On tests of hypotheses about treatment effects and treatmentX places interactions,in two heteroscedastic experiments

Summary Suppose we have two independent experiments conducted with a set of ‘t’ treatments each, at different places. This paper deals with two interesting problems of testing of hypotheses associated with these experiments. The first problem deals with the test of the equality of the respective treatment effects in the two experiments. The second problem is concerned with the testing of the equality of treatment into places interactions. Though we assume normality, the variance σ ₁ ² in one experiment is assumed different from the variance σ ₂ ² in the other experiment. When no information is available aboutR=σ ₁ ² /(σ ₁ ² +σ ₂ ² ) except that 0≦R≦1, tests known as ‘bilateral tests’ are proposed in the literature, to test the hypotheses mentioned above. This paper studies some important small sample properties of these bilateral tests. More specifically we study the probability of the first and second kind of error of these bilateral tests as a function ofR. When the two experiments have the same number of observations on each treatment, the bilateral test is shown to control the first kind of error. Fort=1,2, the level of these tests is a strictly convex function ofR and hence these tests can be very conservative. Some power properties of these tests are also obtained. Two tests which are equivalent to the bilateral tests for large sample sizes, and which are superior to the bilateral tests for small sample sizes, are obtained.     

Global weak solutions of the wave map system to compact homogeneous spaces

We construct global weak solutions of the wave map problem in the class of maps with bounded energy, with values in an arbitrary compact homogeneous space, for arbitrary initial data inH _c ¹ . The proof proceeds by a ‘penalty approximation’ method, which generalizes J.Shatah's [5] argument for the case of maps with values in then-sphere. Supported in part by a grant from the National Science Foundation and Science Alliance.     

Roof duality,complementation and persistency in quadratic 0–1 optimization

The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx _i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.     

Good points for diophantine approximation

Given a sequence (x _n ) _n=1^∞ of real numbers in the interval [0, 1) and a sequence (δ _n ) _n=1^∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x _n | < δ _n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x _n ) _n=1^∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ _n ) _n=1^∞ converges to zero. On the other hand, for any sequence of positive numbers (δ _n ) _n=1^∞ tending to zero, there is a well distributed sequence (x _n ) _n=1^∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.     

Playing with admissibility spectra

Jensen showed that any countable sequenceA ofA-admissibles is the initial part of the admissibility spectrum of a realR. His construction generalizes straightforwardly to Σ _n -admissibles. This adaptation makes admissibles not inA R-inadmissible. We strengthen Jensen’s theorem by requiring that Σ _n (A)-admissibles not inA be Σ _m (R)-admissible or Σ _m (R)-non-projectible, form <n. The contents of this paper formed part of the author’s doctoral thesis. He would like to thank Professor Sy Friedman for his capable advising, and the NSF and MIT for their financial support.     

Sur Prolungamento Delle Funzioni Della Prima Classe

It is proved that a setH inR ⁿ has Lebesgue null measure if any function which is the restriction toH of a Baire function defined inR ⁿ is the restriction toH of a derivative of an interval function.

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Ricerca eseguita nell’ambito del progetto “Analisi Reale” cofinanziato MURST.     

Half-factorial-domains

LetR be a commutative domain with 1. We termR an HFD (Half-Factorial-Domain) provided the equality Π _i=1 ⁿ χ_i=Π{_f=1/^m}y _f impliesm=n, whenever thex’s and they’s are non-zero, non-unit and irreducible elements ofR. The purpose of this note is to study HFD’s, in particular, Krull domains that are HFD’s, and to provide examples of HFD’s, that contradict a conjecture of Narkiewicz.     

Intersection properties of boxes. Part I: An upper-bound theorem

LetP be a family ofn boxes inR ^d (with edges parallel to the coordinate axes). Fork=0, 1, 2, …, denote byf _k (P) the number of subfamilies ofP of sizek+1 with non-empty intersection. We show that iff _r (P)=0 for somer≦n, then where thef _k (n, d, r) are ceg196rtain definite numbers defined by (3.4) below. The result is best possible for eachk. Fork=1 it was conjectured by G. Kalai (Israel J. Math.48 (1984), 161–174). As an application, we prove a ‘fractional’ Helly theorem for families of boxes inR ^d .     

Planar algebra of the subgroup-subfactor

We give an identification between the planar algebra of the subgroupsubfactor R ⋊ H ⊂ R ⋊ G and the G-invariant planar subalgebra of the planar algebra of the bipartite graph ★ _n , where n = [G: H]. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant of R ⋊ H ⊂ R ⋊ G in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor R ^G ⊂ R ^H and the G-invariant planar subalgebra of the planar algebra of the ‘flip’ of ★ _n .     

Finite sets on curves and surfaces

A complete proof is given for Schnirelmann’s theorem on the existence of a square inC ² Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface inR ⁿ,C ³-diffeomorphic toS ⁿ⁻¹, there exist 2n points which are the vertices of a regular 2 ⁿ -cellC _n. 2. Every planeC′ Jordan curve can beC′ approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (anglesπ not excluded). 3. On every planeC ² curve there exist 5 distinct points which are the vertices of an axially symmetric pentagon with given base anglesa, π/2≦a<π. (The angle at the vertex on the axis of symmetry might beπ). Research supported by Grant AF-AFOSR-664-64, Air Force Office of Scientific Research.     

Determination of the poles of the topological zeta function for curves

Tof ∈ℂ[x ₁…,x _n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f ⁻¹{0} inf ⁻¹{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x ₁,x ₂]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution. The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research N.F.W.O.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

Construction of wavelet sets with certain self-similarity properties

A measurable set Q ⊂ R ⁿ is a wavelet set for an expansive matrix A if F ⁻¹ _(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle [7] discovered the existence of wavelet sets in R _n associated with any real n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are, up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with the matrix B = A ^t. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes.     

Costruzione di una soluzione fondamentale per operatori differenziali ( p n )-ipoellittici,p=1, …, n

Riassunto è costruita una soluzione fondamentale per un operatore differenziale ( _p ⁿ )-ipoellittico,p=1, …,n, con coefficienti costanti. Tale soluzione fondamentale soddisfa a condizioni di Gevrey nei due semi-spazi {x _n <0}, {x _n >0}.

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Summary A constructive procedure of a fundamental solution to a linear ( _p ⁿ )-hypoelliptic,p=1, …,n, differential operator inR ⁿ with constant coefficients is exhibited. Such a fundamental solution satisfies Gevrey conditions in the two half-spaces {x _n >0}, {x _n <0}.

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Lavoro eseguito nell’ambito dell’attività del Gruppo Nazionale per l’Analisi Funzionale e le sue Applicazioni.     

Strong approximation ofGSBV functions by piecewise smooth functions

Let Ω be an open and bounded subset ofR ⁿ with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R ^m ) whose jump setS _vis essentially closed and polyhedral and which are of classW ^{k, ∞} (S _v,R ^m) for every integerk are strongly dense inGSBV ^p(Ω,R ^m ), in the sense that every functionu inGSBV ^p(Ω,R ^m ) is approximated inL ^p(Ω,R ^m ) by a sequence of functions {v _k{_j∈N with the described regularity such that the approximate gradients ∇v _jconverge inL ^p(Ω,R ^nm ) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS _{v j} converges to the (n−1)-dimensional measure ofS _u. The structure ofS _v can be further improved in casep≤2.

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Sunto Sia Ω un aperto limitato diR ⁿ con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R ^m ) con insieme di saltoS _v essenzialmente chiuso e poliedrale che sono di classeW ^{k, ∞} (S _v,R ^m ) per ogni interok sono fortemente dense inGSBV ^p(Ω,R ^m ), nel senso che ogni funzioneu∈GSBV ^p(Ω,R ^m ) è approssimata inL ^p(Ω,R ^m ) da una successione di funzioni {v _j}_j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v _jconvergono inL ^p(Ω,R ^nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS _{v j}converge alla misura (n−1)-dimensionale diS _u. La struttura diS _vpuó essere migliorata nel caso in cuip≤2.

     

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].     

A modulus of smoothness on the unit sphere

For functions onS ^d−1 (the unit sphere inR ^d) and, in particular, forf∈L _p(S ^d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL _p(S ^d−1). Supported by NSERC Grant A4816 of Canada.     

The cauchy problem at simply characteristic points andP-convexity

Summary Let Ω cR ⁿ be an open set and let P be a linear partial differential operator with constant coefficients inR ⁿ. Then Ω is said to be P-convex if for each f ε C^∞(Ω) there is a u ε D′(Ω) such that P(D)u=f. A complete geometric characterization of P-convex sets inR ³ is given when P is of principal type and when Ω has C²-boundary. As a step in the proof one also obtains necessary and sufficient conditions for uniqueness in the local Cauchy problem at simply characteristic points inR ³. The tools are a sophisticated use of the author's uniqueness cones on one hand and his semi-global nullsolutions on the other hand. Hints are given on the difficulties that may be encountered inR ⁿ for the same problem. Entrata in Redazione il 7 giugno 1978.     

A metrical characterization of symmetric polytopes

A family of symmetric projections of polytopes inR ²ⁿ toR ⁿ was given in [10]. By determining the invariantn-vectors of the induced action of these symmetry groups on the exterior algebra Λ _n R ²ⁿ , we show that the symmetric projections have the greatest volume among projections with the same combinatorial type.