Transformation Theorem Research Articles

The Central Limit Theorem for Piecewise Linear Transformations

The purpose of the present paper is to give central limit theorems for piecewise linear transformations ([6]), which are generalizations of /^-transformations and which belong to a class of number-theoretical transformations with dependent digits (cf. [4]). The central limit theorems for ones with independent digits are studied by many authors ([!]> [7] etc.). However, the cases of dependent digits seem to be not studied. These cases are more complicated than the cases of independent digits. In [2] it is shown that the ^-transformations have Ornstein's weak Bernoulli property. Then it is easy to see by an analogous way to [2] that our transformations also satisfy the weak Bernoulli condition. Therefore Ornstein and Friedman's theorem implies that the natural extensions of our transformations are isomorphic to the Bernoulli shifts. But we never know how to construct their Bernoulli generators. Hence the classical central limit theorems for the Bernoulli shifts imply no concrete result for our transformations. We modify the method, which is used in [2] to prove the weak Bernoulli property of jS-transforma tions, to show that the natural generators of piecewise linear transformations satisfy Rosenblatt's strong mixing condition. Thus we obtain central limit theorems. By virtue of the good properties of our generators, we obtain concrete results, namely if / is of bounded variation or Holder continuous, we get the central limit

Strong Uniform Distributions and Ergodic Theorems

Let $G$ and $H$ be locally compact $\sigma$-compact abelian groups, $\mathcal {A}$ a mapping from $G$ to $H$, and $\{ {\mu _n}\} _{n = 1}^\infty$ a sequence of measures on $G$. We define the notions: “$\mathcal {A}$ is a uniform distribution with respect ot $\{ {\mu _n}\}$” and “$\mathcal {A}$ is a strong uniform distribution". We give a number of examples of these notions and derive some general individual ergodic theorems for measure-preserving transformations with discrete spectrum.

Strong uniform distributions and ergodic theorems

Let G G and H H be locally compact σ \sigma -compact abelian groups, A \mathcal {A} a mapping from G G to H H , and { μ n } n = 1 ∞ \{ {\mu _n}\} _{n = 1}^\infty a sequence of measures on G G . We define the notions: “ A \mathcal {A} is a uniform distribution with respect ot { μ n } \{ {\mu _n}\} ” and “ A \mathcal {A} is a strong uniform distribution". We give a number of examples of these notions and derive some general individual ergodic theorems for measure-preserving transformations with discrete spectrum.

Further transformable nth order differential equations with transition points of particular order m

Further consideration is given to properties of the Stokes phenomenon associated with a class of nth order differential equations with transition points of order m (a rational fraction) at the origin. The systematic vanishing of the Stokes multipliers leads to the discovery of sets of equations that are transformable from certain specific values of m to particular lower values of m. A series of theorems in the theory of congruences is first proved, from which the main transformation theorem is then deduced. The theory is ifiustrated by various numerical examples.

A transformation theorem for one-dimensional F-expansions

If f is a monotone function subject to certain restrictions, then one can associate with any real number x between zero and one a sequence { a n ( x)} of integers such that x=f(a 1(x) + f(a 2(x) +f(a 3(x) +…))) . In this paper properties of the function F defined by Fx=g(a 1(x) + g(a 2(x) +g(a 3(x) +…))) , where g is any function satisfying the same restrictions as f, are discussed. Principally, F is found to be useful in finding stationary measures on the sequences { a n ( x)}.

Weak Convergence of Certain Vectorvalued Measures

There are two kinds of vectorvalued measures which are involved in the theory of weakly stationary processes: orthogonal (Hilbert space valued) and multiplicative (projection-valued) measures. For both classes we show that weak convergence is equivalent with the convergence of integrals over bounded continuous functions. Moreover we prove continuity theorems for the Fourier transformation as well as for the Laplace transformation of such measures.

Inclusion theorems for matrix transformations

Inclusion theorems for matrix transformations

A fixed point theorem for transformations whose iterates have uniform Lipschitz constant

A fixed point theorem for transformations whose iterates have uniform Lipschitz constant

Remarks on the Isomorphism Theorems for Weak Bernoulli Transformations In General Case

Remarks on the Isomorphism Theorems for Weak Bernoulli Transformations In General Case

Comments on the Scaling Behavior of Currents

Necessary and sufficient conditions in order for all components of a local operator (such as a current, for example) to have the same dimension are given and discussed. In addition, we then note that exact scale invariance is in an apparent formal contradiction with a definite scaling behavior for the currents. The solution of this contradiction is then traced back to the fact that in scale-invariant theories, the Schwinger term must be quadratically divergent. We finally conclude by briefly discussing Coleman's theorem for scale and conformal transformations.

On broken conformal invariance

The equal-time commutator algebra satisfied by the generators of the broken conformal group is derived. These expressions are then used to discuss Coleman's theorem for dilatations and conformal transformations and to give necessary and sufficient conditions in order for all the components of a local operator to have the same dimension.

A Riesz representation theorem

Introduction. This paper deals with a Riesz representation theorem in the following setting: Let H be a compact Hausdorff space, let E and be locally convex Hausdorff topological vector spaces over the real or complex field. Let C(H, E) denote the continuous functions from H into E with the topology of uniform convergence. The purpose of this paper is to improve certain integral representation theorems for continuous linear transformations T from C(H, E) into The well-known Riesz representation theorem [10] gives a Stieltjes integral representation for T when H is a closed interval and when E and are the real numbers. There have been many generalizations of this theorem in the literature, and there have been two essentially different approaches giving rise to two different kinds of representation theorems. In one approach (see [1], [2], [5], [8], [9], [11]) y'T is written as an integral where y' is in the topological dual of The integral converges in the weak topology with a measure defined on the Borel sets with values in L(E, F), the space on continuous linear transformations of E into F. Under various additional assumptions on E, F, or T, T is written as an integral. (For example, this can be done if T is assumed to be compact or is assumed to be reflexive.) The most general of these theorems is due to Swong [11, Theorem 6.1, p. 283]. Another approach in the literature can be found in the papers [6], [12], [13], [14]. In these papers the method is of a constructive nature and T is written as an integral, and the convergence of the integral is in the e00 topology of F (norm topology in the case is a normed space), and T is thought of as a mapping of E into F by the canonical imbedding of into F. This is done by restricting the class of sets the measure is defined on. In [12] the class of sets is the intervals where H = [0, I], and E and are assumed to be normed spaces. In [6] and [13] the class of sets is the ring generated by compact Gj's. Recently in [14] Tucker and Wayment have been able to extend the measure to a larger proper subring of the Baire sets. In this paper it is shown that the restriction of the measure in [61, [12], [13], [14] was not necessary, and in fact T can be represented as an integral with convergence in the e topology with the measure defined on the Borel sets. This result thus implies and strengthens

Inclusion theorems for section-bounded matrix transformations

Inclusion theorems for section-bounded matrix transformations

Poznámka k platnosti věty o substituci u zobecněných integrálů ve smyslu Cauchyho hlavní hodnoty

Poznámka k platnosti věty o substituci u zobecněných integrálů ve smyslu Cauchyho hlavní hodnoty

Y-Δ transformation theorem for reliability-function calculations with application to nonseparable structures

A method is proposed to solve complex system-reliability problems by successively reducing the structural model network to more simple equivalent configurations. Transfiguration theorems are given, valid when the component-failure probabilities are not too high.

Combinatorial Theorem for Graphs on a Lattice

Several problems in lattice statistical mechanics, such as the spin-½ Ising problem and the monomer-dimer problem, can be formulated in terms of the p-generating function for the weak subgraphs of a regular lattice. This paper presents an algebraic transformation theorem which allows the p-generating function for the weak subgraphs of a lattice to be determined from the p-generating function for the far less numerous subset consisting of the closed weak subgraphs. This result will be especially useful in reducing the labor required to obtain exact finite series for various problems. The theorem also enables one to give a straightforward proof of the Ising susceptibility graph theorem due to Sykes.

The theory of absorption in aggregated media

The question of the applicability of the 'diffusion analysis' to transient flows in unsaturated aggregated media has been, till now, an open one. It has been uncertain whether significant deviations from the predictions of the diffusion analysis occur as a consequence of local disequilibrium. The present paper develops the theory of absorption in aggregated media and so provides a means of studying this question. An aggregated medium is regarded as one made up of macroporosity, through which flow on the Darcy scale occurs, and microporosity, which is free to exchange water with the macroporosity. Absorption in such a medium is analysed with the aid of two extreme models of macropore flow, one a wet-front model and the other a linearized model (the mathematical details of which are developed elsewhere). The wet-front analysis leads to an integral equation that is solved by means of the Faltung theorem of the Laplace transformation. It is found that there is little difference between the solutions for the two models. The wet-front model is chosen for further study since it appears to be more realistic and is more versatile. Various models of absorption into the microporosity are studied, and are found to lead to essentially similar results. All models (and general physical considerations) indicate that the aggregated medium behaves initially as a classical medium with sorptivity equal to that of the macroporosity S; and that the apparent sorptivity increases with time, ultimately approaching a limiting value (> S). The cumulative uptake into the microporosity appears to be proportional to time initially, but is finally proportional to (time)1/2. The characteristic time of absorption into the microporosity tends to be small for most media of interest. Ln such cases, deviations from the diffusion analysis due to aggregation occur only during a short initial period. These considerations hold also for other transient phenomena such as infiltration and capillary rise in various geometries. It is concluded that the diffusion analysis may usually be applied to aggregated media. The present work provides means of analysing the exceptional cases where this is not so, such as media containing large aggregates of low sorptivity.

A transformation theorem on spectral measures

A transformation theorem on spectral measures

Spectral analysis of digitized seismic data

abstract Techniques of spectral analysis used for digitized data are discussed in this article from a nonstatistical viewpoint. By a generalization of the theory of linear equations and a truncation of Fourier series, a unique relationship similar to the Fourier Transform Theorem for continuous functions can be derived. Application of this theorem to Fourier series shows how aliasing of frequency occurs. Further application to digitized Fourier transforms indicates loss of aperiodicity in general. Depending upon the choice of fundamental frequencies, we will be able to perform data stacking. Some simple examples shown in the text reveal dependence of phase spectra on different modes of digitization. The phenomenon is associated with analysis of discontinuous functions by a digital method. Only deterministic investigations are conducted in this article.

Limitation Theorems for Certain Non-Negative Triangular Matrix Transformations

Limitation Theorems for Certain Non-Negative Triangular Matrix Transformations