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It is proved that there are no A-stable explicit methods in a general class of "linear" methods. The class contains, for example, Runge-Kutta methods, linear multistep methods, predictor-corrector formulas, cyclic multistep methods and linear multistep methods with higher derivatives.

We give a measurable selection theorem which generalizes von Neumann-Aumann's theorem when the domain of definition is an abstract measurable space and the range space is a Suslin space. As application we give a measurable implicit function theorem and a parametrized version of Choquet's theorem on integral representation.

The author proves a theorem about systems of sets which generalizes the closed graph theorem. To illustrate its application proofs, are given of several generalizations of the closed graph theorem, of selection theorems, a transitivity theorem for C...-algebras and factorization theorems in Banach algebras.

In economics, the Frobenius theorem and its extension are often used in the theory of linear models and the theory of balanced growth. In these contexts optimization does not appear explicitly. Our proof by the Kuhn-Tucker theorem may lead to some interesting economic reinterpretations of the Frobenius root and eigenvectors in these models. As for economic applications, see, for example, Morishima (1964); the extended bibliography of Debreu and Herstein (1953) is also useful.

The main results of this paper are two-fold. The first, Theorem 1, is a generalization of the work of Chow and others concerning the set of locally accessible points of a nonlinear control system. It is shown that under quite general conditions, this set lies on a surface in state space and has a nonemptyinterior in the relative topology of that surface.The second result, Theorem 3, generalizes the bang-bang theorem to nonlinear control systems using higher order control variations as developed by Kelley and others. As a corollary we obtain Halkin’s bang-bang theorem for a linear piecewise analytic control system.

Let f be an entire function, and let β \beta and λ \lambda be positive numbers with β ≤ π \beta \leq \pi and β λ > π \beta \lambda > \pi . Let E ( r ) = { θ : log ⁡ | f ( r e i θ ) | > cos ⁡ β λ log ⁡ M ( r ) } E(r) = \{ \theta :\log |f(r{e^{i\theta }})| > \cos \beta \lambda \log M(r)\} . It is proved that either there exist arbitrarily large values of r for which E ( r ) E(r) contains an interval of length at least 2 β 2\beta , or else lim r → ∞ r − λ log ⁡ M ( r , f ) {\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f) exists and is positive or infinite. For β = π \beta = \pi this is Kjellberg’s refinement of the cos π ρ \pi \rho theorem.

This paper presents a relatively simple proof of the maximum principle. The main objective has been to obtain a proof, similar to that due to Halkin, but replacing the use of Brouwer's fixed point theorem by an easily proven contraction mapping theorem. The first use of Brouwer's fixed point theorem, in proving the convexity of the reachable set. of the linearized system, is eliminated by a simple, constructive proof of Lyapunov's theorem (requiring a slightly more general class of admissible controls than the piecewise continuous class employed by Halkin). The second use, in proving the fundamental lemma, is eliminated by a more powerful Balayage result and the use of an additional property of the linearized model of the non-linear system, permitting a contraction mapping theorem to be employed.

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Beruhrsatz). The same special Pascal theorem now leads to an incidence theorem, π, for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of π contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.

In this paper we consider a class of stochastic linear Ito differential equations with control parameters. It is shown (theorem 2.1) that the set of conditional probability measures, generated by the differential system in response to a class of bounded measurable controls with values in a compact convex subset U of a finite dimensional Euclidean space Rv , is equivalent to those corresponding to controls taking values only on the boundary of the set U. Further this set of conditional probability measures is shown (theorem 2.2) to be weakly compact. These results are then used to prove the existence (theorem 2.4) of an optimal control that satisfies the so called ’ bang-bang ’ principle (Hermes and LaSalle, p. 46).

Contents § 1. Introduction § 2. Statement of results; remarks § 3. The basic approximation § 4. A generalized implicit function theorem § 5. Reformulation § 6. The invertibility of Ae § 7. Proof of Theorem 2.1References

(1974). A Theorem on k-Solubility of Linear Equations. The American Mathematical Monthly: Vol. 81, No. 6, pp. 617-620.

The Kondo-Addison theorem says that every set A of reals contains a real which is implicit (for short, Imp). Any real which is Imp is constructible, and it is the main purpose of this note to show that a close examination of the usual Kondo-Addison argument allows us to compute (in terms of the ordinals associated with A by the classical representation theorem) where in the constructible hierarchy an element of A ∩ Imp must occur. This result is related to a basis theorem of Barwise and Fisher [1] and is accompanied by a strong counterexample which shows that certain tepid improvements of their theorem (and, a fortiori, of the main theorem in this note) are impossible. As a final flourish, the ‘classifying ordinals’ used to compute the locations of the singletons are themselves characterized using an idea from [1].

The purpose of this paper is to prove a common generalization of a theorem due to T. W. Gamelin [3] and a theorem due to Z. Semadeni [5]. Both these results are generalizations of E. Bishop’s abstract version of the well-known RudinCarleson theorem [2].

Using the ’spectral’ order relations ≺ \prec and ≺≺ \prec \prec introduced by Hardy, Littlewood and Pólya, we characterize the uniform integrability of a family of integrable functions. We also prove an extension and a ’converse’ of the classical Lebesgue’s dominated convergence theorem in terms of the ’spectral’ orders ≺ \prec and ≺≺ \prec \prec .

We generalize a theorem of Asplund and Rockafellar concerning the duality between the differentiability of a convex function with respect to a system of setsA and theA rotundity of the conjugate convex function to the case of differentiabilities and rotundities defined by pseudotopologies of a broad class, including pseudotopologies of convergence on systems of filters and their bulblike modifications.

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.

The 6Ψ6 summation theorem was first proved by Bailey1, who deduced it indirectly from a transformation of a well-poised 8Φ7 series into two 4Φ3 series. No direct proof of the theorem has been published, and, since it has interesting applications in the proofs of various identities which occur in combinatory analysis, for example the A series of Rogers2 and some elegant identities due to Ramanujan3, we give two new proofs of the theorem in this paper.

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative number ϕ(P, ), to be defined later, which is called the interior angle of P at the face .

A theorem of Gaschütz is used to prove: Let τ be a homomorphism of the distributively generated near-ring R with identity element and descending chain condition for left modules, τ have finite kernel, and U(R) be the group of units of R; then U(Rτ) = U(R)τ.Furthermore it is shown that the finiteness condition for ker τ can be dropped in the case of R being a ring.

In this study we present an important theorem of the alternative involving convex functions and convex cones. From this theorem we develop saddle value optimality criteria and stationary optimality criteria for convex programs. Under suitable constraint qualification we obtain a generalized form of the Kuhn-Tucker conditions. We also use the theorem of the alternative in developing an important duality theorem. No duality gaps are encountered under the constraint qualification imposed earlier and the dual problem always possesses a solution. Moreover, it is shown that all constraint qualifications assure that the primal problem is stable in the sense used by Gale and others. The notion of stability is closely tied up with the positivity of the lagrangian multiplier of the objective function.

(1951). On the Theorem of Frullani. The American Mathematical Monthly: Vol. 58, No. 3, pp. 158-164.

Two small-gain theorems and a passivity theorem yielding stability results for general feedback systems are shown to be equivalent.

In the paper, a necessary and sufficient condition is given for the strong convergence of minimizing sequences for a differentiable convex functional in a reflexive Banach space.

CONTENTSIntroduction § 1. Preliminary information § 2. Hilbert's Theorem and its associated propositions. The general problem of the paper § 3. Statement of the results § 4. Derivation of the basic equations § 5. Reduction of the basic equations to canonical form § 6. Proof of Theorem 1 § 7. Proof of Theorems 2 and 3 and their Corollaries § 8. Proof of Theorem A § 9. Proof of Lemmas IV and V § 10. Proof of Theorem B and Lemma VI § 11. Proof of Theorem 5 § 12. Remarks and problemsReferences