, n} we define enriched homology and cohomology modules. They are graded modules over k[x ₁,…, x _n ] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein^* complexes Δ, and also orientable homology manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex. We also put strong vanishing conditions on the enriched homology modules and describe the simplicial complexes we then get. They are block designs and include Steiner systems S(c, d, n) and cyclic polytopes of even dimension. This paper is to a large extent a complete rewriting of a previous preprint, “Hierarchies of simplicial complexes via the BGG-correspondence”. Also Propositions 1.7 and 3.1 have been generalized to cell complexes in [11].     

Comonotone approximation of twice differentiable periodic functions

In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y _i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y _i } _i∈ℤ of points y _i = y _i+2s + 2π such that the function f does not decrease on [y _i , y _i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T _n of order ≤n that changes its monotonicity at the same points y _i ∈ Y as f and is such that

Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f _n } ∪L _p , based on the size of the norms of sums of sub-blocks of the firstn functions. The results are aplied to the study of a.e. convergence of series Σ _n a _n T ⁿ g/ n whenT is anL ₂ -contraction,g∃L ₂ , and {a _n } is an appropriate sequence. Given a sequence {f _n }∪L _p (Ω, μ), 1<p≤2, of independent centered random variables, we study conditions for the existence of a set ofx of μ-probability 1, such that for every contractionT on andg∈L ₂ (π), the random power series Σ _n f _n (x)T ⁿ g converges π-a.e. The conditions are used to show that for {f _n } centered i.i.d. withf ₁∃L log⁺ L, there exists a set ofx of full measure such that for every contractionT on andg∃L ₂ (π), the random series Σ _n f _n (x)T ⁿ g/n converges π-a.e. We use Menchoff's own spelling of his name in the papers he wrote in French. Dedicated to Hillel Furstenberg upon his retirement     

Ergodic averages on spheres

LetU ₁,U ₂, …,U _n denoten commuting ergodic invertible measure preserving flows on a probability space (X,Σ,m). LetS _r denote the sphere of radiusr inR ⁿ , and α_r the rotationally invariant unit measure onS _r. WriteU ^tx to denote x wheret=(t ₁ …,t_n). Define the ergodic averaging operator . This paper shows that these averages converge for eachf ∈L ^p(X), p>n/(n−1), n≥3. This is closely related to the work on differentiation by E. M. Stein, S. Wainger, and others. Because of their work, the necessary maximal inequality transfers quite easily. The difficulty is to show that we have convergence on a dense subspace. This is done with the aid of a maximal variational inequality. Partially supported by NSF grant DMS-8910947.     

Longest Paths and Longest Cycles in Graphs with Large Degree Sums

 Let p(G) and c(G) denote the number of vertices in a longest path and a longest cycle, respectively, of a finite, simple graph G. Define σ₄(G)=min{d(x ₁)+d(x ₂)+ d(x ₃)+d(x ₄) | {x ₁,…,x ₄} is independent in G}. In this paper, the difference p(G)−c(G) is considered for 2-connected graphs G with σ₄(G)≥|V(G)|+3. Among others, we show that p(G)−c(G)≤2 or every longest path in G is a dominating path. Received: August 28, 2000 Final version received: May 23, 2002     

A distortion theorem for the class of typically real functions

The paper continues the studies of the well-known class T of typically real functions f(z) in the disk U = {z:|z| < 1}. The region of values of the system {f(z ₀), f(z ₀), f(r ₁), f(r ₂),…, f(r _n )} in the class T is investigated. Here, z ₀ ∈ U, Im z ₀ ≠ 0, 0 < r _j  < 1 for j = 1,…, n, n ≥ 2. As a corollary, the region of values of f′(z ₀) in the class of functions f ∈ T with fixed values f(z ₀) and f(r _j ) (j = 1,…, n) is determined. The proof is based on the criterion of solvability of the power problem of moments. Bibliography: 10 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 33–45.