Let R be an artin ring and Θ = {Θ(1), Θ(2), · · · , Θ(n)} be a family of objects in an artin extriangulated R-category (C, E, s) such that E(Θ(j), Θ(i)) = 0 for all j ≥ i. In this paper, we show that the class P(Θ) of the Θ-projective objects is a precovering class and the class I(Θ) of the Θ-injective objects is a preenveloping one in C. Furthermore, if C has enough projectives and enough injectives, we show that the subcategory F(Θ) of Θ-filtered objects is functorially finite in C. As an appliacation, this generalizes the works by Ringel in a module category case and Mendoza-Santiago in a triangulated category case.

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The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulated category C has Auslander-Reiten E−triangles and the relations of the Grothendieck group K0(C ) are generated by the Auslander-Rieten E−triangles. A partial converse result is given when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category C with a cluster tilting subcategory, the relations of the Grothendieck group K0(C ) are generated by Auslander-Reiten triangles if and only if the triangulated category C is locally finite. It is also shown that there is a one-to-one correspondence between subgroups of K0(C ) containing the image of G and dense G−(co)resolving subcategories of C where G is a generator of C , which generalizes results about classifying subcategories of a triangulated [T] or an exact category C [M] by subgroups of K0(C ).

A novel focusing lens conical horn antenna loaded with dielectric is designed in this paper. The design method of the dielectric surface equation is given. The comparison between the proposed antenna and traditional spot focusing lens antenna is well discussed. Simulations show that the proposed antenna has advantages of small reflection, low cross-polarization, symmetrical beam, and small focal spot. The antenna can be obtained very symmetrical focal spot. What's more, the focal spot is approximately circular. The average electric field strength at the focal point of the antenna can be improved by about IdB V / m than traditional spot focusing lens antenna.This article provides a new focusing lens antenna design ideas, with good reference value.

Extriangulated category was introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (or cotilting) subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (or cotilting) subcategories and obtain an Auslander-Reiten correspondence between tilting (cotilting) subcategories and coresolving covariantly (resolving contravariantly, resp.) finite subcatgories which are closed under direct summands and satisfies some cogenerating (generating, resp.) conditons. Applications of the results are given: we show that tilting (cotilting) subcategories defined here unify many previous works about tilting theory in module categories of Artin algebras and abelian categories admitting a cotorsion triples; we also show that the results work for triangulated categories with a proper class of triangles introduced by Beligiannis.