Integration by Parts for the Denjoy-Bochner Integral

On the universal completion of pointfree function spaces

We prove that if the space $Y$ of Henstock-Lebesgue integrable functions from a compact real interval to a Banach space $E$ is normed by the Alexiewicz norm and ordered by a regular cone $E_+$, then the $E_+$-valued functions of $Y$ form a regular order cone of $Y$. This property is shown to hold also when $Y$ is the space of Henstock-Kurzweil integrable functions if $E$ is weakly sequentially complete and $E_+$ is normal. As an application of the obtained results we prove existence and comparison results for least and greatest solutions of nonlocal implicit initial-value problems of discontinuous functional differential equations containing non-absolutely integrable functions.

Metric Projections in Spaces of Integrable Functions

We characterize all (signed) measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and $BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{1}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. As a consequence of our characterizations, an old issue raised in Meyers-Ziemer [MZ] is resolved by constructing a locally integrable function $f$ such that $f$ belongs to $BV(\mathbb{R}^n)^{*}$ but $|f|$ does not. Moreover, we show that the measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ coincide with the measures in $\dot W^{1,1}(\mathbb{R}^n)^*$, the dual of the homogeneous Sobolev space $\dot W^{1,1}(\mathbb{R}^n)$, in the sense of isometric isomorphism. For a bounded open set $\Omega$ with Lipschitz boundary, we characterize the measures in the dual space $BV_0(\Omega)^*$. One of the goals of this paper is to make precise the definition of $BV_0(\Omega)$, which is the space of functions of bounded variation with zero trace on the boundary of $\Omega$. We show that the measures in $BV_0(\Omega)^*$ coincide with the measures in $W^{1,1}_0(\Omega)^*$. Finally, the class of finite measures in $BV(\Omega)^*$ is also characterized.

The Kac-Baker model describes a 1-dim. classical lattice spin system with exponentially fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to investigate the phenomenon of phase transition in systems with weak long-range interactions like van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of Fredholm determinants of two transfer operators and hence is a meromorphic function. One of the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator acting in some Banach space of holomorphic observables of the system. In this paper we show how the Kac operator can be explicitly related basically through the Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many non-trivial zeros on the real axis. In a special case we can show the existence of also infinitely many trivial zeros on the line Re s = ln 2 in the complex s-plane. Hence some kind of Riemann hypothesis seems to hold for this dynamical zeta function.

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

In this article we characterize certain ultradifferential operators by the condition of being local. First, we examine the continuity properties of local linear operators on spaces of ultradifferentiable functions in the sense of Beurling and of Roumieu. Next, a structure theorem for vector-valued ultradistributions with support at the origin is proved. This result leads to a representation theorem for continuous local operators from spaces of ultradifferentiable functions into various spaces of ultradistributions. In combination with the continuity results we thus obtain in many cases the desired characterization.

A strict dual representation for L1( μ, X)

Universal approximation capability is an important research topic in artificial neural networks. The purpose of this study is to investigate universal approximation capability of a single hidden layer feed forward multiple 2π-periodic approximate identity neural networks in two function spaces. We present the notion of multiple 2π-periodic approximate identity. With respect to this notion, we prove two theorems in the space of continuous multivariate 2π-periodic functions. The second theorem shows that the above networks have universal approximation capability. The proof of the theorem uses a technique based on the notion of epsilon-net. Moreover, we discuss the universal approximation capability of the networks in the space of Lebesgue integrable multivariate 2π-periodic functions. The results of this study will be able to extend the standard theory of the universal approximation capability of feedforward neural networks.

We consider the problem of determining the Fourier integral in the Hilbert space of square integrable func- tions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the Hilbert space of almost periodic functions. Scalar product for different Hilbert spaces defined at the intersection of these spaces, which contains only one zero element. Therefore, the Fourier integral is not defined in the Hilbert space of square integrable functions with nonzero norm.

The problem of creation and investigation of parallel Monte Carlo algorithms for solving LargeScale Air Pollution Problems is studied. We consider a model of Air pollution transport. This model includes diffusion, advection (drift due to the wind), deposition and chemical reactions. For obtaining the initial and boundary conditions of the model it is necessary to deal with Noisy Data. The work consists of three parts. In the first part we obtain a practical efficient Monte Carlo method for estimating the optimum amount of smoothing from the noisy data. In the second part a special Monte Carlo techniques (Monte Carlo with frozen velocities) for calculation of concentrations is developed and studied. An error analysis of this method is realized. The last part of the work deals with parallel realization of Monte Carlo methods on distributed transputer system. A numerical results for estimation of the Speed-Up and Efficiency are obtained. INTRODUCTION Monte Carlo methods are a powerful tool in many fields of mathematics and engineering. It is known that these methods have proved to be very efficient in solving multidimensional problems in composite domains [1,2,3]. Moreover, it is shown that for some problems(including one dimensional problems) in the corresponding functional spaces Monte Carlo methods have better speed of convergence than the optimal deterministic methods in such functional spaces [4,5]. * Supported by the Ministry of Science and Education of Bulgaria, Grand I 210 /1992. Transactions on Information and Communications Technologies vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 480 Applications of Supercomputers in Engineering An important advantage of these algorithms is that Monte Carlo ones allow to find directly the unknown functional of the solution of the problem with a number of operations, necessary to solve the problem in one point of the domain [1,6]. 1 PRELIMINARIES Our approach is based on an integral representation of the solution of the problem. The Fredholm integral equation t u(r, t) = jdtj k(r', f; r, t)u(r', t')dr' + f(r, t), (1) 0 G where re G C R*; t e [0,T]or (x,x')u(x')dx' + f(r,t), (2) D (« = Ku + /) where x = (r, t) £ D = G x [Q,T] and K is an integral operator, is studied. It is assumed that u, / G L, K G [L — •> L], where Lisa Banach space of integrable functions. Monte Carlo method for evaluation of linear functionals of the solution of the following type /t(z)w(z)dz = (12, A) (3) D is considered. The adjoint equation %* = JT w* 4k (4) will be used. In (4) w*, h G L*, K* G [L* — » L*], L* is the adjoint functional space and K* is an adjoint operator. For many applications L = LI and II / lk= / I /(*) I dx; D \\K\\L,<su.p f\k(x,x')\dx' (5) X J D In the case h(x) G Z/oo, sence L\* = L^ and Transactions on Information and Communications Technologies vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 Applications of Supercomputers in Engineering 481 || A||z_=sup|/t(z) |,zED. (6) It is assumed that || K* ||< 1,where n is a natural number, because the studied method uses the Neuman series CO « = £#'/• (7) i=0 But the condition || KTM ||< 1 is not very strong, since as shown by K.Sabelfeld [7] it is possible to construct Monte Carlo method when von Neuman series does not converge. Using method of analytical extension of resolvent through change of spectral parameter [7,p.60] it is possible to create convergent method when von Neuman series does not converge or to accelerate the rate of convergence when it converges slowly. It is easy to show that J = (u,h) = (/,«•). (8) So, consider Monte Carlo method for evaluation of functional (8).(One can see that when fc(x,xi) = 0we have a problem of evaluation of integrals.) Let £ £ D be a random point with a density p(x) and let there be n realizations of the random point £,-(i — 1,2, ...,ra). Let a random variable #(£) be defined in D, such that E6(t) = J Then the computational problem becomes one of calculating repeated realizations of 9 and of combining them into an appropriate statistical estimator of J. Because of the nature of the process every realization of 0 is a Markov chain. So, next we will use the Markov chain as a synonym of the realization of a random variable of this type. As approximate value of the linear functional J is set up

Harmonic analysis on a locally compact group is the study of certain spaces and algebras of functions and measures defined on the group. The function spaces in question are defined by measurability and integrability conditions; manipulation of the relevant measures requires delicate techniques from measure and integration theory. Also, the fundamental theorems of the theory of representations of locally compact groups [see §22] depend wholly upon the study of a certain space of square integrable functions. Thus the theories of measure and integration play a vital role in the present book.

Using compactness properties of bounded subsets of spaces of vector measure integrable functions and a representation theorem for q-convex Banach lattices, we prove a domination theorem for operators between Banach lattices. We generalize in this way several classical factorization results for operators between these spaces, as psumming operators.

The dual space (with respect to the Alexiewicz norm) of the class of ${\cal BV}$-integrable functions on a compact cell ${\mathop{\prod}\limits_{i=1}^{m}} [a_i, b_i] \subset {\mathbb R}^m$ is shown to be isometrically isomorphic to the space of finite signed Borel measures on ${\mathop{\prod}\limits_{i=1}^{m}} [a_i, b_i)$, and the usual integral representation theorem holds. An example is also given to show that Lipschitz functions are not part of this dual space. This answers a question of Thierry De Pauw.

An alternative approach to Privault's discrete-time chaotic calculus