Arbitrary Family Research Articles

Integral-Type Representations for the Subdifferential of Suprema

AbstractAn integral-like representation is provided for the $$\varepsilon $$ ε -subdifferential of the supremum of an arbitrary family of convex functions. Our characterizations are expressed through appropriate discrete sums performed on the data functions $$f_{t}$$ f t ’s together with specific singular measures operating on them. Moreover, as long as the underlying space is a reflexive Banach space or a separable normed space, we substitute these additional measures with related limits that involve the data functions. All the objects involved in our characterizations rely intrinsically on the data functions that are (almost) active at the reference point.

Best approximation theorems via relatively u-continuous and Jaggi nonexpansive mappings

In this paper, we introduced a new type of cyclic (noncyclic) mappings, called Jaggi relatively nonexpansive mappings and use to investigate the existence of best proximity points (pairs) in the framework of reflexive (strictly convex) Banach spaces by using a geometric concept of proximal normal structure. We also present a best proximity version of Schauder’s fixed point problem for non-self mappings. Finally, we prove the existence of a common best proximity point for an arbitrary family of cyclic relatively u-continuous mappings which are compact and then we obtain a generalization of Markov-Kakutani’s theorem.

Extremality of families of sets

The paper proposes another extension of the extremal principle. A new extremality model involving collections of arbitrary families of sets is studied. It generalizes the conventional model based on linear translations of given sets as well as its set-valued extensions. This approach leads to a more general and simpler version of fuzzy separation. The new model is capable of treating a wider range of optimization and variational problems.

Difference analogues of the second main theorem for holomorphic curves and arbitrary families of hypersurfaces in projective varieties

Difference analogues of the second main theorem for holomorphic curves and arbitrary families of hypersurfaces in projective varieties

Towards systematic evaluation of de Sitter correlators via Generalized Integration-By-Parts relations

We generalize Integration-By-Parts (IBP) and differential equations methods to de Sitter correlators related to inflation. While massive correlators in de Sitter spacetime are usually regarded as highly intricate, we find they have remarkably hidden concise structures from the perspective of IBP. We find the factorization of the IBP relations of each vertex integral family corresponding to dτi integration. Furthermore, with a smart construction of master integrals, the universal formulas for iterative reduction and d log-form differential equations of arbitrary vertex integral family are presented and proved. These formulas dominate all tree-level de Sitter correlators and play a kernel role at the loop-level as well.

Koszul Complexes and Relative Homological Algebra of Functors Over Posets

AbstractUnder certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

Schmidt's subspace theorem for arbitrary families of moving hypersurface targets

In this article, we mainly obtain a Schmidt's Subspace Theorem for arbitrary families of moving hypersurface targets in algebraic varieties, which is an extension of the Schmidt's Subspace Theorem due to Son-Tan-Thin [17] and Quang [10], etc.

Improvement of non-integrated defect relation for meromorphic maps from Kähler manifolds

ABSTRACT The purpose of this paper is to establish a non-integrated defect relation for meromorphic mappings from a complete Kähler manifold into a projective variety intersecting an arbitrary family of hypersurfaces with explicit truncation level. In our result, both the total defect and the truncation level are estimated independently of the number of involved hypersurfaces. Our result generalizes and improves the previous results in this topic.

Conjugation-Based Approach to the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document}-Subdifferential of Convex Suprema

We provide new characterizations of the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document}-subdifferential of the supremum of an arbitrary family of convex functions. The resulting formulas only involve approximate subdifferentials of adequate convex combinations of the data functions. Families of convex functions with a concavity-like property are introduced and their relationship with affine models is studied. The role of the lower semi-continuity of the data functions is also analyzed.

Second main theorem for holomorphic curves on annuli with arbitrary families of hypersurfaces

<abstract><p>The aim of this paper is to establish the second main theorem for holomorphic curves from the annulus into a complex projective variety intersecting an arbitrary family of hypersurfaces. This is done by using the notion of "Distributive Constant" for a family of hypersurfaces with respect to a complex projective variety developed by Quang. We also give an explicit estimate for the level of truncation.</p></abstract>

A short note on coproducts of Abelian pro-Lie groups

The notion of conditional coproduct of a family of abelian pro-Lie groups in the category of abelian pro-Lie groups is introduced. It is shown that the cartesian product of an arbitrary family of abelian pro-Lie groups can be characterized by the universal property of the conditional coproduct.

Numberings, c.e. oracles, and fixed points

The Arslanov completeness criterion says that a c.e. set A is Turing complete if and only there exists an A-computable function f without fixed points, i.e. a function f such that W f ( x ) ≠ W x for each integer x. Recently, Barendregt and Terwijn proved that the completeness criterion remains true if we replace the Gödel numbering x ↦ W x with an arbitrary precomplete computable numbering. In this paper, we prove criteria for noncomputability and highness of c.e. sets in terms of (pre)complete computable numberings and fixed point properties. We also find some precomplete and weakly precomplete numberings of arbitrary families computable relative to Turing complete and non-computable c.e. oracles respectively.

Ultradifferentiable classes of entire functions

We study classes of ultradifferentiable functions defined in terms of small weight sequences violating standard growth and regularity requirements. First, we show that such classes can be viewed as weighted spaces of entire functions for which the crucial weight is given by the associated weight function of the so-called conjugate weight sequence. Moreover, we generalize results from M. Markin from the so-called small Gevrey setting to arbitrary convenient families of (small) sequences and show how the corresponding ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated with an evolution equation problem). Finally, we study the connection between small sequences and the recent notion of dual sequences introduced in the Ph.D. thesis of J. Jiménez-Garrido.

Internal Direct Products and the Universal Property of Direct Product Groups

Abstract This is a “quality of life” article concerning product groups, using the Mizar system [2], [4]. Like a Sonata, this article consists of three movements. The first act, the slowest of the three, builds the infrastructure necessary for the rest of the article. We prove group homomorphisms map arbitrary finite products to arbitrary finite products, introduce a notion of “group yielding” families, as well as families of homomorphisms. We close the first act with defining the inclusion morphism of a subgroup into its parent group, and the projection morphism of a product group onto one of its factors. The second act introduces the universal property of products and its consequences as found in, e.g., Kurosh [7]. Specifically, for the product of an arbitrary family of groups, we prove the center of a product group is the product of centers. More exciting, we prove for a product of a finite family groups, the commutator subgroup of the product is the product of commutator subgroups, but this is because in general: the direct sum of commutator subgroups is the subgroup of the commutator subgroup of the product group, and the commutator subgroup of the product is a subgroup of the product of derived subgroups. We conclude this act by proving a few theorems concerning the image and kernel of morphisms between product groups, as found in Hungerford [5], as well as quotients of product groups. The third act introduces the notion of an internal direct product. Isaacs [6] points out (paraphrasing with Mizar terminology) that the internal direct product is a predicate but the external direct product is a [Mizar] functor. To our delight, we find the bulk of the “recognition theorem” (as stated by Dummit and Foote [3], Aschbacher [1], and Robinson [11]) are already formalized in the heroic work of Nakasho, Okazaki, Yamazaki, and Shimada [9], [8]. We generalize the notion of an internal product to a set of subgroups, proving it is equivalent to the internal product of a family of subgroups [10].

Extending the reach of the point-to-set principle

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces Rn. These are classical questions, meaning that their statements do not involve computation or related aspects of logic.In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X. We first extend two algorithmic dimensions—computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim⁡(x) and Dim(x) to individual pointsx∈X—to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families.We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces.

A general approach for improving the Padé iterations for the matrix sign function

The Padé family of iterations is a well-known set of methods used to compute the matrix sign function, which includes classical methods such as Newton’s method, the Newton–Schultz iteration, and Halley’s method. In this paper, we present a general approach to enhance the Padé iterations by choosing an arbitrary three-point family of methods based on weight functions. We determine the weight functions in a way that, for a complex quadratic with distinct roots, the three-point methods are conformally conjugate to a complex polynomial with as many parameters as desired. This approach leads to a multi-parameter family of iterations for the matrix sign function, which allows us to discover many new methods, including the Padé family of iterations as a special case. We provide a convergence and stability analysis of the multi-parameter family and conduct numerical experiments to confirm the improved performance of the new methods. Although the three-point family of methods is arbitrarily chosen, our approach can be easily extended to any other multipoint methods.

Operator formalism for the propagation of paraxial electromagnetic fields

The operator formalism of the time evolution of quantum systems is applied to describe the propagation of light beams in homogeneous and inhomogeneous media, which are governed by the paraxial wave equation. This approach allowed us to address the important case of a Gaussian beam propagating in homogeneous and isotropic space, along with the depiction of relevant beam parameters, such as the beam waist, curvature radius and Gouy phase. By using this method, we show that a special class of operators can be used to generate arbitrary families of Gaussian beams when applied to the fundamental Gaussian mode. We exemplify with the reproduction of the well-known higher-order modes of the Hermite- and Laguerre–Gaussian beams, as well as with the generation of a new family of Gaussian modes, whose propagation properties could also be described. The present method provides a new venue to investigate physical phenomena in the realm of paraxial wave optics.

Direct sum of various generalizations of h-lifting modules

In this work, [Formula: see text]-lifting modules, a special extension of [Formula: see text]-lifting which is a further generalization of lifting modules, are studied. Here, we observed that finite direct sum of [Formula: see text]-lifting modules may not be [Formula: see text]-lifting, so we provided various sufficient conditions for which [Formula: see text]-lifting modules are closed under direct sum. Moreover, we have introduced and studied the properties of a new version of [Formula: see text]-lifting module namely, [Formula: see text]-lifting, [Formula: see text]-lifting and completely [Formula: see text]-lifting modules and develop some more properties of the [Formula: see text]-lifting modules in terms of these modules. Further, we proved that if the direct sum of arbitrary family of hollow modules is [Formula: see text]-lifting, then the arbitrary direct sum of [Formula: see text]-lifting [Formula: see text]-module is again [Formula: see text]-lifting.

Value distribution theory on angular domains for holomorphic mappings and arbitrary families of moving hypersurfaces

Our goal in this paper is to establish some second main theorems for holomorphic mappings from angular domains into projective varieties intersecting arbitrary families of moving hypersurfaces with truncated counting functions. Our results generalize and also improve all previous results for the case of holomorphic mappings from the complex plane.

An effective Schmidt’s subspace theorem for arbitrary hypersurfaces over function fields

We establish an effective version of Schmidt’s subspace theorem on a smooth projective variety [Formula: see text] over function fields of characteristic zero for arbitrary family of hypersurfaces. We also show an application to study a general Thue equation over function fields.