Geometry of the common dynamics of flipped Pisot substitutions

In this article we study the common dynamics of two different Pisot substitutions σ ₁ and σ ₂ having the same incidence matrix. This common dynamics arises in the study of the adic systems associated with the substitutions σ ₁ and σ ₂. Since the adic systems considered here have geometric realizations given by solutions to graph-directed iterated function systems, we actually study topological and measure-theoretic properties of the solution of those iterated function systems which describe the common dynamics. We also consider generalizations of these results to the nonunimodular case, the case of more than two substitutions and the case of two substitutions with different incidence matrices.     

Estimating the ratio of two scale parameters: a simple approach

We describe a simple approach for estimating the ratio ρ = σ ₂/σ ₁ of the scale parameters of two populations from a decision theoretic point of view. We show that if the loss function satisfies a certain condition, then the estimation of ρ reduces to separately estimating σ ₂ and 1/σ ₁. This implies that the standard estimator of ρ can be improved by just employing an improved estimator of σ ₂ or 1/σ ₁. Moreover, in the case where the loss function is convex in some function of its argument, we prove that such improved estimators of ρ are further dominated by corresponding ones that use all the available data. Using this result, we construct new classes of double-adjustment improved estimators for several well-known convex as well as non-convex loss functions. In particular, Strawderman-type estimators of ρ in general models are given whereas Shinozaki-type estimators of the ratio of two normal variances are briefly treated.     

The structure of invertible substitutions on a three-letter alphabet

We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a fi-nite set S of invertible substitutions such that any invertible substitution can be written as Iw^oQ1^oQ2^o……^oQk,,where Iw is the inner automorphism associated with w, and ajЕS for l≤j≤k. As a consequence, M is thematrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.     

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.     

Intuitionistic Frege systems are polynomially equivalent

In a paper by Cook and Reckhow (1979), it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules. (The rule A/B is derivable if the formula A → B is derivable. The rule A/B is admissible if for all substitutions σ, if σ(A) is derivable, then σ(B) is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 129–146.     

Pentadiagonal Oscillatory Matrices with Two Spectrum in Common

We denote the spectrum of an square matrix A by σ(A), and that of the matrix obtained by deleting the first i rows and columns of A by σ_i(A). It is known that a symmetric pentadiagonal oscillatory (SPO) matrix may be constructed from σ, σ₁ and σ₂. The pairs σ, σ₁ and σ₁, σ₂ must interlace; the construction is not unique; and the conditions on the data which ensure that A is oscillatory are extremely complicated. Given one SPO matrix A, the paper shows that operations may be applied to A to construct a family of such matrices with σ and σ₁ in common. Moreover, given one totally positive (TP) matrix A, we construct a family of TP matrices with σ, σ₁ and σ₂ in common.     

Higher dimensional extensions of substitutions and their dual maps

Given a substitution σ ond letters, we define itsk-dimensional extension,E _k (σ), for 0≤k≤d. Thek-dimensional extension acts on the set ofk-dimensional faces of unit cubes inR ^d with integer vertices. The extensions of a substitution satisfy a commutation relation with the natural boundary operator: the boundary of the image is the image of the boundary. We say that a substitution is unimodular (resp. hyperbolic) if the matrix associated to the substitution by abelianization is unimodular (resp. hyperbolic). In the case where the substitution is unimodular, we also define dual substitutions which satisfy a similar coboundary condition. We use these constructions to build self-similar sets on the expanding and contracting space for an hyperbolic substitution.     

Completions of local morphisms and valuations

In this note we study injective local morphisms of local excellent domains. In particular we are interested in the problem when the –adic topology onS restricts to a topology on R that is linearly equivalent to the –adic topology. Using a valuative criterion, we prove this in case R is analytically irreducible and is essentially of finite type, and we recover and extend a weak version of Gabrielov's rank condition. Received July 30, 1999; in final form November 16, 1999 / Published online December 8, 2000     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.     

p-Energia in spazi metrici generalizzati

Given a setS and a function σ:S x S→[0, +∞[ such that σ(x, x)=0, we define the generalizedp-energy of a curve γ: [a, b]→S, so that, ifS is a Hilbert space and σ(x, y)=‖x−y‖ we obtain . Sufficient conditions for the equality of the defined energies are also given. Moreover the case in whichS is a set of measurable parts of ℝⁿ and σ_r is a family of functions used in order to study the generalized minimizing motions, is discussed.

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Conferenza tenuta il 30 ottobre 1995     

Multiphasic Individual Growth Models in Random Environments

The evolution of the growth of an individual in a random environment can be described through stochastic differential equations of the form dY _t  = β(α − Y _t )dt + σdW _t , where Y _t  = h(X _t ), X _t is the size of the individual at age t, h is a strictly increasing continuously differentiable function, α = h(A), where A is the average asymptotic size, and β represents the rate of approach to maturity. The parameter σ measures the intensity of the effect of random fluctuations on growth and W _t is the standard Wiener process. We have previously applied this monophasic model, in which there is only one functional form describing the average dynamics of the complete growth curve, and studied the estimation issues. Here, we present the generalization of the above stochastic model to the multiphasic case, in which we consider that the growth coefficient β assumes different values for different phases of the animal’s life. For simplicity, we consider two phases with growth coefficients β ₁ and β ₂. Results and methods are illustrated using bovine growth data.     

Intermittency and regularized Fredholm determinants

We consider real-analytic maps of the interval I=[0,1] which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the associated Perron-Frobenius operator ℳ has a continuous and residual spectrum contained in the line-segment σ _c =[0,1] and a point spectrum σ _p which has no points of accumulation outside 0 and 1. Furthermore, points in σ _p −{0,1} are eigenvalues of finite multiplicity. We construct a regularized Fredholm determinant d(λ) which has a holomorphic extension to λ∈ℂ−σ _c and can be analytically continued from each side of σ _c to an open neighborhood of σ _c −{0,1} (on different Riemann sheets). In ℂ−σ _c the zero-set of d(λ) is in one-to-one correspondence with the point spectrum of ℳ. Through the conformal transformation the function d∘λ(z) extends to a holomorphic function in a domain which contains the unit disc. Oblatum 10-X-1996 & 31-I-1998 / Published online: 14 October 1998     

A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes σ_λ and σ_μ in the quantum cohomology of a Grassmannian, we construct a partition ν, depending on λ and μ, such that σ_ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ_λ⋆σ_μ. To do this, we show that for any two partitions λ and μ, contained in a k × (n − k) rectangle and such that the 180^∘-rotation of one does not overlap the other, there is a third partition ν, also contained in the rectangle, such that the Littlewood-Richardson number c _λμ ^ν is 1.     

Any two irreducible Markov chains of equal entropy are finitarily Kakutani equivalent

We show here that any two finite state irreducible Markov chains of the same entropy are finitarily Kakutani equivalent. By this we mean they are orbit equivalent by an invertible measure preserving mapping that is almost continuous and monotone in time when restricted to some cylinder set. Smorodinsky and Keane have shown that any two irreducible Markov chains of equal entropy and period are finitarily isomorphic. Hence, all that is necessary to obtain our result is to show that for every entropy h > 0 and period p ∈ ℕ there exists two irreducible Markov chains σ ₁, σ ₂ both of entropy h, where (1) σ ₁ is mixing (2) ς ₂ has period p and (3) σ1 and σ ₂ are finitarily Kakutani equivalent.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Affine Systems: Asymptotics at Infinity for Fractal Measures

We study measures on ℝ ^d which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures μ induced by a finite family of affine mappings in ℝ ^d (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure μ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when μ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of μ for paths at infinity in ℝ ^d . We show how properties of μ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in ℝ ^d , in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.      

On the real exponential field with restricted analytic functions

The model-theoretic structure (ℝ_an, exp) is investigated as a special case of an expansion of the field of reals by certain families ofC ^∞-functions. In particular, we use methods of Wilkie to show that (ℝ_an, exp) is (finitely) model complete and O-minimal. We also prove analytic cell decomposition and the fact that every definable unary function is ultimately bounded by an iterated exponential function. An erratum to this article is available at .     

Additive maps between standard operator algebras compressing certain spectral functions

We characterize the surjective additive maps compressing the spectral function Δ（·） between standard operator algebras acting on complex Banach spaces, where Δ（·） stands for any one of nine spectral functions σ（·）, σl（·）, σr（·）,σl（·） ∩ σr（·）, δσ（·）, ησ（·）, σap（·）, σs（·）, and σap（·） ∩ σs（·）.     

A unified characterization of reproducing systems generated by a finite family,II

By a “reproducing” method forH =L ²(ℝ ⁿ ) we mean the use of two countable families {e _α : α ∈A}, {f _α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e _α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e _α >:f _α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ ⁿ . Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices.     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that and where ω(f,t)_p is the 1st-order modulus of continuity of f in L^p-metric which is defined in a way different than in the classical case of n=2. In Memory of Professor M. T. Cheng Project supported by the NSF of China under the grans # 19771009.