Extension Of Theorem Research Articles

Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Sectorial Extensions, via Laplace Transforms, in Ultraholomorphic Classes Defined by Weight Functions

We prove several extension theorems for Roumieu ultraholomorphic classes of functions in sectors of the Riemann surface of the logarithm which are defined by means of a weight function or weight matrix. Our main aim is to transfer the results of V. Thilliez from the weight sequence case to these different, or more general, frameworks. The technique rests on the construction of suitable kernels for a truncated Laplace-like integral transform, which provides the solution without resorting to Whitney-type extension results for ultradifferentiable classes. As a byproduct, we obtain an extension in a mixed weight-sequence setting in which assumptions on the sequence are minimal.

Representable extensions of positive functionals and hermitian Banach *-algebras

We present a general extension theorem for representable positive linear functionals defined on a *-subalgebra of an arbitrary *-algebra. The case of pure positive functionals is an improvement of the results from some previous works of Maltese [13], and Doran and Tiller [5]. From our statement we obtain characterizations of hermitian Banach *-algebras, among others the classical ones. As applications we prove that H*-algebras and the Lp-algebras of compact groups are hermitian.

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In the paper we consider the following Cauchy problem for the systems of the ordinary dierential equations { ψ′′(t) + K(t,ψ(t),ψ′(t)) = S(t), t ∈ (0,T), ψ(0) = ψ0, ψ′(0) = ψ1, (∗) where K and S are some vector-valued functions, ψ0,ψ1 ∈ Rd are some xed vectors. Such systems appear in various classical mechanics problems because they follow from Newton's second law of the dynamics. If we use the Faedo-Galerkin method for the seeking of the weak solution to the initial boundary value problem for the linear or nonlinear hyperbolic equations, then we also use the problems of type (*) for the construction of the approximation functions. In practice, the Cauchy problems of type (*) often are reduced to the following Cauchy problem of the higher dimension: ψ′(t) = θ(t), t ∈ (0,T), θ′(t) = S(t)−K(t,ψ(t),θ(t)), t ∈ (0,T), ψ(0) = ψ0, θ(0) = ψ1. (∗∗) Then the standard Peano/Caratheodory types existence theorems are used. But these are local theorems and they should be complemented by some extension theorems. On the other hand we can seek the global solutions of problems (*) or (**) directly. In the present paper we and the conditions of the global solvability of the Cauchy problem (*). Corresponding results for problem (**) we got in our previous paper. These conditions contain the Caratheodory existence conditions and the Lasalle extension conditions for (*). Finally, we use these facts to proving the existence theorem for the initial boundary value problem for some nonlinear hyperbolic equations with the variable exponent of the nonlinearity.

On the extension of Whitney ultrajets

We prove necessary and sufficient conditions for the validity of Whitney's extension theorem in the ultradifferentiable Roumieu setting with controlled loss of regularity.

REMARKS ON SOBOLEV-MORREY-CAMPANATO SPACES DEFINED ON C_{0;\gamma} DOMAINS

We discuss a few old results concerning embedding theorems for Campanato and Sobolev-Morrey spaces adapting the formulations to the case of domains of class $C^{0,\gamma}$, and we present more recent results concerning the extension of functions from Sobolev-Morrey spaces defined on those domains. As a corollary of the extension theorem we obtain an embedding theorem for Sobolev-Morrey spaces on arbitrary $C^{0,\gamma}$ domains.

Siu’s lemma, optimal $$L^2$$ extension and applications to twisted pluricanonical sheaves

We prove a general version of Siu’s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal $$L^2$$ extension theorem and an optimal $$L^{\frac{2}{m}}$$ extension theorem for Kahler fibrations. These results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for Kahler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles.

(p, q)-Regular operators between Banach lattices

We study the class of (p, q)-regular operators between quasi-Banach lattices. In particular, a representation of this class as the dual of a certain tensor norm for Banach lattices is given. We also provide some factorization results for (p, q)-regular operators yielding new Marcinkiewicz–Zygmund type inequalities for Banach function spaces. An extension theorem for $$(q, \infty )$$ -regular operators defined on a subspace of $$L_q$$ is also given.

Extensions and Smooth Approximations of Definable Functions in O-minimal Structures

In 1934, H. Whitney presented a series of papers which discussed how to determine whether a function or a jet of order m is the restriction of a C m -function on R n . In the first paper of the series, Whitney's Extension Theorem was proved. In the latter, Whitney answered special cases of the following question Question (Whitney's Extension Theorem, WEPn,m) Let f be a continuous function from a closed subset E of R n to R .How can we determine whether f is the restriction of a C m -function on R n ? In this dissertation, we work in o-minimal expansions of real closed ordered fields. Definable versions of Whitney's Extension Theorem and Whitney's Extension Problems will be discussed in this context. Definable set-valued maps are also studied; a definable version of Michael's Selection Theorem will be proved and used, in combination with a definable version of Whitney's Extension Theorem, to give a positive answer to a definable version of WEPn,1.In addition to the above problems, we also discuss smoothing problems. This is inspired by a series of papers by A. Fischer.In this series, a construction of a definable C m -approximation of a definable locally Lipschitz function is provided. Here, we also work in an o-minimal expansion of a real closed field and relax the condition further to just continuous.

On Partial Galois Algebras

We generalize, in the context of partial group action, the Kanzaki commutator theorem for Galois extensions and the structure theorem for Galois algebras given by Szeto and Xue.

Linear extensions of Baire-alpha functions

In this paper, we extend the main result on linear extensions of some Baire-one functions, obtained recently by W. Sieg, to a larger class of Baire functions.Let A be a nonempty subset of a Hausdorff space X, and let α be an ordinal number <ω1. By Bα(X) we denote the space of all real functions X→R of Baire-class-α. F(A) is the set of functions A→R with a property F, such that F(A) is a linear subspace of Bα(A). We prove Borsuk–Dugundji-type extension theorems: we give an explicit form of a linear extension operator T:F(A)→F(X), where A is an Fσ and Gδ-subset of a normal space X. Our results apply for F=to be piecewise continuous, and F=to be of Baire-class-α. We show that T restricted to the subspace of bounded functions from F(A) is a positive isometry with the supremum norm. In particular, this solves partially an extension problem set in 2005 by Kalenda and Spurný.The same technique of proof can be used to obtain similar results for functions with values in separable normed spaces.

MacWilliams’ extension theorem for infinite rings

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

We prove an analogue of the Lewy extension theorem for a real dimension 2 n 2n smooth submanifold M ⊂ C n × R M \subset \mathbb {C}^{n}\times \mathbb {R} , n ≥ 2 n \geq 2 . A theorem of Hill and Taiani implies that if M M is CR and the Levi-form has a positive eigenvalue restricted to the leaves of C n × R \mathbb {C}^n \times \mathbb {R} , then every smooth CR function f f extends smoothly as a CR function to one side of M M . If the Levi-form has eigenvalues of both signs, then f f extends to a neighborhood of M M . Our main result concerns CR singular manifolds with a nondegenerate quadratic part Q Q . A smooth CR f f extends to one side if the Hermitian part of Q Q has at least two positive eigenvalues, and f f extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.

Whitney differentiability of optimal-value functions for bound-constrained convex programming problems

ABSTRACTIn the spirit of the Whitney Extension Theorem, consider a function on a compact subset of Euclidean space to be ‘Whitney-differentiable’ if it is a restriction of a continuously Fréchet-differentiable function with an open domain. Whitney-differentiable functions have been shown to have useful (yet possibly nonunique) derivatives and calculus properties even on the boundaries of their domains. This article shows that optimal-value functions for bound-constrained convex programmes with Whitney-differentiable objective functions are themselves Whitney-differentiable, even when the linear-independence constraint qualification is not satisfied. This result extends classic sensitivity results for convex programmes, and generalizes recent work. As an application, sufficient conditions are presented for generating continuously differentiable convex underestimators of nonconvex functions for use in methods for deterministic global optimization in the multivariate McCormick framework. In particular, the main result is applied to generate Whitney-differentiable convex underestimators for quotients of functions with known Whitney-differentiable relaxations.

On the extension of binary relations in economic and game theories

Szpilrajn’s extension theorem on binary relations and its strengthening by Dushnik and Miller are fundamental in economic and game theories. Szpilrajn’s result entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn’s theorem to show that each partial order has a realizer. Since then, many authors utilize Szpilrajn’s theorem and the well-ordering principle to prove more general theorems on extending binary relations. The original extension theorems of Szpilrajn, Dushnik-Miller and Moulin-Weymark are called: Szpilrajn extension theorem, Dushnik-Miller extension theorem and Moulin-Weymark’s Pareto extension theorem respectively. The generalizations of these theorems are called: Szpilrajn-type extension theorem, Dushnik-Miller-type extension theorem and Moulin-Weymark’s Pareto-type extension theorem respectively. The presented results generalize well-known extension theorems in the literature.

THE OPTIMAL JET EXTENSION OF OHSAWA–TAKEGOSHI TYPE

We prove the $L^{2}$ extension theorem for jets with optimal estimate following the method of Berndtsson–Lempert. For this purpose, following Demailly’s construction, we consider Hermitian metrics on jet vector bundles.

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

Remainders of extremally disconnected spaces and related objects

In this article, we study remainders of extremally disconnected spaces and extremally disconnected remainders. Extremally disconnected spaces satisfying certain homogeneity type conditions are at the center of it. It is established that the absolute X of a non-discrete separable metrizable space M not only is never homogeneous, but never has a homogeneous extension (Theorem 3.2). This extends a classic theorem of Z. Frolik on nonhomogeneity of any extremally disconnected compactum [17], [18]. It is proved that if X is an extremally disconnected space with a homogeneous extension, then, for any compactification bX of X, the remainder Y=bX∖X is countably compact (Theorem 3.7). Among other new results are Theorem 4.6 on remainders of Banach spaces, Theorem 4.8 on remainders of normal topological groups, and Theorem 2.9 on remainders of k-trivial spaces.

An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds

In this paper, we prove an $L^2$ extension theorem for holomorphic sections of holomorphic line bundles equipped with singular metrics on weakly pseudoconvex Kahler manifolds. Furthermore, in our $L^2$ estimate, optimal constants corresponding to variable denominators are obtained. As applications, we prove an $L^q$ extension theorem with an optimal estimate on weakly pseudoconvex Kahler manifolds and the log-plurisubharmonicity of the fiberwise Bergman kernel in the Kahler case.

The extension theorem for the Lee and Euclidean weights over [formula omitted

The Lee and Euclidean weights have the extension property over the local rings Z/pkZ, p prime. The non-vanishing of certain Fourier coefficients is established by expressing the coefficients in terms of generalized Bernoulli numbers and making use of knowledge of the locations of zeros of Dirichlet L-functions.