CHAPTER III - PARTICULAR TYPES OF GENERALIZED FUNCTIONS

Existence theorems are proved for multidimensional Lagrange control systems, in which the state is a function of several independent variables in a fixed domain, while in contrast, the control is a function of only some of the independent variables (that is, u is a lower-dimensional control). As usual in Lagrange problems, the cost functional is a multiple integral, the state equation is a system of partial differential equations, with assigned boundary conditions, and constraints may be imposed on the values of the state and control variables.

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.

Computed Imaging-Assisted Stereotactic Brain Biopsy: A Risk Analysis of 225 Consecutive Cases

Classical biological control in the past has been characterized by numerous unsuccessful attempts and by relatively few outstanding successes. An analysis of the recent developments in the theoretical and applied aspects of classical biological control reveals, however, that introducing parasites (predators, pathogens, herbivores) against pest organisms apparently has a considerable potential of solving or lessening many important pest problems of the world. In this review, classical biological control was found to be applicable everywhere in the world regardless of temperature or moisture conditions, and native pests were found to be as susceptible to biological control as introduced pests. Biocontrol success was found to be a function of several independent variables (e.g., characteristics of the pest and control organism and the target ecosystem) and many dependent variables (e.g., the methods of selecting, propagating, and introducing the control organism). Knowledge about the interactions between these factors and other, external factors, will help to find probable ways of improving the success rate in introducing natural enemies. This has already been demonstrated by the steady increase in the efficiency of biocontrol work in the last 30 to 40 years.

Estimation of the Probability of an Event as a Function of Several Independent Variables

For a field F of characteristic not 2, let \({\widehat{F}}\) denote the maximal dimension of anisotropic, weakly isotropic, non-degenerate quadratic forms over F. In this paper, we investigate the behavior of this invariant under field extensions.

Discussion of Dr Green's Paper

LetQ i, i=1, …,t, be real nondegenerate indefinite quadratic forms ind variables. We investigate under what conditions the closure of the set $$\left\{ {\left( {Q_1 \left( {\bar x} \right),...,Q_t \left( {\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). As a corollary, we deduce several results on the magnitude of the set Δ ofg ∈ GL(d, ℝ) such that the closure of the set $$\left\{ {\left( {Q_1 \left( {g\bar x} \right),...,Q_t \left( {g\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). Special cases are described when, depending on the mutual position of the hypersurfaces {Q i=0},i=1, …,t, the set Δ has full Haar measure or measure zero and Hausdorff dimensiond 2−(d−2)/2.

Unital quadratic Jordan algebras J(q, I) determined by nondegenerate quadratic forms with basepoints over a field axe called full Jordan Clifford algebras. In characteristic 2 they have ample outer ideals which are also simple; they come in 3 sizes, tiny, small, and large, where the large are full Clifford algebras but the tiny and small algebras are lacking some of their parts. The simple algebras played a role in Zelmanov's solution of the Burnside Problem. In this paper we will analyze these in more detail, determining their centroids and their local algebras; this is important in the classification of prime Jordan triples of Clifford type in arbitrary char-acterstics. In addition we make a careful charcterization of the tiny, small, and large Clifford algebras. We use this to straighten one or two missteps in a proof from the classification of simple algebras. An important role in our characterization is played by commutators, and we describe the Jor­dan commutator products and Bergmann formulas for Clifford algebras in general.

The aims of this paper were to investigate the impacts of the number of registered doctors, number of thana health complexes, and population growth rates on economic growth, to determine the responses of economic growth due to shocks in independent variables, and to illustrate the fluctuations of economic growth due to shocks in independent variables in the short run and long run. Annual time series data were used for the study from 1980 to 2016. Augmented Dicky-Fuller test was applied to check the annual time series data are stationary or not. Johansen test was used to investigate the long-run relationship among the variables. Applying VECM I have seen that in the long run the impacts of the number of registered doctors and population growth rates on economic growth were negative while it was positive for the number of health complexes. On the other hand, in the short run, the impacts of population growth rates and the number of health complexes were positive on economic growth and the impact of the number of registered doctors was negative. Analyzing impulse response function (IRF) I have observed that the responses of gross domestic product due to shocks in the number of registered doctors, number of health complexes, and population growth rates were positive and increasing. Employing the variance decomposition model I have found that in the long run for the number of registered doctors and population growth rates, fluctuations of the gross domestic product increased significantly. But for the number of health complexes, fluctuations of the gross domestic product remained constant for the long run.

1. The main notions.- 2. The main lemmas.- 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution.- 2.2. Proof of lemmas 2.1-2.4.- 3. Theorems on large deviations for the distributions of sums of independent random variables.- 3.1. Theorems on large deviations under Bernstein's condition.- a) Sums of non-identically distributed random variables.- b) Sums of weighted random variables.- 3.2. A theorem of large deviations in terms of Lyapunov's fractions.- 4. Theorems of large deviations for sums of dependent random variables.- 4.1. Estimates of the kth order centered moments of random processes with mixing.- 4.2. Estimates of mixed cumulants of random processes with mixing.- 4.3. Estimates of cumulants of sums of dependent random variables.- 4.4. Theorems and inequalities of large deviations for sums of dependent random variables.- 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates.- 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics.- 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations.- 5.3. Large deviations for estimates of the spectrum of a stationary sequence.- 6. Asymptotic expansions in the zones of large deviations.- 6.1. Asymptotic expansion for distribution density of an arbitrary random variable.- 6.2. Estimates for characteristic functions.- 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables.- 6.4. Asymptotic expansions in integral theorems with large deviations.- 7. Probabilities of large deviations for random vectors.- 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants.- 7.2. Theorems on large deviations for sums of random vectors and quadratic forms.- a) Sums of non-identically distributed random vectors.- b) Sums of weighted random vectors.- c) Sums of random number of random vectors.- d) Quadratic forms.- Appendices.- Appendix 1. Proof of inequalities for moments and Lyapunov's fractions.- Appendix 2. Proof of the lemma on the representation of cumulants.- Appendix 3. Leonov - Shiryaev's formula.- References.

Let X,X 1,X 2,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S N ] of the normalized sums S N =N −1/2(X 1+⋯+X N ) and show that provided that d≥9 and the fourth moment of X exists. The bound ?(N −1) is optimal and improves, e.g., the well-known bound ?(N − d /( d +1)) due to Esseen (1945). The result extends to the case of random vectors taking values in a Hilbert space. Furthermore, we provide explicit bounds for Δ N and for the concentration function of the random variable ℚ[S N ].

We obtain broad conditions under which distributions of homogeneous functions in Gaussian and more general random variables have bounded densities or even densities of bounded variation or densities with finite Fisher information. Analogous results are obtained for convex functions. Applications to maxima of quadratic forms are given.

We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on a continuous-time integration by parts setting and discrete chaos expansions methods. Our approach improves on related results obtained in discrete-time integration by parts settings and applies to U-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate U-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weighted U-statistics, quadratic forms, and random subgraph weighing. In the case of quadratic forms, our results recover and improve the bounds available in the literature, and apply to matrices with non-empty diagonals.

Schrödinger equations with critical nonlinearity, singular potential and a ground state