Extension Of Theorem Research Articles

A general approach to define binders using matching logic

We propose a novel definition of binders using matching logic, where the binding behavior of object-level binders is directly inherited from the built-in exists binder of matching logic. We show that the behavior of binders in various logical systems such as lambda-calculus, System F, pi-calculus, pure type systems, can be axiomatically defined in matching logic as notations and logical theories. We show the correctness of our definitions by proving conservative extension theorems, which state that a sequent/judgment is provable in the original system if and only if it is provable in matching logic, in the corresponding theory. Our matching logic definition of binders also yields models to all binders, which are deductively complete with respect to formal reasoning in the original systems. For lambda-calculus, we further show that the yielded models are representationally complete, a desired property that is not enjoyed by many existing lambda-calculus semantics. This work is part of a larger effort to develop a logical foundation for the programming language semantics framework K (http://kframework.org).

Extension operator for the MIT Bag Model

This paper is devoted to the construction of an extension operator for the MIT bag Dirac operator on a $\mathcal{C}^{ 2,1}$ bounded open set of $\mathbb{R}^3$ in the spirit of the extension theorems for Sobolev spaces. As an elementary byproduct, we prove that the MIT bag Dirac operator is self-adjoint.

From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory

The property of isotonicity of a continuous convex function on the positive cone is characterized via subdifferentials. This is used to illustrate a new generalization of the Hardy–Littlewood–Polya inequality of majorization to the case of functions of a vector variable.

Homogenization of quadratic convolution energies in periodically perforated domains

Abstract We prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The corresponding limit is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem from perforated domains belonging to a wide class containing compact periodic perforations.

A new extension theorem for ternary linear codes and its application

An [n,k,d]q code is a linear code of length n, dimension k and minimum weight d over the field of q elements. An [n,k,d]q code C is called extendable if C can be extended to an [n+1,k,d+1]q code. We give a new extension theorem for ternary linear codes. As an application, we prove the non-existence of [512,6,340]3 code, which is a new result.

More about singular traces on simply generated operator ideals

During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.

Applications of the Ohsawa–Takegoshi Extension Theorem to Direct Image Problems

AbstractIn the 1st part of the paper, we study a Fujita-type conjecture by Popa and Schnell and obtain an effective bound on the generic global generation of direct images of twisted pluricanonical bundles. We also point out the relation between the Seshadri constant and the optimal bound. In the 2nd part, we give an affirmative answer to a question by Demailly, Peternell, and Schneider in a more general setting. As byproducts of our proofs, we extend a result by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal settings and refine a theorem by Broustet and Pacienza on the rational connectedness of the image.

Planarly branched rough paths and rough differential equations on homogeneous spaces

This work studies rough differential equations (RDEs) on homogeneous spaces. We provide an explicit expansion of the solution at each point of the real line using decorated planar forests. The notion of planarly branched rough path is developed, following Gubinelli's branched rough paths. The main difference being the replacement of the Butcher–Connes–Kreimer Hopf algebra of non-planar rooted trees by the Munthe-Kaas–Wright Hopf algebra of planar rooted forests. The latter underlies the extension of Butcher's B-series to Lie–Butcher series known in Lie group integration theory. Planarly branched rough paths admit the study of RDEs on homogeneous spaces, the same way Gubinelli's branched rough paths are used for RDEs on finite-dimensional vector spaces. An analogue of Lyons' extension theorem is proven. Under analyticity assumptions on the coefficients and when the Hölder index of the driving path is one, we show convergence of the planar forest expansion in a small time interval.

PROOFS OF URYSOHN’S LEMMA AND THE TIETZE EXTENSION THEOREM VIA THE CANTOR FUNCTION

AbstractUrysohn’s lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze extension theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn’s lemma and the Tietze extension theorem.

Bergman kernel and oscillation theory of plurisubharmonic functions

Based on Harnack’s inequality and convex analysis we show that each plurisubharmonic function is locally BUO (bounded upper oscillation) with respect to polydiscs of finite type but not for arbitrary polydiscs. We also show that each function in the Lelong class is globally BUO with respect to all polydiscs. A dimension-free BUO estimate is obtained for the logarithm of the modulus of a complex polynomial. As an application we obtain an approximation formula for the Bergman kernel that preserves all directional Lelong numbers. For smooth plurisubharmonic functions we derive a new asymptotic identity for the Bergman kernel from Berndtsson’s complex Brunn–Minkowski theory, which also yields a slightly better version of the sharp Ohsawa–Takegoshi extension theorem in some special cases.

Ultraholomorphic extension theorems in the mixed setting

The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.

Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data

In this paper we derive local L2 error estimates for penalized least-squares approximation on the d-dimensional unit sphere Sd⊆Rd+1, given noisy, scattered, local data representing an underlying function from a Sobolev space of order s>d∕2 defined on a non-empty connected open set Ω⊆Sd with Lipschitz-continuous boundary. The quadratic regularization functional has two terms, one measuring the squared pointwise ℓ2-discrepancy from the local data, the other containing the squared native space norm of a radial basis function (RBF), multiplied by a regularization parameter. The RBF is chosen so that its native space is equivalent to the (global) Sobolev space of order s on Sd. While both the data and the approximated function are local, we minimize the quadratic functional over all functions in the native space of the RBF, and obtain as exact minimizer a (global) radial basis function approximation. By choosing the RBF to be a Wendland function the resulting linear system has a sparse matrix which is easily computed. We consider three different strategies for choosing the smoothing parameter, namely Morozov’s discrepancy principle and two a priori strategies, and derive L2(Ω) error estimates for each strategy. As auxiliary tools for proving the local L2 error estimates we develop both a local L2 sampling inequality and a suitable Sobolev extension theorem. The paper concludes with numerical experiments.

Characterization of the traces on the boundary of functions in magnetic Sobolev spaces

We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field A is differentiable and its exterior derivative corresponding to the magnetic field dA is bounded. In particular, we prove that, for d≥1 and p>1, the trace of the magnetic Sobolev space WA1,p(R+d+1) is exactly WA∥1−1/p,p(Rd) where A∥(x)=(A1,…,Ad)(x,0) for x∈Rd with the convention A=(A1,…,Ad+1) when A∈C1(R+d+1‾,Rd+1). We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.

Local holomorphic mappings respecting homogeneous subspaces on rational homogeneous spaces

Let $G/P$ be a rational homogeneous space (not necessarily irreducible) and $x_0\in G/P$ be the point at which the isotropy group is $P$. The $G$-translates of the orbit $Qx_0$ of a parabolic subgroup $Q\subsetneq G$ such that $P\cap Q$ is parabolic are called $Q$-cycles. We established an extension theorem for local biholomorphisms on $G/P$ that map local pieces of $Q$-cycles into $Q$-cycles. We showed that such maps extend to global biholomorphisms of $G/P$ if $G/P$ is $Q$-cycle-connected, or equivalently, if there does not exist a non-trivial parabolic subgroup containing $P$ and $Q$. Then we applied this to the study of local biholomorphisms preserving the real group orbits on $G/P$ and showed that such a map extend to a global biholomorphism if the real group orbit admits a non-trivial holomorphic cover by the $Q$-cycles. The non-closed boundary orbits of a bounded symmetric domain embedded in its compact dual are examples of such real group orbits. Finally, using the results of Mok-Zhang on Schubert rigidity, we also established a Cartan-Fubini type extension theorem pertaining to $Q$-cycles, saying that if a local biholomorphism preserves the variety of tangent spaces of $Q$-cycles, then it extends to a global biholomorphism when the $Q$-cycles are positive dimensional and $G/P$ is of Picard number 1. This generalizes a well-known theorem of Hwang-Mok on minimal rational curves.

Construction of rogue waves and conservation laws of the complex coupled Kadomtsev–Petviashvili equation

In this work, the solutions of the real coupled Kadomtsev–Petviashvili equations (KPEs) are obtained and they are used to find the ones of the complex system. To this end, the extension theorem is used. The formation of rogue wave (RW) is shown to occur via inelastic collisions of lines-lump solitons and periodic waves and also of two elliptic waves. The quadratic invariant is used to show the phase portrait and bifurcation through the contour plot. Further, the conservation laws (CLs) are constructed by means of new conservation theorem by finding the governing system of equation. The results are given, work consolidated from a previous work, by the first two authors, for suggesting a mechanism for the formation of RW when the complex Korteweg–de Vries equation was studied. Based on the exact solutions found here, numerical computations are carried out and they are represented graphically. Finally, the unified method (UM) can be applied to solve a variety of partial differential equations in science and engineering.

Extension of positive maps

We prove an extension theorem for positive maps from operator systems into matrix algebras

Partitions of matrix spaces with an application to [formula omitted]-rook polynomials

We study the row-space partition and the pivot partition on the matrix space Fqn×m. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial methods, we explicitly compute the Krawtchouk coefficients associated with these partitions. This establishes MacWilliams-type identities for the row-space and pivot enumerators of linear rank-metric codes. We then generalize the Singleton-like bound for rank-metric codes, and introduce two new concepts of code extremality. Both of them generalize the notion of MRD code and are preserved by trace-duality. Moreover, codes that are extremal according to either notion satisfy strong rigidity properties analogous to those of MRD codes. As an application of our results to combinatorics, we give closed formulas for the q-rook polynomials associated with Ferrers diagram boards. Moreover, we exploit connections between matrices over finite fields and rook placements to prove that the number of matrices of rank r over Fq supported on a Ferrers diagram is a polynomial in q, whose degree is strictly increasing in r. Finally, we investigate the natural analogues of the MacWilliams Extension Theorem for the rank, the row-space, and the pivot partitions.

Stability of Lipschitz-type functions under pointwise product and reciprocation

This article provides necessary and sufficient conditions on the structure of a metric space such that for various vector lattices of real-valued Lipschitz-type functions defined on the metric space, the vector lattice is stable under pointwise product, and such that the reciprocal of each non-vanishing member of the vector lattice remains in the vector lattice. In each case the family of metric spaces for which the first property holds contains the family of metric spaces for which the second property holds. At the end we prove some extension theorems for classes of locally Lipschitz functions that complement known results for Cauchy continuous functions and for uniformly continuous functions.

Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories

In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.

A converse of Hörmander’s L2-estimate and new positivity notions for vector bundles

We study conditions of Hormander’s L2-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of the Hormander-type condition, we show a converse of Hormander L2-estimate under some regularity assumptions on an n-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson (1998). We also de_ne new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.