Translation Invariant Measure Research Articles

Detection of Perturbation Phases and Developmental Stages in Organisms from DNA Microarray Time Series Data

Available DNA microarray time series that record gene expression along the developmental stages of multicellular eukaryotes, or in unicellular organisms subject to external perturbations such as stress and diauxie, are analyzed. By pairwise comparison of the gene expression profiles on the basis of a translation-invariant and scale-invariant distance measure corresponding to least-rectangle regression, it is shown that peaks in the average distance values are noticeable and are localized around specific time points. These points systematically coincide with the transition points between developmental phases or just follow the external perturbations. This approach can thus be used to identify automatically, from microarray time series alone, the presence of external perturbations or the succession of developmental stages in arbitrary cell systems. Moreover, our results show that there is a striking similarity between the gene expression responses to these a priori very different phenomena. In contrast, the cell cycle does not involve a perturbation-like phase, but rather continuous gene expression remodeling. Similar analyses were conducted using three other standard distance measures, showing that the one we introduced was superior. Based on these findings, we set up an adapted clustering method that uses this distance measure and classifies the genes on the basis of their expression profiles within each developmental stage or between perturbation phases.

Translation-invariant and positive-homogeneous risk measures and optimal portfolio management in the presence of a riskless component

Risk portfolio optimization, with translation-invariant and positive-homogeneous risk measures, leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. This problem was recently treated by the authors for the case when the portfolio does not contain a riskless component. When it does, however, the initial covariance matrix Σ becomes singular and the problem becomes more complicated. In the paper we focus on this case and provide an explicit closed-form solution of the minimization problem, and the condition under which this solution exists. The results are illustrated using data of 10 stocks from the NASDAQ Computer Index.

Translation-invariant and positive-homogeneous risk measures and optimal portfolio management

The problem of risk portfolio optimization with translation-invariant and positive-homogeneous risk measures, which includes value-at-risk (VaR) and tail conditional expectation (TCE), leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. In this paper, we provide an explicit closed-form solution of this minimization problem, and the condition under which this solution exists. The results are illustrated using the data of 10 stocks from NASDAQ/Computers. The distance between the VaR and TCE optimal portfolios has been investigated.

Statistical Inference in Adaptive Group Sequential Trials with the Standardized Mean Difference as Effect Size

We consider the standardized difference of normal means as the outcome measure for comparing two independent groups, say experimental E and control C. This scale- and translation-invariant effect measure enables a convenient specification of a noninferiority margin in applications. Starting with a family of null hypotheses, we derive adaptive group sequential confidence intervals keeping the predefined confidence coefficient. The interval at the end of the trial determines whether and which null hypotheses can be rejected. During the course of the trial, the sample size can be calculated in a completely adaptive way based on the unblinded data of previously performed stages. Concrete rules for sample size updating are provided. Moreover, in each interim analysis, it is possible to change the planning from showing noninferiority to showing superiority or vice versa without affecting the overall type I error. A real data example is worked out in detail and the change in the sample size planning from showing noninferiority to showing superiority is considered during the ongoing trial. In Monte Carlo simulations, we investigate the practical properties of the proposed intervals.

Q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.

Integration on Valuation Fields over Local Fields

We present elements of a theory of translation-invariant integration, measure, and harmonic analysis on a valuation field with local field as residue field, extending work of I. Fesenko. Applications to zeta integrals for two-dimensional local fields are then considered.

Translation-Invariant and Positive-Homogeneous Risk Measures and Optimal Portfolio Management

The problem of risk portfolio optimization with translation-invariant and positive-homogeneous risk measures, which includes value-at-risk (VaR) and tail conditional expectation (TCE), leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. In this paper we provide an explicit closed-form solution of this minimization problem, and the condition under which this solution exists. The results are illustrated using data of 10 stocks from NASDAQ/Computers. The distance between the VaR and TCE optimal portfolios is investigated.

A Nonmeasurable Set from Coin Flips

To motivate the elaborate machinery of measure theory, it is desirable to show that in some natural space Q one cannot define a measure on all subsets of £2, if the measure is to satisfy certain natural properties. The usual example is given by the Vitali set, obtained by choosing one representative from each equivalence class of R induced by the relation x ~ y if and only ifx-yeQ. The resulting set is not measurable with respect to any translation-invariant measure on R that gives nonzero, finite measure to the unit interval [8]. In particular, the resulting set is not Lebesgue measurable. The construction above uses the axiom of choice. Indeed, the Solovay theorem [7] states that in the absence of the axiom of choice, there is a model of Zermelo-Frankel set theory where all the subsets of R are Lebesgue measurable. In this note we give a variant proof of the existence of a nonmeasurable set (in a slightly different space). We will use the axiom of choice in the guise of the wellordering principle (see the later discussion for more information). Other examples of nonmeasurable sets may be found for example in [1] and [5, Ch. 5]. We will produce a nonmeasurable set in the space Q := {0, 1}Z. Translationinvariance plays a key role in the Vitali proof. Here shift-invariance will play a similar role. The shift T : Z -> Z on integers is defined via Tx := x + 1, and the shift r : Q -> Q on elements co e Q is defined via (xco)(x) := co(x 1). We write xA := {xco : co e A] for A c Q.

Adelic Approach to the Zeta Function of Arithmetic Schemes in Dimension Two

This paper suggests a new approach to the study of the fun- damental properties of the zeta function of a model of elliptic curve over a global field. This complex valued commutative approach is a two-di- mensional extension of the classical adelic analysis of Tate and Iwasawa. We explain how using structures which come naturally from the ex- plicit two-dimensional class field theory and working with a new R((X))- valued translation invariant measure, integration theory and harmonic analysis on various complete objects associated to arithmetic surfaces one can define and study zeta integrals which are closely related to the zeta function of a regular model of elliptic curve over global fields. In the two-dimensional adelic analysis the study of poles of the zeta function is reduced to the study of poles of a boundary term which is an integral of a certain arithmetic function over the boundary of an adelic space. The structure of the boundary and function determines the analytic proper- ties of the boundary term and location of the poles of the zeta function, which results in applications of the theory to several key directions of arithmetic of elliptic curves over global fields.

A Breiman Type Theorem for Gibbs Measures

For a translation invariant Gibbs measure on a suitable translation invariant configuration set X ? S G , where G is an amenable group and S is a finite set, we prove the convergence of the Shannon---McMillan---Breiman ratio on a specific subset of "generic" configurations. Provided that the above Gibbs measure exists, we also prove the convergence in the definition of the pressure and the fact that this Gibbs measure is an equilibrium measure.

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

Abstract: We study the stationary distribution of the standard Abelian sandpile model in the box $Lam_n = [-n, n]^d cap d$ for $d ge 2$. We show that as $n o infty$, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in $d$. This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on $d$. In the case $d > 4$, we also make use of Wilson's method.

Testing the master constraint programme for loop quantum gravity: III. models

This is the third paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. In this work, we analyse models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper. These are systems with an gauge symmetry and the complications arise because non-compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the master constraint does not contain the point zero. However, the minimum of the spectrum is of order ℏ2 which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to ℏ normal ordering constants). The physical Hilbert space can then be obtained after subtracting this normal ordering correction.

On a Two‐Temperature Problem for the Klein–Gordon Equation

We consider the Klein-Gordon equation in R n , n 2, with constant or variable coefficients. The initial datum is a random function with a finite mean density of the energy and satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. We also assume that the random function is close to different space-homogeneous processes as xn →± ∞, with the distributions μ±. We study the distribution μt of the random solution at time t ∈ R. The main result is the convergence of μt to a Gaussian translation-invariant measure as t →∞ that means the central limit theorem for the Klein-Gordon equation. The proof is based on the Bernstein room-corridor method and oscillatory integral estimates. The application to the case of the Gibbs measures μ± = g± with two different temperatures T± is given. It is proved that limit mean energy current density formally is −∞ · (0 ,..., 0 ,T + − T−) for the Gibbs measures, and it is finite and equals −C(0 ,..., 0 ,T + − T−) with some positive constant C> 0 for the smoothed solution. This corresponds to the second law of thermodynamics.

Long-range exclusion processes, generator and invariant measures

We show that if $\mu$ is an invariant measure for the long range exclusion process putting no mass on the full configuration, $L$ is the formal generator of that process and $f$ is a cylinder function, then $Lf\in\mathbf{L}^1(d\mu)$ and $\int Lf d\mu=0$. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on $\mathbb{Z}^d$ when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on $\mathbb{Z}$ when the underlying random walk is irreducible and either has zero mean or allows jumps only to the nearest-neighbors.

Is Lebesgue measure the only σ-finite invariant Borel measure?

S. Saks and recently R.D. Mauldin asked if every translation invariant σ-finite Borel measure on R d is a constant multiple of Lebesgue measure. The aim of this paper is to investigate the versions of this question, since surprisingly the answer is “yes and no,” depending on what we mean by Borel measure and by constant. According to a folklore result, if the measure is only defined for Borel sets, then the answer is affirmative. We show that if the measure is defined on a σ-algebra containing the Borel sets, then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant σ-finite Borel measure (in the wider sense) on R d can be non- σ-finite when we restrict it to the Borel sets.

Zero–infinity laws in Diophantine approximation

It is shown that for any translation invariant outer measure M, the M-measure of the intersection of any subset of Rn that is invariant under rational translations and which does not have full Lebesgue measure with the closure of an open set of positive measure cannot be positive and finite. Analogues for p-adic fields and fields of formal power series over a finite field are established. The results are applied to some problems in metric Diophantine approximation.

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary (J) and binary (J1) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model is studied. We completely describe the set of all periodic Gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type IIIλ then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too.

Borel sets which are null or non- [formula omitted]-finite for every translation invariant measure

We show that the set of Liouville numbers is either null or non- σ -finite with respect to every translation invariant Borel measure on R , in particular, with respect to every Hausdorff measure H g with gauge function g. This answers a question of R.D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch–Eggleston numbers, as well as all Borel subgroups of R that are not F σ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.

Markov approximation of homogeneous lattice random fields

We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular, we prove that every translation invariant Borel probability measure μ on the space X of finite-alphabet configurations on ℤd, d≥1, can be weakly approximated by Markov measures μn with supp(μn)=X and with the entropies h(μn)→h(μ). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.

An Ising model with three competing interactions on a Cayley tree

In this paper we consider an Ising model with three competing restricted interactions on the Cayley tree J2(J3). The translation invariant and periodic Gibbs measures for these models are investigated and the problem of the phase transition in these classes is solved.