Showing papers on "Bounded function published in 1970"
TL;DR: In this paper, a theory of generalized solutions in the large Cauchy's problem for the equations in the class of bounded measurable functions is constructed, and the existence, uniqueness and stability theorems for this solution are proved.
Abstract: In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the "vanishing viscosity method"; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in of the solution of this problem which do not depend on small viscosity.Bibliography: 22 items.
1,799 citations
TL;DR: In this paper, it was shown that convolution with (1xI2)+ and related convolutions are bounded from LI to Lq for certain values of p and q, and that p is a unique choice of p which maximizes the measure of smoothing.
Abstract: We prove that convolution with (1xI2)+ and related convolutions are bounded from LI to Lq for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing il/p-l/q, in contrast with fractional integration where il/p-l/q is constant. We apply the results to obtain a priori estimates for solutions of the wave equation in which we sacrifice one derivative but gain more smoothing than in Sobolev's inequality.
183 citations
165 citations
TL;DR: The positive coding theorem is proved for the possibly nonstationary Gaussian autoregressive source with a constraint on the parameters and it is shown that the rate distortion function of any time-discrete autore progressive source can be bounded below by the rate distortions of the independent-letter identically distributed generating sequence with the same distortion measure.
Abstract: The rate distortion function R(D) is calculated for two time-discrete autoregressive sources--the time-discrete Gaussian autoregressive source with a mean-square-error fidelity criterion and the binary-symmetric first-order Markov source with an average probability-of-error per bit fidelity criterion. In both cases it is shown that R(D) is bounded below by the rate distortion function of the independent-letter identically distributed sequence that generates the autoregressive source. This lower bound is shown to hold with equality for a nonzero region of small average distortion. The positive coding theorem is proved for the possibly nonstationary Gaussian autoregressive source with a constraint on the parameters. Finally, it is shown that the rate distortion function of any time-discrete autoregressive source with a difference distortion measure can be bounded below by the rate distortion function of the independent-letter identically distributed generating sequence with the same distortion measure.
147 citations
TL;DR: In this paper, it was proved that the [N, N + J] Pade approximants to any meromorphic function converge in measure within any bounded region of the complex plane as N approaches infinity.
122 citations
TL;DR: The authors generalize Umegaki's definition of an expectation on the algebra of all bounded operators on a Hilbert space, and classify certain classes of expectations subject to a covariance condition with respect to a unitary representation of a given locally compact group.
111 citations
TL;DR: A characterization of L B as the smallest family of languages which contains the bounded context-free languages and which is closed under the operations of union and intersection is proved.
Abstract: Simple matrix languages and right-linear simple matrix languages are defined as subfamilies of matrix languages by putting restrictions on the form and length (degree) of the rewriting rules associated with matrix grammars. For each n ⩾ 1, let L ( n ) [ ℛ ( n ) ] be the class of simple matrix languages [right-linear simple matrix languages] of degree n, and let L = ⋃ n ⩾ 1 L ( n ) [ ℛ = ⋃ n ⩾ 1 ℛ ( n ) ] . It is shown that L ( 1 ) [ ℛ ( 1 ) ] coincides with the class of context-free languages [regular sets] and that L is a proper subset of the family of languages accepted by deterministic linear bounded automata. It is proved that L ( n ) [ ℛ ( n ) ] forms a hierarchy of classes of languages in L [ ℛ ] . The closure properties and decision problems associated with L ( n ) , L , ℛ ( n ) , and ℛ are thoroughly investigated. Let L B [ ℛ B ] be the bounded languages in L [ ℛ ] . It is shown that L B = ℛ B and that most of the positive closure and decision results which are true for bounded context-free languages are carried over in L B . A characterization of L B as the smallest family of languages which contains the bounded context-free languages and which is closed under the operations of union and intersection is proved.
102 citations
102 citations
TL;DR: In this article, the formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis are discussed, and the usual notations of recursion theory are used formally and informally.
Abstract: Publisher Summary This chapter discusses the formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis. The first order systems express a principle for defining specific sets of natural numbers can be iterated υ times. The systems in the language of classical analysis (containing variables for sets of natural numbers) express roughly that there are hierarchies obtained by iterating the hyperjump operation any number less than υ times. An arithmetic formula is one without set quantifiers or set constants; it may contain set parameters. An arithmetical formula in which all quantifiers are bounded is said to be elementary. The usual notations of recursion theory are used formally and informally. These are special cases of Kreisel's results concerning intuitionistic systems. The chapter suggests definite technical advantages of the reductions and some limitations of their foundational significance.
77 citations
72 citations
TL;DR: In this paper, Carleson's solution of the corona problem for planar open sets and the fibers of the maximal ideal space of H°°(D) are shown to depend only on the local behavior of D.
Abstract: The corona problem for planar open sets D and the fibers of the maximal ideal space of H°°(D) are discussed and shown to depend only on the local behavior of D. Let D be an open subset of the Riemann sphere C*, and let H^iD) be the uniform algebra of bounded analytic functions on D. We will assume always that H^iD) contains a nonconstant function, that is, that C*\D has positive analytic capacity. Our object is to study the maximal ideal space ΛT(D) of H°°(D), and the "fibers" ^&(D) of ^£{Ώ) over points XedD. The basis for our investigation is the observation that the fiber ^tχ(D) depends only on the behavior of D near λ. This localization principle is used to obtain information related to the corona problem. The corona of D is the part of ^£φ) which does not lie in the closure of D. Our main positive results are that D has no corona under either of the following assumptions: (1) that the diameters of the components of C*\D (in the spherical metric, if D is unbounded) be bounded away from zero; or (2) that for some fixed m ^ 0, the complement of each component of D has ίgm components. The proofs rest on the localization principle, and on Carleson's solution of the corona problem for the open unit disc [2]. Each of the above conditions includes the extension of Carleson's theorem to finitely connected planar domains due to Stout [9]. In the negative direction, we present an example, due to E. Bishop, of a connected one-dimensional analytic variety W which is not dense in the maximal space of H°°(W). The construction is similar to that of Rosay [8].
TL;DR: In this article, the authors established a sufficient condition such that the equation Au + g(u) = h has a weak solution u with generalized Dirichlet data zero where A is any self-adjoint strongly and uniformly elliptic linear partial differential operator with real-valued, reasonably smooth coefficients defined on a bounded open set Omega of E superscript n.
TL;DR: Here, a weighted-square error-distortion measure is considered and the class of all stationary random processes that satisfy a certain strong mixing property are considered, that have zero mean, known power, and a bounded fourth moment.
Abstract: In certain situations, a single transmission system must be designed to function satisfactorily when used for any source from a class a of sources. In this situation, the rate-distortion function R_a (d) is the minimum capacity required by any transmission system that can transmit each source from a with average distortion \leq d . One of the most interesting classes of sources is a class of random processes. Here we consider a weighted-square error-distortion measure and the class of all stationary random processes that satisfy a certain strong mixing property, that have zero mean, known power, and a bounded fourth moment, and that satisfy one of the following alternative specifications on the spectrum: 1) the spectrum is known exactly; 2) the amount of power within the band 0 \leq f \leq f_k is known for N -- 1 frequencies f_1 \le f_2 \le \cdots \le f_{N-1} ; or 3) the fraction of power outside some frequency f_l is \leq 1 -- \gamma . For the class of sources determined by each of the above three cases and for an arbitrary error-weighting function we evaluate the rate-distortion function.
TL;DR: The existence theory for problems involving equations involving equations (1) in exterior regions has required either some restriction to finite energy or a fixed rate of decay at infinity as mentioned in this paper.The following result presents a solution that is merely bounded.
Abstract: The previous existence theory for problems involving equations (1) in exterior regions has required either some restriction to finite energy or a fixed rate of decay at infinity The following result presents a solution that is merely bounded THEOREM 1 Let g(x) be a bounded, continuous, divergence free vector field defined on E3 If T is sufficiently small, depending only on g, then there is a solution u,p of (1) on D(T) such that u(x, t) is bounded on D(T) and lim,,o u(x, t) = g(x) for each x E E3 If also limx,O g(x) = uc,, for some vector uc, then limX,C u(x, t) = uc, A general theorem of Kahane [4] shows that these solutions are analytic in the space variables Analyticity in time for a wide class of solutions of (1) satisfying a finite energy condition has been proved by Masuda [6] However, our results [1] concerning the continuous dependence of the solution on the data indicate that time analyticity of bounded solutions is important In this direction we have the following theorem THEOREM 2 The solution assured by Theorem 1 for the Cauchy problem for (1) is analytic in all variables on D(T) Moreover, if u, p is any solution of(1) in D(T), for some T, such that
TL;DR: In this article, the authors prove analogous theorems for a certain class of notions of convergence, which will include, in the case of Banach spaces of functions with the sup norm, bounded, pointwise convergence, and convergence of bounded sequences which is uniform over compact sets.
Abstract: In all of these theorems the notion of convergence used in (1-1) and (1-2) has essentially been strong convergence. It is the purpose of the present paper to prove analogous theorems for a certain class of notions of convergence, which will include, in the case of Banach spaces of functions with the sup norm, bounded, pointwise convergence, and convergence of bounded sequences which is uniform over compact sets. To motivate the eventual abstract formulation of our problem, let us consider the setting, due to Trotter, which was used in [4] and [7]. Here the Ln and L are Banach spaces. We assume there exist continuous linear maps Pn: L -> Ln such that
TL;DR: In this paper, the authors studied bounded quadratic systems, which have all of their trajectories bounded for t > 0, and showed that any such system will have a rest point in the plane and by translating the origin to that rest point it will have the form
01 Jan 1970
TL;DR: Very little is known about the solutions of a Newtonian gravitational system of n mass particles where the mutual distances between the particles are bounded as time, t, approaches infinity as discussed by the authors, and it is not even known whether or not the velocities must be bounded t→∞.
Abstract: Very little is known about the solutions of a Newtonian gravitational system of n mass particles where the mutual distances between the particles are bounded as time, t, approaches infinity. In fact it is not even known whether or not the velocities must be bounded t→∞.
01 Jan 1970
TL;DR: In this article, it is proved that the perceptron error correction procedure stays bounded, even when no solution to the system of linear inequalities exists, and it is shown that the procedure is bounded even when a solution to linear inequalities does not exist.
Abstract: : It is proved that the perceptron error-correction procedure stays bounded, even when no solution to the system of linear inequalities exists. This supplements earlier papers by B. Efron and by M. Minsky and S. Papert. (Author)
TL;DR: The quantization error for a particular value of x (unusually) is defined as the length of the quantizing interval in which x finds itself, and quantizer performance is measured by the r th mean value of thequantizing interval lengths.
Abstract: A quantizer Q divides the range [0, 1] of a random variable x into K quantizing intervals the i th such interval having length \Delta x_i We define the quantization error for a particular value of x (unusually) as the length of the quantizing interval in which x finds itself, and measure quantizer performance (unusually) by the r th mean value of the quantizing interval lengths M_r (Q) = \overline{\Delta x^{r^{1/r}}} , averaging with respect to the distribution function F of the random variable x Q_1 is defined to be an optimum quantizer if M_r (Q_1) \leq M_r (Q) for all Q The unusual definitions restrict the results to bounded random variables, but lead to general and precise results We define a class Q^{\ast} of quasi-optimum quantizers; Q_2 is in Q^{\ast} if the different intervals \Delta x_i make equal contributions to the mean r th power of the interval size so that Pr \{ \Delta x_i \} \Delta x_{i^{r}} is constant for all i Theorems 1, 2, 3, and 4 prove that Q_2 \in Q^{\ast} exists and is unique for given F, K , and r : that 1 \geq KM_r (Q_2) \geq KM_r (Q_1) \geq I_r , where I_r = \{\int_0^{1} f (x)^p dx\}^ {1/q}, f is the density of the absolutely continuous part of the distribution function F of x, p = 1/(1+ r) , and q = r /(1 + r) : that lim KM_r (Q_2) = I_r as K \rightarrow \infty ; and that if KM_r (Q) = I_r for finite K , then Q=Q^{\ast}
01 Dec 1970
TL;DR: A modified Liapunov design technique for model reference adaptive control system is shown to result in improved system convergence and an adaptive rule, derived on the basis of a new Liap unov function, is compared to the previous rule.
Abstract: A modified Liapunov design technique for model reference adaptive control system is shown to result in improved system convergence. An adaptive rule, derived on the basis of a new Liapunov function, is compared to the previous rule. A local stability analysis applied to the modified design shows that the error response is more rapidly convergent. Furthermore, system simulations show that the transient response for the adjustable parameters is also improved. A second result presented is a design technique for a class of plants whose parameters cannot be adjusted directly. This design leads to a system with a set of prefilter and feedback adjustable gains as the adaptive parameters and physically realizable linear time-invariant filter networks in both the feedback and prefilter paths. It eliminates the problem of nonunique adaptive laws previously encountered and requires only n - m - 1 derivative networks for its implementation (nth order plant with m zeros); hence, if m = n - 1, no derivative networks are required for implementation. In order to maintain a bounded plant input signal, the zeros of the plant transfer function must be restricted to the open left-half plane.
01 Mar 1970
TL;DR: In this paper, weak and strong forms of the physical law of diminishing returns are derived for homogeneous and convex technologies, respectively, and it is shown that the classical forms of this law hold if the technology is homogeneous (degree one) and the production possibility sets of the technology are strictly convex.
Abstract: Based on a general mathematical model of a technology, implying certain properties for the production function, weak and strong forms of a physical law of diminishing returns are derived. It is also shown that the classical forms of this law hold if the technology is homogeneous (degree one) and the production possibility sets of the technology are strictly convex, but the latter property violates an essential property of a technology, namely that these sets have bounded efficient subsets.
01 Jun 1970
TL;DR: Some direct and iterative schemes for solving a standard finite-difference scheme for Poisson''s equation on a two-dimensional bounded region R with Dirichlet conditions specified on the boundary R as mentioned in this paper.
Abstract: Some direct and iterative schemes are presented for solving a standard finite-difference scheme for Poisson''s equation on a two-dimensional bounded region R with Dirichlet conditions specified on the boundary $\delta$R. These procedures make use of special-purpose direct methods for solving rectangular Poisson problems. The region is imbedded in a rectangle and a uniform mesh is superimposed on it. The usual five-point Poisson difference operator is applied over the whole rectangle, yielding a block-tridiagonal system of equations. The original problem, however, determines only the elements of the right-hand side which correspond to grid points lying within $\delta$R; the remaining elements can be treated as parameters. The iterative algorithms construct a sequence of right-hand sides in such a way that the corresponding sequence of solutions on the rectangle converges to the solution of the imbedded problem.
01 Dec 1970
TL;DR: In this article, it was shown that if a bounded monotonic sequence is convergent, then it is also convergent with respect to the difference equation, and this was proved by considering the corresponding difference equation.
Abstract: A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence { a n } of real numbers satisfies the inequality then it is convergent. This I was able to prove by considering the corresponding difference equation
TL;DR: In this paper, the authors extend the first-order necessary conditions obtained there to a general form of the control problem of Bolza with inequality state constraints of the type ψα(t, x)≤0, α=1,...,m.
Abstract: This paper serves as the sequel to a recent article by the author concerned with a certain canonical control problem. The present paper extends the first-order necessary conditions obtained there to a general form of the control problem of Bolza with inequality state constraints of the type ψα(t, x)≤0, α=1,...,m.
TL;DR: In this article, a sequence of variational principles for converting a trial solution of a linearized Boltzmann equation into bounds on a transport coefficient is presented, which converges monotonically to the exact transport coefficient.
Abstract: A sequence of variational principles for converting a trial solution of a linearized Boltzmann equation into bounds on a transport coefficient is presented. For systems in which the Boltzmann collision operator has a bounded eigenvalue spectrum, we obtain an infinite sequence of lower bounds which begins with the familiar result of Ziman. For an arbitrary trial function, this sequence converges monotonically to the exact transport coefficient. Application of the first two terms has been made to the lattice thermal conductivity of a model simulating solid argon; the second bound lies considerably higher than the first.
Journal Article•
TL;DR: In this paper, a set of first-order necessary conditions for the general control problem of Bolza with bounded state constraints of the formψα(t, x)≤0, α=1,...,m.
Abstract: A set of first-order necessary conditions is obtained for the general control problem of Bolza with bounded state constraints of the formψα(t, x)≤0, α=1,...,m. With the solution required to satisfy the vector differential equationsx=f(t, x, u), whereu is control, an important feature of this paper is in relaxing the assumption on the rank of the matrixψxαfu generally made in attacking problems of this type. This is accomplished even though the solution may have an infinite number of intervals satisfyingψα(t, x)=0 for various α.